matroid decomposition

MatroidDecompositionREVISED EDITION

K. Truemper

Truem

perM

atroid Decom

position

RevisedEdition

Leibniz

ISBN ��

Matroid Decomposition

Revised Edition

Matroid DecompositionRevised Edition

K� Truemper

University of Texas at Dallas

Richardson� Texas

Leibniz

Plano� Texas

Copyright c� �� by Klaus Truemper

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Cataloging Data

Truemper�K��

Matroid decomposition �revised edition��K� Truemper�

Includes bibliographical references and indexes�

ISBN ��

�� Matroids� �� Decomposition�Mathematics�

I� Title�

Contents

Preface � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ix

Chapter � Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Summary �

�� Historical Notes �

Chapter � Basic De�nitions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Overview and Notation �

�� Graph De�nitions �

�� Matrix De�nitions ��

�� Complexity of Algorithms �

�� References ��

Chapter � From Graphs to Matroids � � � � � � � � � � � � � � � � � � � � � ��

�� Overview ��

�� Graphs Produce Graphic Matroids �

�� Binary Matroids Generalize Graphic Matroids ��

�� Abstract Matrices Produce All Matroids �

�� Characterization of Binary Matroids ��

�� References �

v

vi Contents

Chapter � Series�Parallel and Delta�Wye

Constructions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��


�� Series�Parallel Construction �

�� Delta�Wye Construction for Graphs ��

� Delta�Wye Construction for Binary Matroids � �

�� Applications� Extensions� and References � �

Chapter � Path Shortening Technique � � � � � � � � � � � � � � � � � � � � ��


�� Shortening of Paths ��

�� Intersection and Partitioning of Matroids ��

�� Extensions and References ��

Chapter Separation Algorithm � � � � � � � � � � � � � � � � � � � � � � � � � � ��


�� Separation Algorithm ��

�� Su�cient Conditions for Induced Separations ��

�� Extensions of ��Connected Minors �


Chapter Splitter Theorem and Sequences of

Nested Minors � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��


�� Splitter Theorem ��

�� Sequences of Nested Minors andWheel Theorem ��

� Characterization of Planar Graphs ��


Chapter � Matroid Sums � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��


�� and ��Sums ��

�� General k�Sums ��

�� Finding �� and ��Sums ��

�� Delta�Sum and Wye�Sum ��


Contents vii

Chapter � Matrix Total Unimodularity and

Matroid Regularity � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��


�� Basic Results and Applications ofTotal Unimodularity ��

�� Characterization of Regular Matroids ��

�� Characterization of Ternary Matroids ��

�� Extensions and References � �

Chapter � Graphic Matroids � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �� Overview � �

� �� Characterization of Planar Matroids � �

� �� Regular Matroids with M�K�� Minors ��

� � Characterization of Graphic Matroids ��

� �� Decomposition Theorems for Graphs ��

� �� Testing Graphicness of Binary Matroids ��

� � Applications� Extensions� and References ��

Chapter �� Regular Matroids � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��


�� and ��Sum CompositionsPreserve Regularity ��

�� Regular Matroid Decomposition Theorem ��

�� Testing Matroid Regularity andMatrix Total Unimodularity ��

�� Applications of Regular MatroidDecomposition Theorem ��

�� Extensions and References �

Chapter �� Almost Regular Matroids � � � � � � � � � � � � � � � � � � � � � ��


�� Characterization of Alpha�Balanced Graphs �

�� Several Matrix Classes ��

�� De�nition and Construction ofAlmost Regular Matroids ��

�� Matrix Constructions � �

�� Applications� Extensions� and References ��

viii Contents

Chapter �� Max�Flow Min�Cut Matroids � � � � � � � � � � � � � � � � � ��


�� Sum and Delta�Sum Decompositions ��

�� Characterization of Max�Flow Min�Cut Matroids ��

�� Construction of Max�Flow Min�Cut Matroids andPolynomial Algorithms ��

�� Graphs without Odd�K� Minors �

�� Applications� Extensions� and References ��

References � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Author Index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Subject Index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Preface

Matroids were �rst de�ned in �� as an abstract generalization of

graphs and matrices� In the subsequent two decades� comparatively few

results were obtained� But starting in the mid��s� progress was made

at an everincreasing pace� As this book is being written� a large collection

of deep matroid theorems already exists� These results have been used

to solve di�cult problems in diverse �elds such as civil� electrical� and

mechanical engineering� computer science� and mathematics�

There is now far too much matroid material to permit a comprehen

sive treatment in one book� Thus� we have con�ned ourselves to a part of

particular interest to us� the one dealing with decomposition and compo

sition of matroids� That part of matroid theory contains several profound

theorems with numerous applications� At present� the literature for that

material is quite di�cult to read� One of our goals has been a clear and

simple exposition that makes the main results readily accessible�

The book does not assume any prior knowledge of matroid theory�

Indeed� for the reader unfamiliar with matroid theory� the book may serve

as an introduction to that beautiful part of combinatorics� For the expert�

we hope that the book will provide a pleasant tour over familiar terrain�

The help of many people and institutions has made this book possible�

P� D� Seymour introduced me to matroids and to various decomposition

notions during a sabbatical year supported by the University of Waterloo�

The National Science Foundation funded the research and part of the writ

ing of the book through several grants� Most of the the writing was made

possible by the support of the Alexander von HumboldtFoundation and

of the University of Texas at Dallas� my home institution� The University

of Bonn and Tel Aviv University assisted the search for and veri�cation of

reference material�

ix

x Preface

M� Gr�otschel of the University of Augsburg made the resources of the

Institute of Applied Mathematics available for the editing� typesetting� and

proofreading� He also supported the project in many other ways� P� Bauer�

M� J�unger� A� Martin� G� Reinelt� M� Stoer� and G� Ziegler of the University

of Augsburg were of much assistance�

T� Konnerth most ably typeset the manuscript in TEX� R� Karpelowitz

and C�S� Peng patiently prepared the numerous drawings�

A number of people helped with the collection of reference material�

in particular S� Fujishige and M� Iri�

R� E� Bixby� A� Bouchet� T� J� Reid� G� Rinaldi� P� D� Seymour�

M� Stoer� A� Tamir� and U� Truemper reviewed a �rst draft� Their cri

tique helped considerably to clarify and simplify material�

To all who so generously gave of their time and who lent support in

so many ways� I express my sincere thanks� Without their help� the book

would not have been written�

About the Revised Edition

The transfer of the copyright from Academic Press� Inc�� to the author in

�� made possible the issue of a revised edition that can be distributed in

electronic format and that may be printed for personal use without charge�

Since an extensive revision would have caused a signi�cant delay of

publication� we limited almost all changes to the correction of typographical

errors and to the updating of the publication data of the references�

The change of format forced a reprocessing of the numerous drawings�

R� L� Brooks� G� Qian� G� Rinaldi� and F�S� Sun carried out much of that

work�

A� Bachem� F� Barahona� G� Cornu�ejols� C� R� Coullard� A� Frank�

A� M� H� Gerards� R� Hassin� D� Naddef� T� J� Reid� P� D� Seymour� R�

Swaminathan� F�S� Sun� and G� M� Ziegler assisted with the updating of

the references�

The �nal editing was done by I� Truemper�

We very much thank all who helped with the preparation of the revised

edition� Without that help� we could not have accomplished that task�

Chapter �

Introduction

�� Summary

A matroid may be speci�ed by a �nite set E and a nonempty set of so�called independent subsets ofE� The independent subsets must observe twosimple axioms� We will introduce them in Chapter �� Matroids generalizethe concept of linear independence of vectors� of determinants� and of therank of matrices� In fact� any matrix over any �eld generates a matroid�But there are also matroids that cannot be produced this way�

With matroids� one may formulate rather compactly and solve a largenumber of interesting problems in diverse �elds such as civil� electrical� andmechanical engineering� computer science� and mathematics� Frequently�the matroid formulation of a problem strips away many aspects that atthe outset may have seemed important� but that in reality are quite ir�relevant� Thus� the matroid formulation a�ords an uncluttered view ofessential problem features� At the same time� the matroid formulation of�ten permits solution of the entire problem� or at least of some subproblems�by powerful matroid techniques�

In this book� we are largely concerned with the binary matroids� whichare produced by the matrices over the binary �eld GF�� That �eld hasjust two elements� � and �� and addition and multiplication obey verysimple rules� Any undirected graph may be represented by a certain binarymatrix� The graphic matroid produced by such a matrix is an abstractionof the related graph� Thus� binary matroids generalize undirected graphs�

During the past forty years or so� a large number of profound matroidtheory results have been produced� A signi�cant portion of these results

�

� Chapter � Introduction

concerns properties of binary matroids and related matroid decompositionsand compositions� Unfortunately� much of the latter material is not easilyaccessible� This fact� and our own interest in combinatorial decompositionand composition� motivated us to assemble this book�

As we started the writing of the book� we faced a basic con ict� On onehand� we were tempted to prove all matroid results with as much generalityas possible� On the other hand� we were also tempted to restrict ourselvesto binary matroids� since the proofs would become less abstract� A majorargument in favor of the second viewpoint was that the matroid classesanalyzed here are binary anyway� Thus� that viewpoint won� Nevertheless�we have mentioned extensions of results to general matroids whenever suchextensions are possible�

We proceed as follows� Chapter contains basic de�nitions concerninggraphs and matrices� In Chapter �� we motivate and de�ne binary ma�troids� We also prove a number of basic results� In particular� we classifywhether the elements of a matroid are loosely or tightly bound together�using the idea of matroid separations and of matroid connectivity� In addi�tion� we learn to shrink matroids to smaller ones by two operations calleddeletion and contraction� Any such reduced matroid is called a minor ofthe matroid producing it� Finally� we derive from any matroid another ma�troid by a certain dualizing operation� Appropriately� the latter matroid iscalled the dual matroid of the given one�

Chapters �� contain fundamental matroid constructions� tools� andtheorems� Chapter � is concerned with some elementary constructionsof graphs and binary matroids� The constructions rely on replacementrules called series�parallel steps and delta�wye exchanges� In Chapter ��we introduce a simple yet e�ective method called the path shortening tech�nique� With its aid� we establish basic connectivity relationships and cer�tain results about the intersection and partitioning of matroids� Chapter �contains another elementary matroid tool called the separation algorithm�which identi�es certain matroid separations�

The techniques and results of Chapters �� are put to a �rst use inChapters � and �� In Chapter �� we prove the so�called splitter theorem�which links connectivity of a given matroid with the presence of certainminors� With that theorem� we show that a su�ciently connected matroidalways contains minors that form a sequence with special properties� InChapter �� we establish fundamental notions and theorems about matroiddecomposition and composition�

With Chapter �� we begin the second half of the book� That chapterprovides fundamental facts about a very important property of real matri�ces called total unimodularity� Several translations of the total unimodu�larity property into matroid language are possible� In one such translation�total unimodularity becomes a property of binary matroids called regu�larity� Establishing a real matrix to be totally unimodular then becomes

� � Historical Notes �

equivalent to proving that a certain binary matroid is regular�In Chapters �� we prove a number of decomposition and compo�

sition results about the class of regular matroids and about other� closelyrelated matroid classes� In Chapter �� we begin with an analysis of thegraphic matroids� which are regular� In Chapter �� we examine the re�maining regular matroids� i�e�� the nongraphic ones� In Chapter �� weexplore nonregular matroids that� loosely speaking� have many regular mi�nors� The matroids with that property are called almost regular� Finallyin Chapter �� we investigate ows in matroids by borrowing ideas from ows in graphs� A well�known result about the behavior of ows in graphsis the max� ow min�cut theorem� The matroids whose ows exhibit thenice behavior described in that theorem are called the max� ow min�cutmatroids� The investigation of Chapter �� focuses on these matroids�

For each of the classes of matroids mentioned so far �graphic� regu�lar� almost regular� max� ow min�cut� Chapters �� provide polynomialtesting algorithms� representative applications� and� except for the almost�regular case� characterizations in terms of excluded minors� In addition�excluded minor characterizations of the binary matroids and of the ternarymatroids are given in Chapters � and �� respectively� The ternary matroidsare the matroids produced by the matrices over GF��

The book may be read as follows� First� one should cover Chapters through �� During a �rst reading� one may skip the proofs of the chapters�Chapters �� are largely independent� Thus� one may read them in anyorder� provided one is willing to occasionally interrupt the reading of achapter for a quick glance at some auxiliary result of an earlier chapter� Inthe �rst section of each chapter� we list relevant earlier chapters� if any�

�� Historical Notes

In �� H� Whitney realized the mathematical importance of an abstrac�tion of linear dependence� His pioneering paper �Whitney �� containsa number of equivalent axiomatic systems for matroids� and thus laid thefoundation for matroid theory� In the ��s and ��s� W� T� Tutte builtupon H�Whitney�s foundation a remarkable body of theory about the struc�tural properties of matroids� In the ��s� J� Edmonds connected matroidswith combinatorial optimization� Within a few years� he produced severalkey results� In the process� he popularized matroid theory�

From �� on� an ever growing number of researchers became inter�ested in matroids� In �� D� J� A� Welsh published a book �Welsh ��that contained essentially all results known at that time� As these notesare written� a comprehensive treatment of matroid theory in one book isno longer possible� Selected topics are covered in Crapo and Rota ��

� Chapter � Introduction

Lawler �� Aigner �� Lov�asz and Plummer �� White �� Kung ��c� Schrijver �� Murota �� Recski�� Fujishige �� Oxley �� Bachem and Kern �� and Bj�o�rner� Las Vergnas� Sturmfels� White� and Ziegler �� Kung ��cincludes an excellent historical survey�

Central to this book is the work of P� D� Seymour� W� T� Tutte� andK� Wagner� In historical order� the key results are as follows� K� Wagner�sdecomposition of the graphs without minors isomorphic to the completegraph on �ve vertices �Wagner ��a� W� T� Tutte�s characterization ofthe regular and graphic matroids �Tutte �� and his e�cient test ofgraphicness �Tutte �� P� D� Seymour�s characterization of the max� ow min�cut matroids �Seymour ��a� his decomposition of the regularmatroids �Seymour ��b� and his results on matroid ows �Seymour��a�

Some years ago� these results motivated us to start a systematic in�vestigation using graph and matroid decomposition and composition asmain tools �Truemper ��a� ��b� �� a� ��b� �� a� ��b� Tseng and Truemper �� Except for someminor modi�cations and simpli�cations� this book comprises a large portionof that e�ort�

Due to space constraints� the book does not include details of severalimportant matroid results that are related to the material covered here� Inparticular� we have omitted the principal partitioning results and relatedearlier material by M� Iri� N� Tomizawa� and others �Kishi and Kajitani�� Tsuchiya� Ohtsuki� Ishizaki� Watanabe� Kajitani� and Kishi ��Iri �� Bruno and Weinberg �� Ozawa �� Tomizawa ��b�Iri �� Nakamura and Iri �� Narayanan and Vartak �� To�mizawa and Fujishige �� Murota and Iri �� and Murota� Iri� andNakamura �� Tomizawa and Fujishige �� provide a detailed his�torical survey of the work on principal partitions� We should also mentionL� Lov�asz� matroid matching results �Lov�asz �� see also Lov�asz andPlummer �� That work is not really related to the contents of thisbook� We are compelled to mention it here since it is one of the veryprofound achievements in matroid theory� We also have not included� butshould mention here� work on oriented matroids� These matroids were in�dependently de�ned by Bland and Las Vergnas �� and Folkman andLawrence �� Two recent books cover most of the known results fororiented matroids �Bachem and Kern �� and Bj�orner� Las Vergnas�Sturmfels� White and Ziegler �� Finally� many important matroidapplications are described in Iri and Fujishige �� Iri �� Murota�� and Recski ��

Chapter �

Basic De�nitions

�� Overview and Notation

This chapter covers basic de�nitions about graphs and matrices� and thecomputational complexity of algorithms� For a �rst pass� the reader mayjust scan the material�

We �rst introduce notation and terminology connected with sets� Anexample of a set is fa� b� cg� the set with a� b� and c as elements� With twoexceptions� all sets are assumed to be �nite� The exceptions are the set ofreal numbers IR� and possibly the set of elements of an arbitrary �eld F �

Let S and T be two sets� Then S � T is fz j z � S or z � Tg� theunion of S and T � The set S � T is fz j z � S and z � Tg� the intersectionof S and T � The set S � T is fz j z � S and z �� Tg� the di�erence of Sand T � The set �S � T � � �S � T � is the symmetric di�erence of S and T �

Let T contain all elements of a set S� We denote this fact by S � Tand declare S to be a subset of T � We write S � T if S � T and S �� T �The set S is then a proper subset of T � The set of all subsets of S is thepower set of S� We denote by jSj the cardinality of S� The set is the setwithout elements and is called the empty set�

The terms �maximal and �minimal are used frequently� The meaning depends on the context� When sets are involved� the interpretation isas follows� Let I be a collection� each of whose elements is a set� Then aset Z � I is a maximal set of I if no set of I has Z as a proper subset�Z � I is a minimal set of I if no proper subset of Z is in I�

�

� Chapter � Basic Definitions

�� Graph De�nitions

An undirected graph is customarily given by a set of nodes and a set ofedges� For example� the graph

��

1

4 5 62

3

Graph with node labels

has nodes �� and various edges connecting them� The nodes aresometimes called vertices or points� The edges are sometimes referred to asarcs�

Unless stated otherwise� we rely on a slightly di�erent graph notation�We start with a nonempty set E of edges� say E � fe�� e�� eng� Thenwe declare certain subsets of E to be the nodes� Each such subset speci�esthe edges incident at the respective node�

Let us apply this idea to the graph of �� That graph has tenedges� Thus� we may choose E � fe�� e�� e��g� The graph of ��becomes the following graph G�

�� e10

e8

e6

e4

e1

e2

e3

e5

e7

e9

Graph G with edge labels

Node � of �� is now the subset fe�� e�� e�� e�� e�g of E� and node �has become the subset fe�� e�� e�� e�� e��g� Observe that each edge occursin at most two nodes� The latter sets are the endpoints of the edge� An

� � Graph Definitions �

edge occurring in just one node is a loop� For example� e� occurs only inthe node fe�� e�� e�� e�� e�g and thus is a loop� On the other hand� e� occursin fe�� e�� e�� e�� e�g� as well as in fe�� e� e�g� We also say that an edge isincident at a node� meaning that it is an element of that node�

There are two special cases where the set notation cannot properlyrepresent the nodes� The �rst instance involves nodes that have no edgesincident� We call such nodes isolated� According to our de�nition� eachisolated node produces a copy of the empty set� Thus� if a graph has atleast two isolated nodes� then we encounter multiple copies of the empty set�In the second case� the graph has two nodes that are connected with eachother by any number of edges� but that have no other edges incident� Thenthe two nodes produce identical edge subsets� In principle� one may handlethe two special cases with some auxiliary notation� say using labels on edgesubsets� However� for almost all graphs discussed in this book� the specialcases never arise� Indeed� in a moment we will see how isolated nodes maybe avoided altogether� Thus� we use the abovede�ned set notation andimplicitly assume that a more sophisticated version is employed if needed�

At �rst glance� our notation has little appeal even when one ignores thetrouble caused by the above exceptional cases� But the utility of the ideawill become apparent when we discuss graph minors and related reductionand extension operations� At any rate� we can avoid the cumbersome setnotation by the introduction of additional symbols� For example� we maydeclare i to be the vertex fe�� e�� e�� e�� e�g� and j to be fe�� e�� e�� e�� e��g�We usually work with symbols such as i and j� and in graph drawingswe may write them next to the nodes they reference� By this device� weapproach the compactness of notation inherent in the customary notation�At times� we want to emphasize that a subset of edges de�nes a vertex�We then refer to that subset as a star� or more speci�cally� as a kstar if itcontains k edges� The degree of a vertex is its cardinality�

We avoid isolated nodes as follows� We always start with graphs havingno isolated nodes� Suppose we remove edges from a graph so that a nodebecomes isolated� Then we also remove that node as well� From nowon� whenever we mention the removal of some edges from a graph� weimplicitly assume the removal of isolated nodes� Note that the reducedgraph is unique regardless of the order in which edges are removed� As anexample� removal of the edges e�� e� and e� from the graph G of ��includes removal of the node fe�� e� e�g�

Subgraph

A subgraph is obtained from a given graph by the removal of some edges�A subgraph is proper if at least one edge is removed� Let J be a subsetof the node set of a graph� Delete from the graph all edges that have atleast one endpoint not in J � The resulting graph is the subgraph induced


by J � For example� let i and j be the earlier de�ned nodes of G of ��i�e�� i � fe�� e�� e�� e�� e�g and j � fe�� e�� e�� e�� e��g� The subgraph of Ginduced by J � fi� jg is

�� e8

e1

e9

Induced subgraph of graph G of ��

The nodes of the induced subgraph may have fewer edges incident thanthe corresponding nodes of the original graph� This is so for the aboveexample� As a consequence� the set J need not be the vertex set of theinduced subgraph�

Path� Cycle� Tree� Cocycle� Cotree

Suppose we walk along the edges of a graph starting at some node s� neverrevisit any node� and stop at a node t �� s� The set P of edges we havetraversed is a path from s to t� The nodes of the path are the nodes of thegraph we encountered during the walk� The nodes s and t are the endpointsof P � The length of the path P is jP j� For the graph G of �� let i andj be the previously de�ned nodes� Then P � fe�� e� e�g is a path from ito j� The length of P is � Two paths with equal endpoints are internallynode�disjoint if they do not share any node except for the endpoints� Laterin this section� the statement of Menger�s Theorem relies on a particularfact about the number of internally nodedisjoint paths connecting twonodes� If the two nodes� say i and j� are adjacent� then that number isunbounded since we may declare any number of paths to consist of just theedge connecting i and j� Evidently� these paths are internally nodedisjoint�

Imagine another walk as described above� except that we return to s�The set C of edges we have traversed is a cycle� The length of the cycle isjCj� A set containing just a loop is a cycle of length �� For the graph G of�� C � fe�� e� e�� e�g is a cycle of length �� Let H be a path or cycleof a graph G� An edge e of G is a chord for H if the endpoints of e haveedges of H incident but are not connected by an edge of H�

In a slight abuse of language� we say at times that a graph G is acycle or a path� meaning that the edge set of G is a cycle or path of G�We employ terms of later de�ned edge subsets such as trees and cotreessimilarly� The reader may wonder why we introduce such inaccuracies� Wemust describe a number of diverse graph operations that are not easilyexpressed with one simple set of terms� So either we tolerate a slight abuse


of language� or we are forced to introduce a number of di�erent terms andsets� We have opted for the former solution in the interest of clarity�

Analogously to the use of �maximal and �minimal for sets� we usethese terms in connection with graphs as follows� Suppose certain subgraphs of a graph G have a property P� while others do not� Then asubgraph H of G is a maximal subgraph of G with respect to P if no othersubgraph of G has P and has H as proper subgraph� A subgraphH of G isa minimal subgraph of G with respect to P if no proper subgraph of H hasP� We later use �maximal and �minimal in connection with matrices ina similar fashion� The above de�nitions become the appropriate ones when�graph is replaced throughout by �matrix�

A graph is connected if for any two vertices s and t� there is a path froms to t� The connected components of a graph are the maximal connectedsubgraphs�

A tree T of a connected graph G is a maximal subset of edges notcontaining any cycle� Note that a tree is the empty set if and only if theconnected G has just one node and all edges of G are loops� A tip node orleaf node of a tree is a node of G having just one edge of T incident� Thatedge is a leaf edge of the tree� For G of �� T � fe�� e� e� e�� e��gis a tree� It is easy to show that the cardinality of a tree is equal to thenumber of nodes of G minus �� A cotree of G is E � T for some tree T ofG� Suppose we select for each connected component of a graph G one tree�The union of these trees is a forest of G� An edge of a graph G that is notin any cycle is a coloop� In graph theory� such an edge is sometimes calleda bridge or isthmus� It is easy to see that a coloop is in every forest of G�On the other hand� a loop cannot be part of any forest of G� The rank ofa graph G is the cardinality of any one of its forests�

As a matter of convenience� we introduce the empty graph� That graphdoes not have any edges or nodes� and its rank is �� We consider the emptygraph to be connected�

As one removes edges from a graph� eventually the number of connected components must increase or nodes must disappear� or the emptygraph must result� In each case� the rank is reduced� A minimal set of edgeswhose removal reduces the rank is a cocycle orminimal cutset� In the graphG of �� the set fe�� e�� e�� e�g is a cocycle for the following reason� Itsremoval reduces the rank from � to �� while removal of any proper subsetof fe�� e�� e�� e�g maintains the rank at �� Recall that a coloop is containedin every forest� Hence� removal of a coloop leads to a drop in rank� Weconclude that a set containing just a coloop is a cocycle� The de�nitions offorest and cocycle imply that a cocycle is a minimal subset of edges thatintersects every forest�

A subset of nonloop edges of a given graph G forms a parallel classif any two edges form a cycle and if the subset is maximal with respect tothat property� We also say that the edges of the subset are in parallel� A

�� Chapter � Basic Definitions

subset of edges forms a series class �or coparallel class� if any two edgesform a cocycle and if the subset is maximal with respect to that property�We also say that the edges of the subset are in series or coparallel� In theexample graph G of �� the edges e� and e� are in parallel� and eand e� are in series� In the customary de�nition of �series� a series classof edges constitutes either a path in the graph all of whose intermediatevertices have degree �� or a nonloop cycle all of whose vertices� save atmost one� have degree �� Our de�nition allows for these cases� but it alsopermits a slightly more general situation� For example� in the graph

��e

f

Graph G

the edges e and f are in series since fe� fg is a cocycle� A graph is simpleif it has no loops and no parallel edges� It is cosimple if it has no coloopsand no coparallel edges�

Deletion� Addition� Contraction� Expansion

We now introduce graph operations called deletion� addition� contraction�and expansion� It will become evident that these operations are easilyaccommodated by the graph notation where nodes are edge subsets� This isdecidedly not so for the traditional notation displayed in �� Before weprovide details of the operations� let us examine their goals� Since additionsand expansions are inverse to deletions and contractions� it su�ces for usto state the goals for deletions and contractions� So let G be a connectedgraph� and e be an edge of G� Then the deletion �resp� contraction� of edgee is to result in a connected graph whose trees �resp� cotrees� are preciselythe trees �resp� cotrees� of G that do not contain edge e� The reader shouldhave no trouble verifying that the following de�nitions achieve these goals�

We start with the deletion of an edge e� If e is a coloop� then the deletion is carried out as a contraction� to be described in a moment� Otherwise�the deletion is the removal of the edge e from the graph� Accordingly� weremove e from the edge set E and from the one or two nodes containingit� Addition of an edge is the inverse of deletion� We consider the additionoperation only if the corresponding deletion involves an edge that is not acoloop�

We de�ne the contraction operation� If e is a loop� then the contractionof e is carried out as a deletion� If e is not a loop� then the contraction maybe imagined to be a shrinking of the edge e until that edge disappears� InG� let the edge e have endpoints i and j� Accordingly� in the contraction

� � Graph Definitions ��

of e we remove e from the edge set of G� and replace i and j by a newnode �i � j� � feg� For example� in a contraction of the edge e� of thegraph G of �� we replace the endpoints i � fe�� e�� e�� e�� e�g andj � fe�� e�� e�� e�� e��g by �i � j� � fe�g � fe�� e�� e�� e�� e�� e�� e��g� Notethat the edge e� is an element of i and j� Thus� e� becomes a loop bythe contraction� Expansion by an edge e is the inverse of contraction�We consider the expansion only if the corresponding contraction operationinvolves an edge that is not a loop�

We emphasize that the removal of an edge e may not be the same asits deletion� Indeed� the two operations produce di�erent graphs if andonly if e is a coloop both of whose endpoints have degree of at least ��

A reduction is a deletion or a contraction� An extension is an additionor an expansion�

Uniqueness of Reductions

Suppose a given sequence of reductions for a given graph G produces agraph G�� One would wish that the same G� results if one changes theorder in which the reductions are carried out� Unfortunately� this is notso� For example� suppose in the graph of �� we �rst delete e and thendelete f � After the deletion of e� the edge f is a coloop� Hence� the deletionof f becomes a contraction� and the resulting graph G� is given by ��below�

��

Graph G�

If we reverse the sequence of deletions� we get the graph G�� of ��below� Clearly� G� and G�� are di�erent graphs� The di�erence is producedby the fact that one of the deletions of e and f is actually carried out as acontraction�

��

Graph G��

For obvious reasons� we want to avoid the nonuniqueness of reductions� Aparticularly simple method for achieving that goal relies on the followingconvention� Let G be a graph� When we consider reductions of G� weimplicitly order the edges of G� and perform the reductions in the sequencethat is compatible with that order� The same order of the edges of Gis assumed for all reductions involving G� Indeed� that order is assumed


to induce the related order of edges in all graphs producible from G byreductions� Trivially� this convention induces a unique outcome when agiven subset U of edges is contracted and a given subset W of edges isdeleted�

We denote deletion by �n and contraction by �� Let G be a graph�and suppose U and W are disjoint edge subsets of G� Then G� � G�UnWdenotes the graph produced from G by contraction of U and deletion ofW � We implicitly assume that uniqueness of G� is achieved by the aboveconvention� or possibly by some other device� G� is called a minor of G�As a matter of convenience� we consider G itself to be a minor of G� WhenU or W is empty� we may write GnW or G�U � respectively� instead ofG�UnW � Suppose U contains just one element u� We then write G�uinstead of G�fug to unclutter the notation� Similarly� we write Gnw andG�unw�

Cycle�Cocycle Condition

We should mention that our simple resolution of the nonuniqueness ofreductions may be inappropriate in some settings� For such situations� asecond method for achieving uniqueness of reductions may be a good choice�That method relies on a cycle�cocycle condition� which demands that theset U �resp� W � of edges to be contracted �resp� deleted� does not includea cycle �resp� cocycle�� The cycle�cocycle condition guarantees uniquenessof reductions� since the order of any two successive reduction steps can bereversed without a�ecting the outcome of those two reductions� We leaveit to the reader to �ll in the elementary arguments� For the cycle�cocyclecondition to be useful� we must still prove that any minor can be producedunder that condition� The following arguments establish that fact� Let G�

be any graph producible from a given graphG by some sequence of deletionsand contractions� We claim that G� is G�UnW for some disjoint U and Wthat obey the cycle�cocycle condition� The following construction provesthe claim� Start with U � W � � One by one� perform the reductionsthat transform G to G�� Consider one such reduction� say involving edge z�Let G�� be the graph on hand at that time� Suppose z is to be contracted�If z is not �resp� is� a loop of G�� then add z to U �resp� W �� Supposez is to be deleted� If z is not �resp� is� a coloop of G�� then add z to W�resp� U�� A simple inductive proof establishes that the �nal set U �resp�W � does not contain a cycle �resp� cocyle� of G� We omit the details� Bythe de�nition of U and W � G�UnW is the graph G� as desired� Thus�all minors of G producible under all possible implicit edge orderings areobtainable as minors under the cycle�cocycle condition�

The cycles and cocycles of a minor G�UnW can be readily deducedfrom those of G� We claim that the cycles of G�UnW are the minimal nonempty members of the collection fC � U j C � E � W � C �


cycle of Gg� The proof consists of two easy steps� the details of which weleave to the reader� First� one shows that every cycle of G�UnW occursin the collection� Second� one establishes that each nonempty member ofthe collection contains a cycle of G�UnW � An analogous proof procedureveri�es that the cocycles of G�UnW are the minimal nonempty membersof the collection fC� �W j C� � E �U � C� � cocycle of Gg�

Addition is denoted by �� and expansion by �� Recall that addition of an edge is carried out only if that edge does not become a coloop�Similarly� expansion of an edge is done only if that edge does not become aloop� The latter operation is thus accomplished as follows� We split a nodeinto two nodes and connect them by the new edge� Suppose we add theedges of a set U and expand by the edges of a setW � In the resulting graph�the sets U andW obey the cycle�cocycle condition� Thus� the earlier arguments about that condition imply that the same graph results� regardlessof the order in which the additions and expansions are performed� Thatgraph is denoted by G�U�W � Analogously to G�UnW � we simplify thatnotation at times� That is� we may write G�U � G�W � G�u when U � fug�etc�

Subdivision� Isomorphism� Homeomorphism

In a special case of expansion� we replace an edge e by a path P thatcontains e plus at least one more edge� We say that the edge e has beensubdivided� The substitution process by the path is a subdivision of edge e�

Two graphs are isomorphic if they become identical upon a suitablerelabeling of the edges� They are homeomorphic if they can be made isomorphic by repeated subdivision of certain edges in both graphs�

At times� a certain graph� say G� may be a minor of a graph G� or mayonly be isomorphic to a minor of G� In the �rst case� we say� as expected�that G is a minor of G� or that G has G as a minor� For the second� ratherfrequently occurring case� the terminology �G has a minor isomorphic toG is technically correct but cumbersome� So instead� we say that G hasa G minor�

Planar Graph

A graph is planar if it can be drawn in the plane such that the edges donot cross� The drawing need not be unique� Thus� we de�ne a plane graphto be a drawing of a planar graph in the plane� Consider the followingexample�


�� e10

e8

e6

e4

e1

e2

e3

e5

e7

e9

Graph G

Suppose one deletes from the plane all points lying on the edges or verticesof the plane graph� This step reduces the plane to one or more �topologically� open and connected regions� which are the faces of the plane graph�For example� the edges e�� e� e� and e� and their endpoints border oneface of the graph of �� Note that each plane graph has exactly oneunbounded face�

A connected plane graph has a dual produced as follows� Into theinterior of each face� we place a new point� We connect two such points byan edge labeled e if the corresponding two faces contain the edge e in theirboundaries� We use the asterisk to denote the dualizing operation� Thus�G� denotes the dual of a plane graph G�

As an example� we derive G� from the connected graph G of ��Below� the dashed edges are those of G�� We place each edge label nearthe intersection of the edge of G and the corresponding edge of G��

��e10

e8

e6

e4

e1

e2

e3

e5

e7

e9

Graph G of �� and its dual G�


Note that according to our de�nition� the drawing of the dual graph G�

may not be unique� For example� in the drawing of G� of �� we couldreroute the edge e� to change the unbounded face� One can avoid thedefect of nonuniqueness of G� by drawing the original graph G on thesphere instead of in the plane� Then each face is bounded� and the drawingof the dual graph G� on the sphere is unique� Furthermore� one can showthat the dual of G� is G again� i�e�� G�� G� Using either type ofdrawing� one may verify the following relationships� Coloops of G becomeloops of G�� Indeed� every cocycle of G is a cycle of G�� Any cotree of Gis a tree of G�� Parallel edges of G are series �� coparallel� edges of G��

Vertex� Cycle� and Tutte Connectivity

There are several ways to specify the connectivity of graphs� Two commonlyused concepts of graph theory are vertex connectivity and cycle connectivity�But here we employ Tutte connectivity� We de�ne all three types� thenjustify our choice�

We need an auxiliary process called node identi�cation of two nodes�Informally speaking� node identi�cation amounts to a merging of two givennodes into just one node� For the precise de�nition� let G� and G� be twoconnected graphs� Then we identify a node i of G� with a node j of G� byrede�ning the nodes i and j to become just one node i � j� One extendsthis de�nition in the obvious way for the pairwise identi�cation of k �nodes of G� with k nodes of G��

Let �E�� E�� be a pair of nonempty sets that partition the edge set Eof a connected graph G� Let G� �resp� G�� be obtained from G by removalof the edges of E� �resp� E�� Assume G� and G� to be connected� Supposepairwise identi�cation of k nodes of G� with k nodes of G� produces G�These k nodes of G� and G�� as well as the k nodes of G they create� wecall connecting nodes� Since G is connected� and since both G� and G�

are nonempty� we have k �� If k � �� the single connecting node ofG is an articulation point of G� For general k �� E�� E�� is a vertexk�separation of G if both G� and G� have at least k � � nodes� The pair�E�� E�� is a cycle k�separation if both E� and E� contain cycles of G�Finally� �E�� E�� is a Tutte k�separation if E� and E� have at least k edgeseach� Correspondingly� we call G vertex k�separable� cycle k�separable� orTutte k�separable� For k �� the graph G is vertex k�connected �resp� cyclek�connected� Tutte k�connected� if G does not have any vertex lseparation�resp� cycle lseparation� Tutte lseparation� for � � l � k� Note that theempty graph is vertex� cycle� and Tutte kconnected for every k �� Thesame conclusion holds for the connected graph with just one edge�

It is easy to see that any vertex lseparation or cycle lseparation is aTutte lseparation� Thus� Tutte kconnectivity implies vertex kconnectivity and cycle kconnectivity� The converse does not hold� i�e�� in gen


eral� vertex kconnectivity plus cycle kconnectivity do not imply Tuttekconnectivity� A counterexample is the simple graph on four nodes whereany two nodes are connected by an edge� For any k �� that graph isreadily checked to be both vertex kconnected and cycle kconnected� Butit has a Tutte separation �E�� E�� where E� is one of the stars� andwhere E� contains the remaining three edges� In �� below� we declarethe graph of the counterexample to be the wheel W� There are not manyother counterexamples� Indeed� it is not di�cult to show that the wheelsW� and W� of �� constitute the only other counterexamples�

To summarize� Tutte connectivity implies vertex and cycle connectivity� while vertex and cycle connectivity imply Tutte connectivity except forthe graphs W�� W�� and W�

Suppose G is a plane graph� We claim that �E�� E�� is a vertex kseparation of G if and only if it is a cycle kseparation of G�� The prooffollows by duality once one realizes the following� Each of the graphs G�

and G� de�ned earlier from E� and E� has at least k � � vertices if andonly if each one of E� and E� contains a cocycle of G�

Each one of vertex� cycle� and Tutte connectivity has its advantagesand disadvantages� Thus� one should select the connectivity type depending on the situation at hand� In our case� we prefer a connectivity conceptthat is invariant under dualizing� That is� a plane graph should be kconnected if and only if its dual is kconnected� Tutte connectivity satis�esthis requirement� while vertex and cycle connectivity do not� This featureof Tutte connectivity is one reason for our choice� A second� much moreprofound reason is the fact that Tutte kconnectivity for graphs is in pleasant agreement with Tutte kconnectivity for matroids� as we shall see inChapter �

By Menger�s Theorem �Menger �� a connected graph G is vertexkconnected if and only if every two nodes are connected by k internallynodedisjoint paths� Equivalent is the following statement� G is vertexkconnected if and only if any m � k nodes are joined to any n � k nodesby k internally nodedisjoint paths� One may demand that the m nodesare disjoint from the n nodes� but need not do so� Also� the k paths canbe so chosen that each of the speci�ed nodes is an endpoint of at least oneof the paths� By the above observations� the �only if part remains validwhen we assume G to be Tutte kconnected instead of vertex kconnected�Menger�s Theorem also implies the following result� A graph is �connectedif and only if any two edges lie on some cycle�

From now on we abbreviate the terms �Tutte kconnected� �Tuttekseparation� and �Tutte kseparable to k�connected� k�separation� andk�separable�

The maximal �connected subgraphs of a connected graph G are the��connected components of G� Consider the following process� At a nodeof one of the components� attach a second component� At a node of the


resulting graph� attach a third component� and so on� Then the componentsand nodes of attachment can be so selected that this process creates G�

Complete Graph

The simple graph with n � vertices� and with every two vertices connected by an edge� is denoted by Kn� It is the complete graph on n vertices�Small cases of Kn are as follows�

��

K� K K K�

Small complete graphs

Bipartite Graph

A graph G is bipartite if its vertex set can be partitioned into two nonemptysets such that every edge has one endpoint in each of them� Note that abipartite graph cannot have loops� The complete bipartite graph Km�n issimple� hasm nodes on one side and n on the other one� and has all possibleedges� Small cases are as follows�

��

K�� K�� K�� K�� K�� K�

Small complete bipartite graphs

Evidently� K�� is the complete graph K��

Wheel Graph

A wheel consists of a rim and spokes� The rim edges de�ne a cycle� and thespokes are edges connecting an additional node with each node of the rim�The wheel with n spokes is denoted by Wn� Small wheels are as follows�

��

W� W� W W

Small wheelsEvidently� W is the complete graph K�

In the next section� we introduce de�nitions for matrices over �elds�


�� Matrix De�nitions

In this section� we de�ne a few elementary concepts and tools of matrixtheory� We make much use of the binary �eld GF�� The ternary �eldGF� � and the �eld IR of real numbers are also employed� Occasionally� werefer to a general �eld F �

The binary �eld GF�� has only the elements � and �� Addition isgiven by� � � � � �� and � � � � �� Multiplication is speci�edby� � � � � �� and � � � � �� Note that the element � is also theadditive inverse of �� i�e�� Thus� we may view a f�� g matrix to beover GF�� Each �� then stands for the � of the �eld�

The ternary �eld GF� � has �� and �� Instead of the �� we couldalso employ some other symbol� say �� but will never do so� Addition is asfollows� �� and�� Multiplication is given by� � �� and ��

Submatrix� Trivial�Empty Matrix� Length� Order

A row �resp� column� submatrix is obtained from a given matrix by thedeletion of some rows �resp� columns�� A submatrix is produced by row orcolumn deletions� A submatrix is proper if at least one row or column hasbeen deleted from the given matrix� Subvectors are similarly de�ned�

We allow a matrix to have no rows or columns� Thus� for some k ��a matrix A may have size k� � or �� k� Such a matrix is trivial� We evenpermit the case � � �� in which case A is empty� The length of an m � nmatrix ism�n� The order of a square matrix A is the number of rows of A�We denote any column vector containing only �s by �� Suppose a matrixA has been partitioned into two row submatrices B and C� say A � �B

C��

For typesetting reasons� we may denote this situation by A � �B�C�� Inthe special case where A� B� and C are column vectors� say a� b� and c�respectively� we may correspondingly write a � �b�c��

Frequently� we index the rows and columns of a matrix� We write therow indices or index subsets to the left of a given matrix� and the columnindices or index subsets above the matrix� For example� we might have

��

c

ab

0

0

11

11

gd e f

1

10

1

11B =

Example matrix B

Matrix Isomorphism

We consider two matrices to be equal if up to permutation of rows and

� � Matrix Definitions ��

columns they are identical� Two matrices with row and column indices areisomorphic if they become equal upon a suitable change of the indices�

We may refer to a column directly� or by its index� For example� ina given matrix B let b be a column vector with column index y� We mayrefer to b as �the column vector b of B� We may also refer to b by saying�the column y of B� In the latter case� we should say more precisely �thecolumn of B indexed by y� We have opted for the abbreviated expression�the column y of B since references of that type occur very often in thisbook� We treat references to rows in an analogous manner�

Characteristic Vector� Support Matrix

Suppose a set E indexes the rows �resp� columns� of a column �resp� row�vector with f�� g entries� Let E� be the subset of E corresponding to the�s of the vector� Then that vector is the characteristic column �resp� row�vector of E�� We abbreviate this to characteristic vector when it is clearfrom the context whether it is a row or column vector� The support of amatrix A is a f�� g matrix B of same size as A such that the �s of B occurin the positions of the nonzeros of A� Sometimes� we view B to be a matrixover GF�� or over some other �eld F �

Occasionally� we append an identity to a given matrix� In that casethe index of the ith column of the identity is taken to be that of the ithrow of the given matrix� From the matrix B of �� we thus may derivethe following matrix A�

��

g

1

11

c d

c

ab

0

0

01

10

a b

1

00

0

11

e

1

01

f

1

10A =

Matrix A produced from B of ��

We often view a given f�� g or f�� g matrix at one time to be overGF�� at some other time to be over GF� �� and later still to be over IRor over some other �eld F � Thus� a terminology is in order that indicatesthe underlying �eld� For example� consider the rank of a matrix� i�e�� theorder of any maximal nonsingular submatrix� If the �eld is F � we referto the F�rank of the matrix� For determinants we use �detF � but in thecase of GF�� and GF� � we simplify that notation to �det� and �det��respectively� There is another good reason for emphasizing the underlying�eld� In Chapter � we introduce abstract matrices� which have abstractdeterminants� abstract rank� etc� In connection with these matrices� wejust use �determinant or �det� �rank� etc�


Pivot

Customarily� a pivot consists of the following row operations� to be performed on a given matrix A over a �eld F � First� a speci�ed row a is scaledso that a � is produced in a speci�ed column d� Second� scalar multiples ofthe new row a are added to all other rows so that column d becomes a unitvector� In this book� the term F�pivot refers to a closely related process�We explain the pivot operation using the GF�� case� Let B be a matrixwith row index set X and column index set Y � A GF��pivot on a nonzeropivot element Bxy of a matrix B over GF�� is carried out as follows�

��

�� We replace for every u � �X � fxg� and everyw � �Y �fyg�� Buw by B�

uw � Buw��Buy �Bxw��

�� We exchange the indices x and y� That is� y becomes the index of the row originally indexed byx� and x becomes the index of the column originally indexed by y�

For example� view B of �� to be a matrix over GF�� A GF��pivoton Bad � � may be displayed as follows�

�� GF(2)-pivot

c

ab

0

0

11

11

d e f

1

10

g

1

11B =

c

db

0

0

11

11

ga e f

1

11

1

10B' =

E�ect of GF��pivot in B of ��

Here and later we use a circle to highlight the pivot element�Suppose we append an identity matrix I to B� getting A of �� and

do row operations in A to convert column d to a unit vector� We achievethis by adding row a to row b� Next� we exchange the columns a and d�including indices� Finally� we replace the row index a by d� Let A� be theresulting matrix� Below we display A and A��

�� row operations

andcolumn exchange

g

1

11

c d

c

ab

0

0

01

10

a b

1

00

0

11

e

1

01

f

1

10A =

1g

1

10

c a

c

db

0

0

01

10

d b

1

00

01

e

1

01

f

1

11A' =

E�ect of row operations and column exchangein A of ��

By de�nitionA � �I j B�� and evidently A� � �I j B�� The latter conclusionholds in general� provided the row operations in A replace� in the pivot

� � Matrix Definitions ��

column� the nonzeros other than the pivot element by �s� The pivot rulesof �� are thus nothing but an abbreviated method for obtaining thesubmatrix B� of A� directly from B� Since A and A� are related by rowoperations� every column index subset of A corresponding to a basis of Aalso indexes a basis of A�� and vice versa�

The above operations and conclusions can be extended to arbitrary�elds F as follows� Let B be a matrix over F with row index set X andcolumn index set Y � An F�pivot on a nonzero pivot element Bxy of B isde�ned as follows�

��

�� We replace for every u � �X � fxg� and everyw � �Y � fyg�� Buw by B�

uw � Buw � �Buy �Bxw��Bxy��

�� We replace Bxy by �Bxy� and exchange the indices x and y�

Clearly� the GF��pivot of �� is a special case of the Fpivot� We havethe following result for Fpivots�

�� Lemma� Let B� be derived from B by an F�pivot as described

by �� Append identities to both B and B� to get A � �I j B� andA� � �I j B�� Declare the row index sets of B and B� to become the column

index sets of the identity submatrices I of A and A�� respectively� Then

every column index subset of A corresponding to a basis of A also indexes

a basis of A�� and vice versa�

The proof proceeds along the lines of the GF�� case� except for a simple scaling argument� We omit the details� Pivots have several importantfeatures� For the discussion below� let B� Bxy� and B

� be the matrices justde�ned�

First� when we Fpivot in B� on B�

yx� we obtain B again�

Second� the pivot operation is symmetric with respect to rows versuscolumns� Thus� the Fpivot operation and the operation of taking thetranspose commute�

Third� we may use Fpivots to compute determinants as follows� Suppose that B is square� If we delete row y and column x from B�� thenthe resulting matrix� say B�� satis�es jdetF B��j � j�detF B��Bxyj� Thus�B is nonsingular if and only if this is so for B�� Obviously� this way ofcomputing determinants is nothing but the wellknown method based onrow operations�

Fourth� pivots lead to a simple proof of the submodularity of the rankfunction� to be covered next�


Submodularity of Matrix Rank Function

Let f be a function that takes the matrices over F to the nonnegativeintegers� Suppose for any matrix B over F � and for any partition of B ofthe form

�� B =

B11 B12 B13

B21 B23B22

B31 B32 B33

Partitioned version of B

the values of f for the submatrices

��

B11 B12

B21 B22 ;D1 =B23B22

B32 B33D2 =

;B21 B23B22D3 =B12

B22

B32

D4 =

Submatrices D�� D�� D� D of B

satisfy the inequality

�� f�D�� f�D�� f�D� � f�D��

Then f is submodular� We have the following result�

�� Lemma� The F�rank function is submodular�

Proof� If B�� is nonzero� pivot in B�� From the resulting matrix� say C�delete the pivot row and pivot column� This step converts the submatricesD�� D�� D� and D of B to� say� C�� C�� C� C� Evidently for k �� Frank Ck � Frank Dk�� Thus� the desired conclusion followsby induction once we handle the case where B�� In that situation� wehave

��

Frank D� Frank B�� Frank B��

Frank D� Frank B� �Frank B�

Frank D � Frank B�� Frank B�

Frank D � Frank B�� Frank B�

� � Matrix Definitions �

Thus�

��

Frank D� � Frank D� Frank B�� Frank B��

� Frank B� � Frank B�

Frank D � Frank D

as desired�

The reader may want to prove the following result of linear algebrausing the submodularity of the Frank function�

�� Lemma� Let A be a matrix over a �eld F � with F�rank A �k� If both a row submatrix and a column submatrix of A have F�rank

equal to k� then they intersect in a submatrix of A with the same F�rank�

In particular� any k F�independent rows of A and any k F�independent

columns of A intersect in a k � k F�nonsingular submatrix of A�

Bipartite Graph of Matrix

Let A be any matrix over any �eld� Then BG�A� is the following bipartitegraph� Each row and each column of A corresponds to a node� Eachnonzero entry Axy leads to an edge connecting row node x with columnnode y� In contrast to the earlier graph de�nitions� we do allow isolatednodes in connection with BG�A�� We can a�ord to do so since we neverattempt to apply reductions or extensions to BG�A��

Connected Matrix

We say that A is connected if BG�A� is connected� Suppose A is trivial�i�e�� A is k � � or � � k for some k �� Then BG�A� and hence A areconnected if and only if k � �� Suppose A is empty� i�e�� of size �� ThenBG�A� is the empty graph� By the earlier de�nition� the empty graph isconnected� Thus� the empty matrix is connected� A connected block of amatrix is a maximal connected and nonempty submatrix�

Parallel Vectors

Two column or row vectors of A are parallel if they are nonzero and if oneof them is a scalar multiple of the other one� Equivalently� the two vectorsmust be nonzero and must form a rank � matrix�


Eulerian Matrix

De�ne a f�� g matrix to be column �resp� row� Eulerian if the columns�resp� rows� sum to ��mod �� or equivalently� if each row �resp� column�of the matrix has an even number of nonzeros� Declare a f�� g matrixto be Eulerian if it is both column and row Eulerian�

Display of Matrices

We employ a particular convention for the display of matrices� If in someregion of a matrix we explicitly list some entries but not all of them� thenthe omitted entries are always to be taken as zeros� This convention unclutters the appearance of matrices with complicated structure�

�� Complexity of Algorithms

We cover elementary notions of the computational complexity of algorithmsin a summarizing discussion� De�ne a problem to be any question aboutm�n f�� g matrices that is answered each time by �yes or �no� Any suchmatrix represents a problem instance� Suppose some algorithm determinesthe correct answer for each problem instance� In the case of an a�rmativeanswer� a second m� n f�� g matrix is possibly part of the output�

We measure the size of each problem instance by the size of a binaryencoding of the input matrix and� if applicable� of the output matrix� Denote by s that measure� Up to constants� m � n or the total number ofnonzeros in the input and output matrices constitutes an upper bound ons�

We may imagine the algorithm to be encoded as a computer program�The algorithm is polynomial if the run time of the computer program can�for some positive integers �� and �� be uniformly bounded by a polynomial of the form � � s� � �� We also say that the algorithm is of order ��and we denote this by O�s��

Suppose there are positive integers � � and � such that the followingholds� For each problem instance of size s and with an a�rmative answer�a proof of �yes exists whose binary encoding is bounded by �s�� Thenthe problem is said to be in NP�

A problem P is polynomially reducible to a problem P � if there is apolynomial algorithm that transforms any instance of P into an instanceof P ��

The class NP has a subclass of NP�complete problems� which in somesense are the hardest problems of NP� Speci�cally� a problem is NP�complete if every problem in NP is polynomially reducible to it� Thus� existence of a polynomial solution algorithm for just one of the NPcomplete

� � References ��

problems would imply existence of polynomial solution algorithms for every problem in NP� It is an open question whether or not such polynomialalgorithms exist�

Let P be a given problem� If some NPcomplete problem is polynomially reducible to P � then P is NP�hard�

A polynomial algorithm is not necessarily useable in practice� Theconstants � and � of the upper bound � � s� on the run time may be huge�and the algorithm may require large run times even for small problem instances� The de�nition of �polynomial completely ignores the magnitudeof these constants�

The polynomial algorithms of this book always involve constants �and exponents � that are small enough to make the schemes useable inpractice� Despite this fact� a note of caution is in order� We frequentlyaccept some algorithmic ine�ciency in the interest of simplicity and clarityof the exposition� Thus� most schemes of this book can be speeded up� Therequired modi�cations can be complex� but they also yield substantiallyfaster algorithms�

In the next section� we provide references for the material of this chapter�

�� References

The introductory material of almost any book on graph theory � for example� Ore �� Harary �� Wilson �� or Bondy and Murty �� covers most of the graph de�nitions of Section �� The view of nodesas edge subsets is used in Truemper �� most computer programs forgraph problems rely on the same viewpoint� The Tutte connectivity is dueto Tutte ��a�� Most matrix de�nitions of Section �� are included inany book on linear algebra� see for example Faddeev and Faddeeva �� Strang �� or Lancaster and Tismenetsky�� The de�nition of matrix submodularity is a translation of matroid submodularity �see Truemper��a�� Details about the computational complexity de�nitions may befound in Garey and Johnson ��

Chapter �

From Graphs to Matroids

�� Overview

In this chapter� we construct matroids from graphs and matrices� In Sec�tion �� we start with graphs� We encode them by certain binary matricesthat lead to matroids we call graphic� For these matroids� we adapt anumber of the graph concepts and operations of Chapter �� for examplethe operation of taking minors� In Section �� we generalize the construc�tion of Section �� by starting with arbitrary binary matrices� not just thosearising from graphs� From the binary matrices� we deduce the binary ma�troids� In Section �� we carry the generalization one step further� Thistime� we begin with abstract matrices� which represent a proper general�ization of matrices over �elds� From the abstract matrices� we de�ne allmatroids�

It is easy to determine matroids that cannot be produced from anygraph� or from any binary matrix� or even from any matrix over any �eld�On the other hand� compact characterizations of the matroids that cannotbe obtained from the matrices over some given �eld are usually dicult to�nd� We meet an exception in Section �� There we characterize the ma�troids producible from the binary matrices by excluding a certain ��elementmatroid� called U�

� � as a minor� Sections �� and �� may be skipped by thereader who is only interested in binary matroids� Later� we occasionallyrefer to the material on general matroids to point out extensions� The �nalsection� �� lists references�

The chapter requires knowledge of the de�nitions of Chapter �� Toassist the reader� we will repeat certain de�nitions�

��

� � Graphs Produce Graphic Matroids ��

�� Graphs Produce Graphic Matroids

In this section� we deduce the graphic matroids from graphs by the fol�lowing two�step process� In the �rst step� we encode graphs G by certainbinary matrices called node�edge incidence matrices F � In the second step�we derive from the matrices F the graphic matroids M � For insight into thestructure of the matroids M so created� we transform the node�edge inci�dence matrices F by elementary row operations into certain binary matricesB� The latter matrices contain in compact form all facts about M � Thus�we say that the matrices B represent M � We translate a number of graphde�nitions and concepts for G into statements about the matrices B andthe graphic matroids M � In particular� we link trees� cycles� cotrees� andcocycles of G to features of B and M � We also describe the e ect of takingminors in the graphs G on B and M � and characterize k�connectivity of Gin terms of partitions of B and M � Finally� we establish the relationshipbetween graphs that give rise to the same graphic matroid� and concludewith a handy procedure for deciding whether a certain ��element binaryextension of a graphic matroid is graphic�

Throughout this section� we assume G to be a connected graph withedge set E� The following graph will serve as an example�

�� e10

i1

i2

i3

i4i5 i6

e8

e6

e4

e1

e2

e3

e5

e7

e9

Graph G

Observe the node symbols i�� i�� i�� Recall that each one of thesesymbols stands for the subset of edges incident at the respective node�

Node�Edge Incidence Matrix

We may represent G by a binary matrix F called the node�edge incidencematrix� Each node of G corresponds to a row of that matrix� and each edgeto a column� Suppose an edge e connects nodes i and j in G� Into columne of the matrix� we place one � into row i� a second � into row j� and �sinto the remaining rows� This rule accommodates all edges of G except

�� Chapter � From Graphs to Matroids

for loops� There are several ways to treat loops� Here we decide to placeonly �s into the columns of loops� For the example graph G of �� theresulting matrix F is

��i2i3i4i5i6

i1

F =

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10

0 1 0 0 1 0 0 1 1 0

0 00 0 0 0 0 0 0 10 00 0 0 1 1 1 1 1

0 100 01 0 0 0 00 00110 1 0 0 00 1 1 0 0 1 0 0 0 0

Node�edge incidence matrix F of G of ��

Except for the endpoints of loops� we can reconstruct G from F � Thus�modulo that small defect� F represents G�

Recall from Chapter � the following de�nitions� A cycle of G is theedge set of a walk where all nodes are distinct and where one returns tothe departure node� A tree of G is a maximal edge subset of E withoutcycles� A cocycle of G is a minimal set of edges that intersects every treeof G� A cotree of G is the set E �X for some tree X of G� The rank of Gis the cardinality of any one of its trees� Thus� it is the number of nodesof G minus ��

In the discussion to follow� we always assume that the graph G has acycle and a cocyle� Toward the end of this section� we address the special�actually elementary� situation where G has no cycle or no cocycle�

Over GF�� the linear dependence of n � � vectors� say of f�� f�� fn� is characterized as follows� That set is GF��dependent if there existsa nonempty subset J � f�� ng such that

Pj�J f

j � � �summation in

GF�� of course�� Declare the vectors f�� fn to be minimally GF��dependent� for short GF��mindependent� if they are GF��dependent�and if every proper subset of these vectors is GF��independent�

For example� column e� of F of �� is GF��mindependent� So arethe columns e�� e�� e�� e�� The �rst case corresponds to the loop e� of G� thesecond one to the cycle fe�� e�� e�� e�g� Indeed� by the just�given de�nition�a set of GF��mindependent columns of the node�edge incidence matrixmust correspond to a subgraph G of G with the following two properties�First� each node of G has even degree� Second� there is no subgraph ofG but the empty one where each node has even degree� Evidently� thecycles of G are the only subgraphs of G with these two properties� Thisimplies that the subgraphs of G without cycles correspond to the GF��independent column subsets of F � In particular� the trees of G correspondto the bases of F � and the rank of G is the GF��rank of F �

So far� we have interpreted cycles and trees of G in terms of F � How dococycles and cotrees manifest themselves in F� We will answer that ques�


tion in a moment� In the meantime� the reader may want to try obtainingan answer�

Graphic Matroid

Suppose we are just interested in the collection� say I� of trees of G andtheir subsets� That interest may be surprising� But the set I contains asigni�cant amount of information about G� Exactly how much� we will seelater in this section� For example� with I we can decide whether or not anedge subset C of E is a cycle� The answer is �yes� if and only if C is notin I and every proper subset of C is in I� On the other hand� we cannotdecide with I which nodes are the endpoints of loops�

For the graph G of �� I includes the sets fe�� e�� e�� e�� e��g andfe�� e�� e�g� We know that each set in I is the index set of a columnsubmatrix F � of F with GF��independent columns� Conversely� everysuch index set is recorded in I�

Still assume that we are just interested in I� We are tempted tocombine the edge set E of G and the set I to the ordered pair M � �E�I��The set E is the groundset of M � and I is the collection of independent setsof M � Sometimes� we want to emphasize that M is deduced from G anddenote it by M�G�� In subsequent sections of this chapter� we will see thatM�G� is a special case of what then will be called binary matroids or evenjust matroids� In the spirit of those de�nitions� we call M�G� the graphicmatroid of G�

Representation Matrix

We have established the collection I from the node�edge incidence matrixF of G� Elementary row operations performed on F do not a ect GF��independence of columns� Thus� we may determine I from any matrixderived from F by such operations� We discuss a special case of such rowoperations next�

First� we delete one row from F getting a matrix F �� Since each columnof F has an even number of �s� the sum of the rows of F is the zero vector�Hence� F and F � have same GF��rank�

Second� we perform binary row operations to convert the column sub�matrix of F � corresponding to some tree of G to an identity matrix�

For the example matrix F of �� we select fe�� e�� e�� e�� e��g astree of G� When we apply the preceding two�step procedure to F � we getthe matrix of �� below� Note the row indices of the matrix� They arein agreement with the notation introduced in Section ��


�� 010

00

00

1 0 0 1 0 0 1 1 0

00 0 0 0 0 0 0 0 100 0 1 1 1 1 0

0 010 0 0 0 00 11 1 0 1 1 0

columns of identity matrix

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10

e3

e7

e4

e10

e2

Matrix obtained from F of ��

The matrix of �� is a bit dicult to read� We thus permute itscolumns to collect the identity submatrix at the left end� This changeresults in the matrix A below�

��

e2e3e4A =e7e10

e2 e3 e4 e7 e10 e1 e5 e6 e8 e9

1

11

11

0 1 0 1 1

0 0 0 0 00 0 1 1 10 1 0 0 00 1 1 1 1

Matrix A deduced from the matrix of ��

The unspeci�ed entries in the identity submatrix of A are to be taken aszeros� in agreement with the convention introduced in Section �� aboutunspeci�ed matrix entries�

In the case of a general graph G� let X be some tree of G� and Y be thecorresponding cotree E �X� Then for some binary matrix B� the matrixA is of the form

��BA = IX

X Y

Matrix A for general graph G with tree X

As explained in Section �� the same information is conveyed by B withrow index set X and column index set Y � that is� by

�� B

Y

X

Matrix B for general graph G with tree X


For our example case�

��B = X

e2e3e4e7e10

e1 e5 e6 e8 e9

0 1 0 1 1

0 0 0 0 00 0 1 1 10 1 0 0 00 1 1 1 1

Y

Matrix B for graph G of ��

The matrix B may be viewed as a binary encoding of the matroid M ��E�I�� Since M was de�ned to be a graphic matroid� we call B a graphicmatrix� We also say that B represents M over GF�� or that B is arepresentation matrix of M � In the literature� the term standard represen�tation matrix is sometimes used� The node�edge incidence matrix is thena nonstandard representation matrix� We omit �standard� since we almostalways employ matrices like B to represent M �

Tree� Subgraph Rank

We show how trees� cycles� cotrees� cocycles� and the rank of subgraphs ofG manifest themselves in B of �� We repeatedly make use of somepartition �X��X�� of X� and of some partition �Y�� Y�� of Y � Typically�we just specify one set of X�� X�� and one set of Y�� Y�� For any suchpartitions� we assume B to be partitioned as

�� X1B =

Y1

B1

D1X2

Y2

B2

D2


Let Z be a tree of G� De�ne X� � Z �X and Y� � Z � Y � In A of ��the submatrix A indexed by Z � X� � Y� is thus a GF��basis of A of theform

��

X2

A =X1

X2

0

Y1

D1

B1

...11

Submatrix of A indexed by Z � X� � Y�


The submatrix B� of A is square� and� by cofactor expansion� GF��nonsingular� Conversely� any square and GF��nonsingular submatrix B�

of B de�ned by �X��X�� and �Y�� Y�� corresponds to a tree Z � X� �Y� ofG� More generally� let the submatrix B� of B of �� be of any size andwith GF��rank B� � k� Then the subgraph of G containing precisely theedges of X� � Y� has rank equal to jX�j� k�

Cycle

We know that a cycle C of G corresponds to a column submatrix A ofA with GF��mindependent columns� Analogously to the tree case� letsuch a column submatrix be indexed by X� � C � X and Y� � C � Y �Once more� �� displays the general case� GF��mindependence of thecolumns of A implies that they must add up to � in GF�� Thus� each rowof B� �resp� D�� must contain an even �resp� odd� number of �s� Note thatthis parity condition on the row sums of B� and D� uniquely determinesX� and X� when Y� is speci�ed� Furthermore� by GF��mindependence�no nonempty proper column submatrix of A� say indexed by X �

� � X� andY �� Y�� satis�es the analogous parity condition�

Fundamental Cycle

The special case of a cycle with jY�j � �� say Y� � fyg� is of particularinterest� If column y contains no �s� then y is a loop� Otherwise� X� isthe index set of the rows with a � in column y� The cycle is X� � fyg�It is the fundamental cycle that y forms with a subset of the tree X� Byreversing these arguments� we obtain a quick way of constructing B� Letthe tree X be given� We take each edge y � Y �� E �X� in turn and �ndthe fundamental cycle C that y forms with a subset of X� Then column yof B is the characteristic vector of C � fyg� That is� the entry in columny and row x � X is � if x � �C � fyg�� and is � otherwise�

Parallel Elements

A cycle of cardinality �� say fy� zg with y � Y � manifests itself in Bby two parallel columns indexed by y and z� or by a unit vector columnindexed by y and with the � in row z� We say that y and z are in parallelin M � We de�ne any element to be parallel to itself� Then �is parallel to�is easily checked to be an equivalence relation� The equivalence classes arethe parallel classes of M � They are precisely the parallel classes of G�


Cotree

Recall that a cotree of G is the set E�Z for some tree Z of G� Considerthe transpose of B of �� i�e��

�� (D

Y2

Y1 (B1)t 1)

t

)(D2 t

)(B2 tBt =

X1 X2

Transpose of B of ��

Suppose Z � X� � Y�� The cotree X� � Y� corresponds to the GF��nonsingular submatrix �B��t of B in the same way in which the tree Z isrelated to the GF��nonsingular B� of B�

Append an identity to Bt� getting

��Y IA*=

Y X

Bt

Bt with additional identity matrix

Evidently� every cotree indexes a GF��basis of A�� and vice versa�

Cocycle

We know that a cocycle is a minimal set that intersects every tree� Putdi erently� a cocycle is a minimal set that is not contained in any cotree�The latter de�nition shows that cocycles are related to cotrees in the sameway that cycles are related to trees� Thus� for the interpretation of cocyclesin terms of B� the previous discussion for cycles becomes applicable once wemake suitable substitutions� The special cycle case with jY�j � � becomesthe special cocycle case with jX�j � �� say with X� � fxg� If row x ofB contains no �s� then x is a coloop� Otherwise� let Y� be the index setof the columns with a � in row x� Then Y� � fxg is a cocycle� It is thefundamental cocycle that x forms with a subset of the cotree Y � We may usefundamental cocycles to construct the rows of B� The process is analogousto the construction of the columns of B via fundamental cycles�

Coparallel or Series Elements

A cocycle of cardinality �� say fx� zg with x � X� manifests itself in B bytwo parallel rows indexed by x and z� or by a unit vector row indexed by

� Chapter � From Graphs to Matroids

x and with the � in column z� We also say that x and z are coparallel� orin series� Declare any element to be in series with itself� Then �is in serieswith� is an equivalence relation� The equivalence classes are the coparallelor series classes of M � They are precisely the series classes of G� accordingto the nonstandard de�nition of �series� for graphs in Section ��

Dual of Graphic Matroid

For a given graph G� let I� be the set of cotrees and their subsets� The factthat cycles are related to I in the same way in which cocycles are relatedto I� suggests that we tie I and I� together by duality� Speci�cally� wede�ne the pair M� � �E�I�� to be the dual matroid of M � The pre�x �co��dualizes a term� For example� each set in I� is co�independent� Consistentwith the de�nition of graphic matroid� we call M� the cographic matroid ofG� and denote it by M�G�� By the above discussion� Bt represents M��For this reason� we call Bt cographic�

Let G be a connected plane graph� and G� be its dual� In Section�� the following is shown� The cotrees of G are the trees of G�� and thecocycles of G are the cycles of G�� Thus� in this special case� M� is thegraphic matroid of G�� Consistent with graph terminology� we call M � aswell as all of its representation matrices B� planar� This de�nition and thepreceding observation imply that planarity of M implies graphicness of Mand M�� or equivalently� graphicness and cographicness of M �

Is it possible that the dual of a graphic matroid is not graphic� Theanswer is �yes�� Toward the end of this section� we include two examplesof a graphic M where M� is not graphic� In Chapter �� it is proved thatM� is graphic if and only if G is planar�

Pivot

According to the pivot rule �� of Section �� a GF��pivot in B of�� say with pivot element Bxy� is carried out as follows�

��

�� We replace for every u � �X�fxg� and every w ��Y �fyg�� Buw by Buw � �Buy �Bxw� �operationsin GF��

�� We exchange the indices x and y�

Let B� be the matrix produced from B by the pivot� From Section�� we know that this change corresponds to elementary row operationsand one column exchange that transform the matrix A composed of B and

� � Graphs Produce Graphic Matroids �

an identity matrix� i�e��

��BA = IX

X Y

Matrix A derived from B

to

��X' IA' =

X' Y'

B'

Matrix A� derived from B�

where X � � �X � fxg� � fyg and Y � � �Y � fyg� � fxg� Hence� by pivotswe can deduce from B any matrix B�� that corresponds to a speci�ed treeX �� of G� This fact is very useful� It permits us always to select a B�� thatis particularly convenient for our purposes� One such case we see next�when we discuss the e ects of edge deletions� additions� contractions� andexpansions� We brie�y review these graph operations�

A deletion of a noncoloop edge is the removal of that edge� A deletionof a coloop is accomplished by a contraction� which is de�ned next� Acontraction of a nonloop edge is the contraction of the edge so that itsendpoints become one vertex� A contraction of a loop is a deletion� Anaddition of an edge is the addition of an edge that does not become acoloop� An expansion by an edge involves splitting of a node into twonodes� which are joined by the new edge� The new edge cannot be a loop�A reduction is a deletion or a contraction� An extension is an addition oran expansion� For disjoint U�W � E� the minor G�UnW is the graphproduced by contraction of the edges of U and deletion of the edges of W �The process is well de�ned because of an implicit ordering of the edges ofG� No such ordering needs to be assumed when U contains no cycle and Wno cocycle� Any graph producible from G by some sequence of reductionsis obtainable as G�UnW � where U and W obey the cycle�cocycle conditionjust stated�

Matroid Deletion� Addition� Contraction� Expansion�Minor

We translate the above graph operations into matroid language� We startwith the taking of minors� So let G�UnW be one such minor� As juststated� we may assume U and W to obey the cycle�cocycle condition� Weclaim that G has a tree X and a cotree Y � E �X such that U � X and


W � Y � The proof is as follows� Since W contains no cocycle of G� theminor GnW has the same rank as G� Evidently� GnW contains U � SinceU does not contain a cycle of G� U also does not contain a cycle of GnW �Thus� in GnW we may extend U to a tree X of GnW � Since GnW and Ghave same rank� X is also a tree of G� By the construction� the tree X ofG contains U � and the cotree Y � E �X of G contains W �

Let B be the matrix of �� corresponding to X and Y � i�e�� B is

�� B

Y

X

Matrix B for general graph G with tree X

Derive B� from B by deleting the rows indexed by U and the columns in�dexed by W � We claim that B� represents the graphic matroid of G�UnW �We denote that matroid by M�UnW � and we call it the minor of M ob�tained by contraction of U and deletion of W � The proof is as follows�

Each tree Z � of G�UnW is contained in E � �U �W �� Indeed� Z � �Umust be a tree Z � �E�W � of G� Conversely� for every tree Z � �E�W �of G with U � Z� the set Z � � Z �U is a tree of G�UnW � We know fromthe earlier discussion that Z � X��Y�� with X� � X and Y� � Y � is a treeof G if and only if the square submatrix B� of B indexed by X� � X �X�

and Y� is GF��nonsingular� Thus� the trees Z � of G�UnW correspondprecisely to the square GF��nonsingular submatrices B� of B�� Hence�B� represents M�UnW � the graphic matroid of G�UnW �

Recall from Section �� that the cycles and cocycles of G undergo thefollowing changes as we go from G to G�UnW � The cycles of G�UnW arethe minimal nonempty sets of the collection fC � U j C � E �W � C �cycle of Gg� The cocycles of G�UnW are the minimal nonempty sets of the

collection fC� �W j C� � E � U � C� � cocycle of Gg� Correspondingly�the circuits and cocircuits of M�UnW may be derived from those of M �

We just proved that each deletion �resp� contraction� of an edge of Gthat is not a coloop �resp� loop� produces the removal of a column �resp�row� from a properly selected matrix B� Each addition �resp� expansion��the operation inverse to deletion �resp� contraction�� corresponds to theadjoining of a column �resp� row� to B�

Suppose G is a plane graph� and G� its dual� We know that Bt

represents the graphic matroid of G�� Now row vectors of B appear ascolumn vectors in Bt� and column vectors of B appear as row vectors inBt� A deletion in G induces a column removal in B� and thus a row removalin Bt� The latter removal corresponds to a contraction in G�� By matroidarguments� we have proved that deletions in G correspond to contractionsin G��


Up to this point� we have assumed that G has a cycle and a cocycle�Now suppose that this is not the case� Then G consists only of loopsincident at one node� or of coloops� or is empty� In the �rst case� we de�neB to have no rows and as many columns as G has loops� In the secondcase� B is to have no columns and as many rows as G has coloops� In thethird situation� B is to be the �� matrix� By the de�nitions of Chapter ��in the �rst two situations B is a trivial matrix� and in the third one theempty matrix� A trivial �resp� empty� B represents a trivial �resp� empty�matroid� The empty matroid has E � � I � I� � fg� and has no circuitsor cocircuits� We leave it to the reader to verify that the above reductionand extension results are valid for the special case of a trivial or empty B�

Separations and Connectivity

We turn to Tutte k�separations� for short k�separations� of G� We showhow such separations manifest themselves in B of �� Recall fromChapter � that a k�separation of G with k � � is a pair �E�� E�� ofnonempty sets that partition E and that have the following properties�First� jE�j� jE�j � k� Second� the subgraph G� �resp� G�� obtained fromG by removal of the edges of E� �resp� E�� must be connected� Third�identi�cation of k nodes of G� with k nodes of G� must produce G� Fork � �� the graph G is Tutte k�connected� for short k�connected� if G has nol�separation for � l � k�

Given a k�separation �E�� E�� of G and given the matrix B of ��de�ne for i � �� Xi � Ei�X and Yi � Ei�Y � Thus� B can be partitionedas

�� X1B =

Y1

B1

D1X2

Y2

B2

D2


We claim that

�� GF��rank D� � GF��rank D� � k � ��

For a proof� append to B an identity matrix� getting

��A =

X2

X1

X1 X2

0

0

Y1

D1

B1

Y2

B2

D2

...11

...11

Matrix B of �� with additional identity matrix


Derive from A the matrix A� �resp� A�� by deleting the columns indexedby X� � Y� �resp� X� � Y�� Then

��GF��rank A� � jX�j� GF��rank D�

GF��rank A� � jX�j� GF��rank D�

The matrix A may be obtained from the node�edge incidence matrix Fof G by row operations� Thus� for i � �� the rank of Gi is equal toGF��rank Ai� We combine this result with �� and get

��

�rank of G�� rank of G��

� GF��rank A� � GF��rank A�

� jX�j� jX�j� GF��rank D� � GF��rank D�

� �rank of G� � GF��rank D� � GF��rank D�

Finally� the graphs G� and G� are connected� and identi�cation of k nodesof these graphs produces G� For i � �� let the graph Gi have ni � knodes� Then G has n� � n� � k nodes� With these de�nitions�

��

�rank of G�� rank of G�� n� � k � �� n� � k � ��

� �n� � n� � k � �� k � ��

� �rank of G� � �k � ��

Then �� follows directly from �� and ��

De�nition of k�Separation and k�Connectivity

The preceding discussion motivates the following de�nitions� For any k � ��the matrix B of �� is k�separable if B can be partitioned as in ��such that

��jX� � Y�j� jX� � Y�j � k

GF��rank D� � GF��rank D� k � �

The pair �X� � Y��X� � Y�� is a k�separation of B� For k � �� the matrixB is k�connected if B has no l�separation for � l � k�

Connectivity in M is de�ned via that of B� That is� for k � �� Mis k�separable if B is k�separable� For k � �� M is k�connected if M�equivalently� B� has no l�separation for � l � k� In particular� theempty matroid is k�connected for all k � ��

The above de�nitions and observations validate the following lemma�


�� Lemma� Let G be a connected graph� and M be the graphicmatroid of G� For k � �� any k�separation of G induces a k�separation ofM �

Let us na��vely attempt to prove the converse� That is� we assume ak�separation �X� �Y��X� �Y�� of M as given by �� and �� Welet G� �resp� G�� be the subgraph created from G by removal of the edgesof E� � X� � Y� �resp� E� � X� � Y�� Now we try to reverse the order ofthe arguments made earlier about ��

The inequality GF��rank D� � GF��rank D� k � � of ��creates a �rst diculty� We would like equality� This problem is easilyavoided� We simply restrict ourselves to a k�separation of M with minimalk� Equivalently� we may demand M � and hence B� to be k�connected andk�separable�

By �� and �� we have

�� rank of G�� rank of G�� rank of G� � k � �

We also know that jE�j� jE�j � k by �� We could prove �E�� E�� tobe a k�separation of G if we could show G� and G� to be connected� Try aswe might� this we cannot do� For good reason� since �E�� E�� is not alwaysa k�separation of G� as we shall see�

Link between Graph and Matroid Separations

So let us scale down our goal� Let us simply strive to obtain a detaileddescription of the structure of G with k�separable M � k being minimal�To this end� we temporarily abandon our notion of nodes as edge subsets�Instead� we adopt the customary notion of nodes as points� Thus� a nodeof G may occur in several subgraphs of G� We retain the assumption thatG is a connected graph with edge set E�

We need a few de�nitions to describe and prove the structure of G�Let H�� H�� Hp� p � �� be subgraphs of G whose edge sets partition E�Each Hi contains at least one edge� The connecting vertices of Hi are thevertices of Hi that also occur in some Hj � j �� i� The remaining verticesof Hi are internal� The vertices that occur in both Hi and Hj � i �� j� arethe common vertices of Hi and Hj � The sum of H�� H�� Hr � r p�written as H� � H� � � � � � Hr � is the not necessarily connected subgraphof G whose edge set is the union of the edge sets of H�� H�� Hr�

Let L�H��H�� Hp� be the following connected graph� Its nodesare labeled H�� H�� Hp� As many parallel arcs connect node Hi withnode Hj of L�H��H�� Hp� as Hi and Hj have vertices in common�We declare the graphs H�� H�� Hp to be connected in tree fashion ifL�H��H�� Hp� is a tree� They are connected in cycle fashion if the


latter graph is a cycle� To avoid confusion� we use �vertex� and �edge�in connection with G� and �node� and �arc� when L�H��H�� Hp� isinvolved�

We are now ready to describe and prove the structure of G�

�� Theorem� Let M be the graphic matroid of a connected graphG� Assume �E�� E�� is a k�separation of M with minimal k � �� De�neG� �resp� G�� from G by removing the edges of E� �resp� E�� from G� LetR�� R�� Rg be the connected components of G�� and S�� S�� Sh bethose of G��

�a� If k � �� then the Ri and Sj are connected in tree fashion��b� If k � �� then the Ri and Sj are connected in cycle fashion��c� If k � �� either �c�� or �c�� below holds�

�c�� Each of G� and G� is connected �thus G� � R� and G� � S��and contains a cycle or an internal vertex� The two graphs haveexactly k vertices in common�

�c�� One of g and h is equal to �� and the other one is equal to k�Without loss of generality assume g � � and h � k� Then S��S�� Sh contain exactly one edge each� The union of the edgesets of the Si �which is E�� is a cocycle of G of cardinality k� Thetwo connected components R� and R� of G� contain at least onecycle each�

Proof� The edge sets of R�� R�� Rg and S�� S�� Sh partition E�Let mi �resp� nj� be the number of internal vertices of Ri �resp� Sj�� De�nepij to be the number of vertices Ri and Sj have in common� We accomplishthe proof via a series of claims�

Claim �� The graph L�R�� Rg� S�� Sh� has g � h nodes and

��X

i�j

pij � g � h � k � �

arcs� and thus is a tree plus k � � arcs�

Proof� The number of nodes of L�R�� Rg� S�� Sh� is by de�nitiong�h� Since the graphsRi� Sj � and G are connected� we have �rank of Ri� �mi�P

j pij�� rank of Sj� � nj �P

i pij�� and �rank of G� �P

imi�Pj nj �

Pi�j pij � �� Furthermore� �rank of G��

Pi�rank of Ri� �P

imi �P

i�j pij � g� and �rank of G�� P

j�rank of Sj� �P

j nj �Pi�j pij�h� By �� rank of G��rank of G�� rank of G��k��

and thus �P

imi �P

i�j pij � g� � �P

j nj �P

i�j pij � h� � �P

imi �Pj nj �

Pi�j pij��k�� Solving the latter equation for

Pi�j pij � which

is the number of arcs of L�R�� Rg� S�� Sh�� we obtainP

i�j pij �g � h� k � �� Q� E� D� Claim �


Claim �� Parts �a� and �b� of the theorem hold�

Proof� Suppose k � �� By Claim �� L�R�� Rg� S�� Sh� is a tree�and �a� follows� Let k � �� Then L�R�� Rg� S�� Sh� has exactlyone cycle� If there are additional arcs� then L�R�� Rg� S�� Sh� hasa degree � node� and thus� G and M are ��separable� a contradiction of theminimality of k� Thus� �b� follows� Q� E� D� Claim �

As a result of Claim �� we may assume from now on that k � ��

Claim �� At least one of the graphs R�� Rg� S�� Sh has at leastas many edges as it has connecting vertices�

Proof� Assume otherwise� Thus� each subgraph Ri �resp� Sj� of G isa tree on

Pj pij �resp�

Pi pij� vertices� Hence� G has

Pi�j pij vertices

andP

i�P

j pij � �� P

j�P

i pij � �� P

i�j pij � g � h edges� UsingPi�j pij � g�h� k� � of �� we see that G has g�h� k� � vertices

and g � h � ��k � �� edges� Since G is k�connected� the degree of eachvertex of G is at least k� Thus�

�� k � �number of vertices� � � �number of edges�

Accordingly� k�g � h� k� �� g � h� ��k � �� which implies g � h �� k �� a contradiction� Q� E� D� Claim �

Claim �� For all i and j�

�� rank of Ri� � �rank of Sj� � �rank of Ri � Sj�

The inequality is strict if and only if pij � ��

Proof� Direct computation veri�es the claim� Q� E� D� Claim �

By Claim �� we may assume from now on that R� has at least as manyedges as it has connecting vertices� Equivalently� R� has an internal vertexor a cycle�

Claim �� If g � �� i�e�� if G� � R�� then case �c�� applies�

Proof� If each Sj has no internal vertex and is a tree� then G� has jE�j �Pj p�j � h edges� Using �� with g � �� we have jE�j � k � �� which

contradicts jE�j � k� Thus� without loss of generality� S� has at least asmany edges as it has vertices� Assume h � �� Since G does not have a ��separation� S� has at least two vertices in common with G�� Shift the edgeset of S� from E� to E�� The resulting pair �E�

� � E�� corresponds to the

subgraphs R��S� and S��S�� Sh of G and� with the aid of ��is easily checked to be an l�separation of M with l � k� But this contradictsthe minimality of k� Thus� h � �� By �� G� � R� and G� � S� havek vertices in common� so case �c�� holds� Q� E� D� Claim

From now on� we assume g � ��


Claim � If in L�R�� Rg � S�� Sh� the node Sj has degree �� thenthe subgraph Sj of G contains exactly one edge�

Proof� By the assumption� the subgraph Sj of G has two vertices in com�mon with R�� Rg� Suppose the edge set of Sj� say E�� has at least twoedges� Then �E�� E�E�� is a ��separation of M � which is not possible sincek � �� Q� E� D� Claim �

Claim � Case �c�� applies�

Proof� Shift the edge sets of R�� Rg from E� to E�� The resulting par�tition �E�

� � E�� of E corresponds to graphs R� and G�

� � R�� Rg�S�� Sh� By �� that partition is easily seen to be an l�separation ofM with l k� By the minimality of k� we have l � k� Then by Claim � R�

and G�� have exactly k vertices in common� and G�

� is connected� SupposeS�� S�� St are the Sj graphs that have vertices in common with R��Since G�

� is connected� so is the graph L�R�� Rg� S�� Sh�� Indeed�the latter graph is obtained from L�R�� Rg� S�� Sh� by the removalof node R�� and thus� by Claim �� has g � h � � nodes and g � h � �arcs� Hence� L�R�� Rg� S�� Sh� is a tree� Any tip node of that treemust correspond to some Sj � � j t� since otherwise G and M are��separable� For the same reason� t � �� Let S� be a tip node of thetree� Then the subgraph S� of G has exactly one vertex in common withR� � � � �� Rg � S� � � � �� Sh�

Similar arguments show that the graphs R� � S� and R� � � � ��Rg �S� � � � � � Sh also correspond to a k�separation of G and M � Thus� weconclude the following� R� and R� � � � ��Rg �S� � � � �� Sh have exactlyk vertices in common� R� � S� and R� � � � � � Rg � S� � � � � � Sh alsohave exactly k vertices in common� S� and R� � � � � � Rg � S� � � � �� Shhave exactly one vertex in common� The last three statements imply thatR� and S� have exactly one vertex in common� Hence� the node S� inL�R�� Rg� S�� Sh� has degree �� By Claim �� the subgraph S� of Gcontains just an edge�

Inductively� we now show that each of the subgraphs S�� St con�tains just one edge� Speci�cally� we assume for some r � t� that S�� Sr have just one edge each� We then prove that one of the subgraphs Sjof G� r � � j t� say Sr�� has just one edge� For the case r � � � t�we also show that t � h � k and g � �� We omit the detailed argumentssince they are very similar to the above ones�

It remains to be shown that each of R� and R� contains a cycle� Weknow that each subgraph S�� Sh has just one edge� So if R� or R� hasno cycle� i�e�� is a tree� then G has a degree � vertex� and G and M are��separable� a contradiction� Thus� �c�� holds� Q� E� D� Claim �

The above claims clearly establish Theorem ��


Theorem �� has several important results as corollaries� Theyare the main reason why� in this book� we prefer Tutte connectivity oververtex or cycle connectivity for graphs�

�� Corollary� LetM be the graphic matroid of a connected graphG� For any k � �� M is k�connected if and only if this is so for G�

Proof� By Lemma �� any l�separation of G is an l�separation of M �Via Theorem �� we argue that any l�separation of M with minimal limplies that G has an l�separation� as follows� The cases �a�� b�� and �c��of the theorem are straightforward� For case �c�� the graphs R� � S� �� Sh and R� produce the desired l�separation of G�

�� Corollary� Let G be a connected plane graph� Then for anyk � �� G is k�connected if and only if this is so for the dual graph G��

Proof� By Corollary �� and matroid duality� the graphs G and G�

and the matroids M and M� are all k�connected if this is so for any one ofthese graphs and matroids�

Recall from Chapter � that a matrix A is connected if the associatedbipartite graph BG�A� is connected� It is easily seen that A is not connectedif and only if A is a trivial matrix� i�e�� of size m�� or ��m� for some m � ��or A has a row or column without �s� or A has a block decomposition� Letus apply this result to a binary matrix B representing a graphic matroidM � It is easily checked via �� that B is ��connected if and only if Bis connected� Correspondingly� we de�ne the matroid M to be connected ifit is ��connected� Note that �G is connected� is a statement quite di erentfrom �B is connected� or �M is connected�� The latter two statements areby Corollary �� equivalent to �G is ��connected�� Admittedly� the useof �connected� has become a bit confusing by the above de�nitions� Butthat use is so well accepted in matroid theory that we employ it here� too�The next corollary summarizes the above relationship for future reference�

�� Corollary� Let G be a connected graph and B be a representa�tion matrix of the graphic matroid M of G� Then G is ��connected if andonly if B �and hence M� is connected�

Finally� we have the following characterization of a ��connected graphin terms of any representation matrix of its graphic matroid�

�� Corollary� The following statements are equivalent for a con�nected graph G with edge set E and for any representation matrix B ofthe graphic matroid M of G�

�i� G is ��connected��ii� If jEj � � G has no loops or coloops�

If jEj � � G has no parallel or series edges� Furthermore� deletion ofat most two stars does not produce a disconnected graph�

Chapter � From Graphs to Matroids

�iii� M is ��connected��iv� B is connected� has no parallel or unit vector rows or columns� and

has no partition as in �� with GF��rank D� � �� D� � �� andjX� � Y�j� jX� � Y�j � ��

�v� Same as �iv�� but jX� � Y�j� jX� � Y�j � �

Proof� Corollary �� proves �i� � �iii�� The implications �i� � �ii�and �iii� � �iv� are easily seen� and �iv� � �v� is trivial� The possiblysurprising �v� � �iv� is established by a straightforward checking processas follows� In B of �� assume GF��rank D� � � and D� � �� Ifthe length of B� is � or �� then B can be seen to have a zero column orrow� or parallel or unit vector rows or columns� Any such case is alreadyexcluded by the �rst part of �iv�� Thus� it suces to require jX� � Y�j � �and by duality� jX� � Y�j � �

In the remainder of this section� we address the following questions�How are the graphs related that produce a given graphic matroid� Howcan one obtain a graph that generates a given graphic matroid� When doesa binary matrix correspond to a graphic matroid�

Graph ��Isomorphism

We begin with the �rst question� So let M be a graphic matroid� Declareany two connected graphs G and H that produce M to be ��isomorphic�Necessarily� G and H have the same edge set� say E� Each one of thefollowing sets of edge subsets of G or H completely determines M � andthus must be the same for G and H� the set of trees� the set of cycles�the set of cotrees� and the set of cocycles� For the same reason� G and Hhave the same rank function� the same k�separations for any k � �� andthe same connectivity� Despite the numerous relationships between G andH� these graphs may be quite di erent� For example� in �� below�the �rst graph and the last graph of the sequence are quite distinct� yetwill be shown to be ��isomorphic�

We start with the case where M is ��separable� We claim that the��connected components of G� say G�� G�� Gt� are connected in treefashion� By Theorem �� G consists of p � � connected subgraphsthat are connected in tree fashion� Select a case with p maximum� If one ofthe subgraphs is not ��connected� then that subgraph itself consists of q � �subgraphs that are connected in tree fashion� Evidently� this contradictsthe maximality of p� By ��isomorphism� the edge set of each ��connectedcomponent Gi of G must be that of a ��connected component� say Hi� ofH� Thus� H�� H�� Ht are the ��connected components of H� whichare also connected in tree fashion� We emphasize that the tree structureproduced by G�� Gt may be entirely di erent from that of H�� Ht�

� � Graphs Produce Graphic Matroids

If for all i� we had Gi � Hi� then we would have completely explainedthe di erence between G and H� But this may not be so� Thus� we mustunderstand the relationships between ��connected but di erent Gi and Hi�To simplify the notation� we assume M � G� and H to be themselves ��connected� The next lemma shows that G is equal to H if we strengthenthat assumption to ��connectedness�

�� Lemma� Let G andH be ��connected and ��isomorphic graphs�Then G � H�

Proof� By trivial checking� we may assume that G has at least six edges�Let Z be any star of G� By Corollary �� ii�� Z is a cocycle of G andGnZ is ��connected� Thus� Z is a cocycle of H and HnZ is ��connected�This is only possible if Z is a star of H� We conclude that each star of Gis one of H� and vice versa� Then G � H�

By Lemma �� and the earlier discussion� just one case remains�where M � G� and H are ��connected and ��separable� Take a ��separationof G� It induces two subgraphs G� and G�� Assume that identi�cation ofnodes k and l of G� with nodes m and n� respectively� of G�� produces G�Instead� let us identify k of G� with n of G�� and l of G� with m of G��Here is an example of this operation�

��

fn

m

G'turned over

c

i h

b

a

d

f g

g

i h

G''

G''

j

j

G'

G

new graph

a

d

b

l

k

c

c

d

a

b

k

l

n

mj g

i h

e

e

f

f

i h

a

d

e

e

jb gc

Example of switching operation

Roughly speaking� we have switched the nodes of attachment of G� withG�� For this reason� we say that the new graph is obtained from G by a


switching� It is easy to see that G and the new graph have the same set ofcycles� Thus� they are ��isomorphic�

Let G� and G�� be the just�de�ned subgraphs of a ��separation ofG� The graph G� may not be ��connected� By Theorem �� the��connected components of G� are connected in tree fashion� That treemust be a path� since otherwise G is ��separable� Apply the same argu�ments to G�� Combine the two observations� Thus� G has ��connectedsubgraphs� say G�� G�� Gt� for some t � �� that are connected in cyclefashion� In the notation of Theorem �� G��G�� Gt � G� By ��isomorphism�H also has ��connected subgraphs� say H�� H�� Ht� wherefor all i� Gi and Hi have the same edge set� Clearly� H��H�� Ht � H�We now establish how the Hi are linked in H�

�� Lemma� H�� H�� Ht are connected in cycle fashion�

Proof� Each ��connected Gi with at least two edges constitutes one sideof a ��separation of G� By ��isomorphism and Theorem �� this alsoholds for the ��connected Hi and H� Thus� Hi has exactly two nodes incommon with the remaining subgraphs Hj of H� The same conclusionholds trivially if Gi and Hi have just one edge� Let C be any cycle of Gthat includes at least one edge of Gi and at least one edge of some Gj �j �� i� Then C must include at least one edge each from G�� G�� Gt�The analogous fact holds for the Hi� These observations imply that H��H�� Ht are connected in cycle fashion�

We now link ��isomorphism and switching�

�� Theorem� Let G and H be ��connected and ��isomorphicgraphs� Then H may be obtained from G by switchings�

Proof� If G is ��connected� Lemma �� applies� Otherwise� as ex�plained above� for some t � �� let G�� G�� Gt be ��connected subgraphsof G linked in cycle fashion and satisfying G� � G� � � � � � Gt � G� ByLemma �� the corresponding ��connected subgraphs H�� H�� Ht

of H are connected in cycle fashion as well�Consider G� by itself� Join the two nodes that G� has in common with

the remaining Gi� i � �� by an edge u� Let G�� be the resulting ��connected

graph� Analogously� add an edge u to H�� getting H �� By the structure of

G and H� the graphs G�� and H �

� are ��isomorphic� By induction� G�� can by

switchings be transformed to H �� The same switchings can be performed

in G when we view the edge u of G�� as representing G� � � � � �Gt� Thus�

by certain switchings� every Gi of G becomes Hi� Finally� the subgraphsHi of the new G can by switchings be so positioned that H results�

Chapter �� includes a polynomial algorithm that for a given graphicmatroid M �nds a graph G producing M � The algorithm can even de�termine whether an arbitrary binary matroid is graphic� The algorithm


essentially consists of two subroutines� One of them is called the graphic�ness testing subroutine� We describe and validate it next� using Theorem��

Graphicness Testing Subroutine

The input to the subroutine consists of a matrix B� given by

��b

z

B' = BX

Y

Input matrix for graphicness testing subroutine

The submatrix B indexed by X and Y is known to be graphic� Also givenis a graph G that produces B� That graph is known to be ��connected orto be a subdivision of a ��connected graph� In the subroutine� we want todecide whether B� is graphic� We �rst analyze the relationships among B��B� b� and G�

We know that the row index set X of B is a tree of G� Let Z be theset of rows x � X for which bx � �� In the tree X of G� paint each edge ofZ red� We leave the remaining edges of X unpainted�

Suppose B� is graphic� Let H � be a graph for B�� with the edges ofZ painted red as in G� In that graph� the set X is also a tree� Moreover�according to the column vector b and the painting rule� the red edges mustform a fundamental cycle with the edge z� Thus� the red edges form apath in H � as well as in H �nz � H� The graph H generates B� as does G�Hence� the ��connected G and H are ��isomorphic� By Theorem ��H is obtainable from G by switchings� Thus� graphicness of B� implies thatG can by switchings be transformed to a graph where the red edges forma path�

Conversely� assume that G can by switchings be changed into a graphH where the red edges form a path� Add an edge z to H whose endpointsare those of the path� Let H � be the resulting graph� Then in H � thefundamental cycle that z forms with X is fzg � Z� and thus� H � generatesB�� Therefore� B� is graphic�

Recall that any graph G to be processed by the graphicness testingsubroutine either is ��connected or is a subdivision of a ��connected graph�In the �rst case� no switchings are possible� Thus� B� is graphic if andonly if the red edges form a path in G� Assume the second case� i�e�� G isa subdivision of a ��connected graph� Then in any ��separation of G� theedge set of one side is readily seen to be a subset of some series class ofG� Evidently� any switching amounts to a resequencing of some edges ofthe series class� Conversely� any resequencing of the edges of a series class


can clearly be accomplished by switchings� It is a trivial matter to checkwhether resequencing of series class edges can result in a red path� Thus�we can readily decide whether B� is graphic� We leave it to the reader towrite down the rules formally�

Suppose B� is found to be graphic� The subroutine then outputs thegraph H � and stops� If B� has been determined to be not graphic� then thesubroutine says so and stops�

We demonstrate the subroutine with four examples� In the �rst case�we have

��1

f gY

e

0

0cd

ab

0

0

10

11

h i z

11

0

01

1 00 1

1 1 00 0 1

B' = X

Example � for graphicness test

The graph G

��

f

hea

b

c gd i

Graph G for Example �

produces the submatrix B of B� indexed by X and Y � Evidently� G isisomorphic to the wheel W� plus one additional edge and is ��connected�According to our graphicness test for B�� the edges of X corresponding tothe �s in column z of B� must be painted red� Thus� we paint the edges band d� These red edges form a path� so B� is graphic� In G� we join thetwo endpoints of that path by an additional edge z to obtain the followinggraph for B��

��

f

hea

b

c gd i

z

Graph H � for Example �


That graph is isomorphic to K�� the complete graph on �ve vertices� Thus�B� of �� represents up to index changes M�K�� the graphic matroidof K��

The second example involves the matrix

��

0 1

1

f

e 1 1

g hY

0ab

011

z0 1

0B' = X

0cd

0 1 1 01

1 1


A graph G for the submatrix B indexed by X and Y is given by

��

f

d

a

b

c

g

h

e


Evidently� the graph is a subdivision of the ��connected wheel W�� Thistime� we must paint the edges a� d� and e red� The red edges do not forma path� but we can obtain a path by resequencing a and f as well as d andh� Following such sequencing� we join the endpoints of the resulting redpath by a new edge z to obtain the graph

��f

h

e

a

b

c

g

dz

Graph H � for Example �

Thus� B� is graphic� We leave it to the reader to verify that the graph forB� is isomorphic to K�� the complete bipartite graph with three verticeson either side� We conclude that B� of �� represents up to indexchanges M�K�� the graphic matroid of K��


For the third example� we re�index and repartition the transpose ofthe matrix of �� to get

��

010

0d

0

1

g h iY

0ab

011

z1 0

0

B' = X c 0 1 1 00 0

e1 1

1f 1 1


A graph G producing B is

��

i

f

he

a

b cg

d


The graph is clearly a subdivision of the ��connected wheel W�� This timewe must paint the edges d� e� and f red� Obviously� no resequencing of dand g� of e and h� and of f and i can result in a red path� Thus� B� is notgraphic� Since it is up to indexing the transpose of the matrix of ��it represents up to a change of indices M�K�� Thus� that matroid is notgraphic�

For the fourth case� we re�index and repartition the transpose of thematrix of �� as

��

f g h

0

1

eY

ab

011

B' = X

0cd

0 11

00 1

01 0

1

z1111


The graph G of �� below produces the submatrix B of B� indexed byX and Y � Evidently� G is isomorphic to the ��connected wheel W��


��

f

hea

b

cg

d


According to column z of B�� we paint in G all edges of X red� The rededges do not form a path in G� Thus� B� is not graphic� Since B� is up toa re�indexing the transpose of the matrix of �� it represents up to achange of indices M�K�� Thus� that matroid is not graphic�

The last two examples establish the following result�

�� Lemma� The matroidsM�K�� andM�K�� are not graphic�

There are easier ways to prove Lemma �� The matroid M�K��

has ten elements and rank �� Thus� a graph for that matroid has ten edgesand seven nodes� Since M�K�� is ��connected� the degree of each nodeof the graph is at least �� But then the graph must have at least elevenedges� a contradiction� The matroid M�K�� has nine elements and rank�� Since contraction of any edge in K�� reduces that graph to the ��connected wheel graph W�� deletion of any element from M�K�� mustresult in a ��connected minor� We conclude that a graph for M�K�� musthave nine edges and �ve nodes� and deletion of any edge from that graphmust produce a ��connected minor� There is only one candidate graph� Itis K� minus one edge� But that graph has two vertices of degree �� andthe deletion of some edge produces a ��separable minor� a contradiction�

We conclude this section by recording the existence of the polynomialgraphicness testing subroutine�

�� Lemma� There is a polynomial algorithm� called the graphic�ness testing subroutine� for the following problem� Input is a binary matrixB� � B j b!� where B is graphic� Also given is a graph G for B� It is knownthat G is ��connected or is a subdivision of a ��connected graph� The al�gorithm decides whether B� is graphic� In the armative case� a graph Hfor B� is also produced�

We mentioned earlier that the graphicness testing subroutine will inChapter �� be combined with a second subroutine to a polynomial testfor graphicness of binary matroids� That second subroutine carries out thefollowing task� It analyzes the connectivity of the binary matroid for whichgraphicness is to be decided� In doing so� the subroutine converts the given


problem into a sequence of subproblems� each of which can be solved bythe above graphicness testing subroutine�

In the next section� we extend the class of graphic matroids to theclass of binary matroids�

�� Binary Matroids Generalize Graphic

Matroids

So far� we have used graphs to create the graphic matroids� In the process�we have developed quite a few matroid concepts� In this section� we gener�alize the graphic matroids to the binary ones� Indeed� the de�nitions andconcepts for graphic matroids introduced in the preceding section are al�ready so rich that most of them need at most a trivial adjustment to makethem suitable for binary matroids� Thus� this section is long on de�nitionsand short on motivation of concepts and explanations�

The reader familiar with matroid theory will surely notice that manyresults of this section hold for general matroids� not just binary ones� Inthe next section� �� we include a list of these results� We cover binarymatroids here in such detail for two reasons� First� we want to exhibit howfeatures and properties of matroids are motivated by elementary linearalgebra arguments� Binary matrices and matroids are the perfect vehicleto display that relationship� Second� the techniques and arguments of thissection are used in much more complicated settings in subsequent chapters�The discussion of this chapter thus sets the stage and prepares the readerfor that material�

We proceed as follows� We de�ne the binary matroids via binarymatrices� along with fundamental concepts such as base� circuit� cobase�cocircuit� rank function� and representation matrix� We introduce matroidminors via the reduction operations of deletion and contraction and explainthe e ect these operations have on bases� circuits� cobases� and cocircuits�

Next� we describe �Tutte� k�separations and k�connectivity� We linkthese concepts to the related ones for graphs� At that time� we are ready fora census of the ��connected binary matroids with at most eight elements�

In this book� the presentation relies heavily on what we call the matrixviewpoint of binary matroids� In the remainder of the section� we showthat other viewpoints are just as important� In particular� we introducethe submodularity of the rank function and prove with that concept a basic��connectivity result� We now begin the detailed presentation�

Binary Matroid

Let F be a binary matrix with a column index set E� Let I be the collection

� � Binary Matroids Generalize Graphic Matroids �

of subsets Z � E such that the column submatrix of F indexed by Z hasGF��independent columns� We consider Z � to be in I� The setsZ of I are independent� Declare M � �E�I� to be the binary matroidgenerated by F � The set E is the groundset of the matroid� A base Xof M is a maximum cardinality subset of I� Equivalently� X indexes thecolumns of a GF��basis of F � A circuit C of M is a minimal subset of Ethat is not contained in any base of M � Equivalently� C indexes a GF��mindependent column submatrix of F � A cobase Y of M is the set E �Xfor some base X� A cocircuit C� of M is a minimal subset of E that is notcontained in any cobase of M � The rank of a subset Z � E� denoted byr�Z�� is the cardinality of a maximal independent subset contained in Z�The function r�� is the rank function of M � Collect in a set I� all cobasesY of M and all their subsets� The sets Z� of I� are co�independent� Thepair M� � �E�I�� is the dual matroid of M �

Suppose M is the graphic matroid of a connected graph G with edgeset E� Then a base �resp� a circuit� a cobase� a cocircuit� the rank function�of M is a tree �resp� a cycle� a cotree� a cocycle� the rank function� of G�

Representation Matrix

Suppose a matrix A is deduced from F by elementary row operations�Clearly� GF��independence of columns is not a ected by such a change�Thus� we may determine I from A instead of F � A special case is as follows�First� we delete GF��dependent rows from F � getting� say� F �� Second�we perform binary row operations to convert the column submatrix of F �

indexed by some base X to an identity� With Y � E �X� we thus havefor some binary matrix B�

��BA = IX

X Y

Matrix A for matroid M with base X

We allow the special cases X � or Y � � B is then a trivial or emptymatrix� The information contained in A is also conveyed by the submatrixB of A� which has the form

��B

Y

X

Matrix B for matroid M with base X

The binary B is a representation matrix of M � We also say that B rep�resents M over GF�� In the literature� the term standard representation

Chapter � From Graphs to Matroids

matrix is sometimes used for A or B� The matrix F is then a nonstan�dard representation matrix� But we almost always work with B� so theabbreviated terminology suces�

Bases� circuits� and the rank function of M manifest themselves in Bas follows� For any partition of X into X� and X� and of Y into Y� andY�� we assume B to be partitioned as

�� X1B =

Y1

B1

D1X2

Y2

B2

D2


Base� Rank Function

A set Z � E is a base of M if and only if X� � Z�X and Y� � Z�Y inducea partition in B where B� is square and GF��nonsingular� More generally�let Z be an arbitrary subset of E� Then B� de�ned via X� � Z � X andY� � Z � Y has GF��rank k if and only if Z has rank r�Z� � jX�j� k inM �

Circuit

Let C � E� De�ne X� � C �X and Y� � C � Y � Then C is a circuit of Mif and only if in B of �� the number of �s in the rows of B� �resp� D��is even �resp� odd�� and for any proper subset C � � C� the corresponding�B�� and �D�� de�ned by X �

� � C � �X and Y �� C � � Y � do not satisfy

that parity condition� Note that Y� � C � Y and the parity conditionuniquely determine X�� and thus B� and D��

Recall that a f�� g matrix is column Eulerian if each row contains aneven number of �s� or equivalently� if the columns sum �in GF�� to ��By the above discussion� any circuit of M indexes a column submatrix ofA � I j B! that is column Eulerian� Conversely� any column submatrixof A that is column Eulerian and that does not contain a proper columnsubmatrix with that property corresponds to a circuit of M �

We describe three special cases of circuits using the above notation�

Loop

Suppose jCj � �� say C � fyg� The element y is a loop of M � Necessarily�Y� � fyg� and column y of B must be a zero vector� Conversely� any zerocolumn vector of B corresponds to a loop�

� � Binary Matroids Generalize Graphic Matroids

Parallel Elements� Triangle

Suppose jCj � �� The two elements of C� say y and z� are said to beparallel� Both elements of C cannot be in X� so we may assume y � Y �Two cases are possible� We have either X� � fzg and Y� � fyg� or X� � and Y� � fy� zg� In the �rst case� column y of B is a unit vector with � inrow z� In the second case� columns y and z of B are parallel� Conversely� acolumn unit vector or two parallel columns of B correspond to two parallelelements of M � If jCj � �� then C is a triangle�

Fundamental Circuit

Suppose jCj � � and jY�j � �� say Y� � fyg� Then X� is the index setof the rows of B with �s in column y� and C � X� � fyg� The circuit Cis called the fundamental circuit the element y forms with the base X ofM � The fundamental circuits Cy that the elements y � Y form with Xallow a fast construction of B� Indeed� each column y � Y of B is thecharacteristic vector of Cy � fyg� That is� column y has �s in the rowsindexed by Cy � fyg� and �s elsewhere�

Cobase� Cocircuit

Cobases and cocircuits of M are exhibited by B as follows� Let Z � E� Asbefore� de�ne X� � Z �X� X� � X �X�� Y� � Z � Y � and Y� � Y � Y��By the earlier de�nition of base and cobase� the set Z is a base of M andZ� � E � Z � X� � Y� is a cobase of M if and only if in the transpose ofB�

�� (D

Y2

Y1 (B1)t 1)

t

)(D2 t

)(B2 tBt =

X1 X2

Transpose of B of ��

the submatrix �B��t is square and GF��nonsingular� Append to Bt anidentity matrix� getting

��Y IA*=

Y X

Bt

Bt with additional identity matrix


Evidently� Z� is a cobase of M if and only if Z� indexes the columns of aGF��basis of A��

Let C� � E� X� � C� � X� X� � X � X�� Y� � C� � Y � andY� � Y � Y�� Then C� is a cocircuit of M if and only if the earlierdescribed circuit condition holds for C� and Bt instead of C and B� Wedescribe three special cases in terms of B�

Coloop

Suppose jC�j � �� say C� � fxg� The element x is a coloop of M � Neces�sarily� X� � fxg� and row x of B must be a zero vector� Conversely� anyzero row vector of B corresponds to a coloop�

Coparallel or Series Elements� Triad

Suppose jC�j � �� The two elements of C�� say x and z� are said to becoparallel or in series� Both elements cannot be in Y � so we may assumex � X� Two cases are possible� We have either X� � fxg and Y� � fzg�or X� � fx� zg and Y� � � In the �rst case� row x of B is a unit vectorwith � in column z� In the second case� rows x and z of B are parallel�Conversely� a row unit vector or two parallel rows of B correspond to twoseries elements of M � If jC�j � �� then C� is a triad�

Fundamental Cocircuit

Suppose jC�j � �� and jX�j � �� say X� � fxg� Then Y� is the index set ofthe columns of B with �s in row x� and C� � Y� � fxg� The cocircuit C�

is called the fundamental cocircuit the element x forms with the cobase Yof M � Analogously to the circuit case� the fundamental cocircuits permita fast construction of B�

Binary Spaces

We relate circuits and cocircuits of M to binary spaces on E� i�e�� to thelinear subspaces of GF��E � Speci�cally� consider the nullspace S of A of�� which is given by fs j A � s � �g� Evidently� the vectors s � S withminimal support are exactly the characteristic vectors of the circuits of M �Any basis of these vectors generates S�

By de�nition of M� from A� of �� the minimal support vectorsof the nullspace S� � fs� j A� � s� � �g of A� are exactly the characteristicvectors of the cocircuits of M��

So� in a way� the circuits of M generate S� Correspondingly� thecocircuits produce S�� which is well known �and easily proved� to be theorthogonal complement of S�


Intersection of Circuits and Cocircuits

Frequently� a judicious choice of the baseX and the cobase Y of M producesa B that simpli�es the proof of some result� Equivalently� we may want toproceed by GF��pivots from a given representation matrix of M to someother one to exhibit a particular aspect of M � The pivots were coveredin Section �� and also in �� Thus� we omit details about thatoperation�

Here is an example result that is easily proved with a clever choice ofX and Y � That result ties circuits to cocircuits by a parity condition�

�� Lemma� Let C �resp� C�� be a circuit �resp� cocircuit� of abinary matroid M � Then jC � C�j is even�

Proof� Choose a cobase Y that contains all elements of C� save one� say x�Let B be the related matrix� Thus� row x of B is the characteristic vectorof C� � fxg� If x � C �resp� x �� C�� then by the previously describedparity condition for circuits� row x of B has an odd �resp� even� numberof �s in the columns indexed by C � Y � Either case proves jC � C�j to beeven�

Of course� Lemma �� also follows trivially from the just�cited or�thogonality of the binary spaces S and S� produced by the circuits andcocircuits� respectively� of M �

Symmetric Di�erence of Circuits

We should mention the following result about circuits�

�� Lemma� Let C� and C� be two circuits of a binary matroid M �Then �C� � C�� C� � C�� the symmetric di�erence of C� and C�� is adisjoint union of circuits of M �

Proof� Given B� de�ne A � I j B!� As observed earlier� the columnsubmatrix of A indexed by C� �resp� C�� is column Eulerian� The samefact holds for the column submatrix indexed by Z � �C� �C�� C� �C��Thus� the columns indexed by Z are GF��dependent� Hence� Z containsa circuit C� Then Z�C is also column Eulerian� and the desired conclusionfollows by induction�

Note that the proof of Lemma �� remains valid when each of C�

and C� is a disjoint union of circuits� Thus� we have the following seeminglymore general result�

�� Lemma� Let each of C� and C� be a disjoint union of circuitsof a binary matroid M � Then the symmetric di�erence of C� and C� is adisjoint union of circuits of M �


Deletion� Contraction� Reduction

We turn to deletions and contractions of binary matroids� Any such oper�ation is a reduction�

The operations are de�ned as follows� Let B be the matrix of ��representing M � Thus� B is

�� B

Y

X


A deletion of an element w � E from M leads to a matroid M representedas follows� If w is not a coloop of M � select any representation matrix Bof M where w � Y � Delete column w from B� The resulting matrix Brepresents M � It is easy to verify that the same M results� regardless ofwhich speci�c B is selected to determine B� The proof consists of showingthat all possible B are obtainable from each other by GF��pivots� If wis a coloop of M � we declare the deletion to be a contraction� which iscovered next� A contraction of an element u � E in M leads to a matroidM represented as follows� If u is not a loop of M � select any representationmatrix B of M where u � X� Delete row u from B� The resulting matrixB represents M � Here� too� the outcome does not depend on the selectionof the speci�c B� If u is a loop� declare the contraction to be a deletion�

Even after the discussion of deletions and contractions for graphs andgraphic matroids in Sections �� and �� the reader may still be a bit puz�zled that we have declared the deletion of a coloop of M to be a contraction�and the contraction of a loop of M to be a deletion� Below� we motivatethese rules using the matrix A � I j B!� Suppose we intend to delete anelement w from M � Our goal is to transform A to a matrix A � I j B! sothat the index sets of independent column submatrices of A are preciselythe index sets of independent column submatrices of A that do not includew� Note that this goal is a generalization of the goal for edge deletions ingraphs� as covered in Section �� There we relied on tree subsets insteadof independent column submatrices� It turns out that we must considerthree cases of A to determine the desired A� In the �rst case� w indexes acolumn of B� Then deletion of column w from A� and thus from B� givesthe desired A and B� This case is covered by the deletion rule given above�In the second case� w indexes a column of the identity I� and row w of Bis nonzero� Then by row operations in A� or equivalently by a GF��pivotin B� we achieve the �rst case� again in conformance with the deletion rulegiven above� The third case is like the second one� except that row w of Bis zero� Note that w is then a coloop of M � Suppose we delete column w


from A� Then row w of the resulting matrix A� is zero and has no in�u�ence on the independence of column subsets of A�� Thus� we might as welldrop that row from A�� But these two steps� deletion of column w fromA followed by deletion of row w� correspond precisely to the deletion ofrow w from B� and thus to the contraction of the coloop w� The situationfor the contraction of a loop may be explained in the same manner usingA� � I j Bt! instead of A�

Uniqueness of Reductions

Let U and W be two disjoint subsets of E� Suppose we contract theelements of U and delete the elements of W � In a moment� we outlinea proof showing that the outcome is not a ected by the order in whichthe reductions are carried out� Assuming that result� we are justi�ed indenoting the resulting unique matroid by M�UnW � Any such matroid isa minor of M � For convenience� we consider M itself to be a minor ofM � Analogously to the case of G�UnW in Section �� we may simplifythe notation for M�UnW � Thus� we may write M�U � MnW � M�u whenU � fug� etc�

That the order of the reductions is irrelevant may be shown by in�duction as follows� One reverses the sequence of two successive reductionsteps and proves that this change has no e ect on the outcome� We leaveit to the reader to carry out the elementary case analysis� Note that thepreceding uniqueness result for reduction sequences is at variance with thesituation for graphs� According to Section �� the ordering of reductionsequences does matter when graphs are involved� Indeed� in that section weintroduced a technical device to enforce a certain ordering� We have justseen that the sequence of the reductions is irrelevant in binary matroids�and thus in graphic matroids� This implies that the graphs producible bydi ering sequences must all correspond to the same matroid minor� Thus�all such graphs are ��isomorphic� In this book� we almost always look atgraphs from the matroid standpoint� and can a ord to ignore di erencesbetween ��isomorphic graphs� Indeed� any one graph of a collection of ��isomorphic candidates may be used to carry out proofs or to develop ideasabout matroids� These facts motivated our choice of the technical deviceof Section ��

Circuit�Cocircuit Condition

Analogously to the case for graphs� for any minorM of M � there are disjointU�W � E such that U contains no circuit� W contains no cocircuit� andM � M�UnW � We say that such U and W satisfy the circuit�cocircuitcondition� The proof of the existence of U and W is almost trivial� We


know that a representation matrix for M can be deduced from one for Mby a sequence of operations� each of which is a GF��pivot or the deletionof a row or column� We could perform all GF�� pivots initially� then carryout the deletion of rows and columns� Let U be the index set of the rowsso deleted� and W be that of the columns� Clearly� U and W satisfy thecircuit�cocircuit condition� and M � M�UnW as desired�

Bases� Cobases� Circuits� and Cocircuits of Minors

Sometimes� it is convenient to assume the circuit�cocircuit condition� Aninstance comes up next� We want to express bases� cobases� circuits� andcocircuits of a minor M � M�UnW in terms of the related sets of M �We claim that the formulation below suces if U and W satisfy the cir�cuit�cocircuit condition�

��

The set of bases of M � M�UnW isfX � U j U � X � E �W � X � base of Mg�the set of cobases of M isfX� �W jW � X� � E � U � X� � cobase of Mg

��

The set of circuits of M � M�UnW consists ofthe minimal members offC � U j C � E �W � C � circuit of Mg�the set of cocircuits of M consists ofthe minimal members offC� �W j C� � E � U � C� � cocircuit of Mg

Validation of �� and �� is not dicult� By the circuit�cocircuitcondition on U and W � contraction of U and deletion of W may be trans�lated to submatrix�taking in some matrix B of �� representing M �Thus� one only needs to examine the e ect of such submatrix�taking onbases� cobases� circuits� and cocircuits� We omit the arguments since theyclosely follow the presentation of Section �� about minors of graphic ma�troids�

Display of Minor� Visible Minor

Let M with representation matrix B have M as a minor� Suppose Brepresents M � If B is a submatrix of B� then we say that B displays Mvia B� or more brie�y� that B displays M � Note that the submatrix ofB claimed to be B must match not only the numerical entries� but alsothe row and column index sets of B� We also say that the minor M isvisible by the display of B in B� By the de�nition of minor via pivots androw�column deletions� we have the following simple but useful lemma�

� � Binary Matroids Generalize Graphic Matroids ��

�� Lemma� Let M be a binary matroid with a minor M � and Bbe a representation matrix of M � Then M has a representation matrix Bthat displays M via B and thus makes the minor M visible�

Two special cases of Lemma �� involve so�called contraction anddeletion minors� to be discussed next�

Contraction Minor

De�ne a minor M of M on a set E to be a contraction minor of M iffor some U � E� M � M�U � If M is the graphic matroid of a graph G�then any minor of G corresponding to a contraction minor of M is called acontraction minor of G� The next lemma shows how a contraction minorM manifests itself in representation matrices of M displaying M �

�� Lemma� The following statements are equivalent for any binarymatroid M and any minor M of M � Let B be a representation matrix ofM �

�i� M is a contraction minor of M ��ii� M has a representation matrix B displaying M via B� where B is of

the form

��X

U

B 0

WY

B =0 1

X

Y

Matrix B displaying contraction minor M

�iii� Every representation matrix B ofM displayingM via B is of the formgiven by ��

Proof� Assuming �i�� we deduce �iii� as follows� Let B display M via B�Thus� B is of the form given by �� except that possibly the submatrixof B indexed by X and W may not be zero� Assume that submatrix tobe nonzero� Now M � M��U �W � by assumption� Then by the rule forcontractions� a matrix for M is produced from B by deletion of the rowsof U � followed by one or more pivots and deletion of one or more rows�and �nally by deletion of some zero columns� But the resulting matrix hasfewer rows than B and cannot represent M � a contradiction� Trivially� �iii�implies �ii�� Finally� the contraction rule proves that �ii� implies �i��

Suppose we have a representation matrix B of a binary matroid M �Assume that B displays a minor M of M � Then parts �ii� and �iii� ofLemma �� provide a simple way of ascertaining whether or not M isa contraction minor of M � In the notation of �� the answer is �yes�if and only if the submatrix of B indexed by X and W is ��


Deletion Minor

A minor M � MnW is a deletion minor of M � Correspondingly to thecontraction case� we de�ne deletion minors of a graph G via those of thegraphic matroid of G� By duality� Lemma �� implies the followingresult�

�� Lemma� The following statements are equivalent for any binarymatroid M and any minor M of M � Let B be a representation matrix ofM ��i� M is a deletion minor of M ��ii� M has a representation matrix B displaying M via B� where B is of

the form

��

Y

B =

Y

U

X B

0

W

X 01

Matrix B displaying deletion minor M

�iii� Every representation matrix B ofM displayingM via B is of the formgiven by ��

Addition� Expansion� Extension

Addition and expansion are the inverse operations of deletion and contrac�tion� Thus� an addition �resp� expansion� corresponds to the adjoining of acolumn �resp� row� to a given B� An extension is an addition or expansion�We denote additions by �� and expansions by �"�� For example� a ma�trix for M"U�W is obtained from one for M by adjoining rows indexedby U and columns indexed by W � Suppose the matroid so created is aminor of some other matroid� Then without further speci�cation� the en�tries in the added rows and columns are well de�ned� For example� supposeM � M�UnW where U and W observe the circuit�cocircuit condition� LetU � U and W �W � Then M"U�W is taken to be M��U�U�n�W �W ��We use a simpli�ed notation for extensions analogously to that for reduc�tions� Thus� we may write M"U � M�W � M"u when U � fug� etc�

Deletion� Contraction� Addition� Expansion in DualMatroid

By de�nition� deletions �resp� contractions� of M correspond to the re�moval of columns �resp� rows� from an appropriately chosen B� Recall that


Bt represents M�� Thus� deletions �resp� contractions� of M correspondto contractions �resp� deletions� in M�� Put di erently� for any disjointsubsets U and W of E� �M�UnW �� M��WnU � Furthermore� additions�resp� expansions� in M correspond to expansions �resp� additions� in M��

Matroid Isomorphism

Two matroids are isomorphic if they become equal upon a suitable rela�beling of the elements� Analogously to the case of graph minors� a givenmatroid M may be a minor of a matroid M � or may only be isomorphic toa minor of M � In the �rst situation� we say� as expected� that M is a minorof M � or that M has M as a minor� But in the second case� we frequentlyabbreviate �M has a minor isomorphic to M� to �M has an M minor��

k�Separation and k�Connectivity

We turn to separations and the connectivity of M � Let B be partitionedas in �� i�e��

�� X1B =

Y1

B1

D1X2

Y2

B2

D2


If for some k � ��

��jX� � Y�j� jX� � Y�j � k

GF��rank D� � GF��rank D� k � �

then �X� � Y��X� � Y�� is a Tutte k�separation� for short k�separation� ofB and M � The k�separation is exact if the rank condition of �� holdswith equality� B and M are Tutte k�separable� for short k�separable� if theyhave a k�separation� For k � �� B and M are Tutte k�connected� for shortk�connected� if they have no l�separation for � l � k� When M is ��connected� we also say that M is connected� Let M be the graphic matroidof a graph G� If M is connected �i�e�� connected�� then by Corollary�� G is ��connected� Thus� connectedness of M is not the same asconnectedness of G� We mentioned this problem previously in Section ��

The two connectivity corollaries �� and �� for graphs andgraphic matroids have easy extensions to the general binary case� as follows�


�� Lemma� Let M be a binary matroid with a representationmatrix B� Then M is connected if and only if this is so for B�

Proof� Via �� and �� it is easily checked that B is connected ifand only if it is ��connected� Thus� M is ��connected� and hence connected�if and only if B is connected�

�� Lemma� The following statements are equivalent for a binarymatroid M with set E and a representation matrix B of M �

�i� M is ��connected�

�ii� B is connected� has no parallel or unit vector rows and columns� andhas no partition as in �� with GF��rank D� � �� D� � �� andjX� � Y�j� jX� � Y�j � ��

�iii� Same as �ii�� but jX� � Y�j� jX� � Y�j � �

Proof� �i� � �ii� follows directly from the de�nition of ��connectivity�and �ii� � �iii� is trivial� Finally� �iii� � �ii� is established as follows�In B of �� assume GF��rank D� � � and D� � �� If the length ofB� is � or �� then B can be seen to have a zero column or row� or parallelor unit vector rows or columns� Any such case is already excluded by the�rst part of �ii�� Thus� it suces to require jX� � Y�j � � and by duality�jX� � Y�j � �

Census of Small Binary Matroids

It is instructive that we include a census of small ��connected binary ma�troids� say on n � elements� In that census� we refer to graphic andcographic matroids and their graphic and cographic representation matri�ces� We have discussed such matroids in detail in Section �� We alsorefer to regular and nonregular matroids� which are de�ned via a propertyof real matrices termed total unimodularity� Let us take up the latter con�cepts one by one� A real matrix A is totally unimodular if every squaresubmatrix D of A has detIRD � � or �� In particular� all entries of atotally unimodular matrix must be � or �� A binary matroid M is regularif in some binary representation matrix B of M the �s can be so signed sothat a f��g real totally unimodular matrix results�

We shall motivate these de�nitions in Chapter �� For the time being�we just state without proof two of the many important properties of reg�ular matroids� First� a binary matroid M is regular if and only if everyrepresentation matrix B can be signed to become a real totally unimodularmatrix� Second� every graphic or cographic matroid is regular� Becauseof the �rst property� it makes sense that we de�ne a binary matrix to beregular if its �s can be signed so that a real totally unimodular matrix re�sults� That property also implies that the dual matroid and every minor


of a regular matroid are regular� Finally� we know from Section �� thatgraphicness and cographicness are maintained under minor�taking�

Here is the promised census of the ��connected binary matroids withn � elements� It may be veri�ed by straightforward enumeration of cases�

n � �� M is the empty matroid�

n � �� M consists of a loop or a coloop� the representation matrix B isthe �� or �� trivial matrix� respectively�

n � �� M consists of two parallel elements� We may also consider thetwo elements to be in series� At any rate� B � � !�

n � �� M is a triangle� i�e�� a circuit with three elements� or a triad� i�e��a cocircuit with three elements� In the �rst case� B � �� !� Inthe second case� B � � � !�

n � �� There is no ��connected binary matroid with four or �ve elements�

n � �� M is the graphic matroid M�W�� where W� is the wheel withthree spokes� Up to pivots� B is the matrix

��01 1 01 0 1

11

Matrix representing graphic matroid M�W��

Note that up to this point� all matroids have been graphic and cographic�

n � �� M is the Fano matroid F� given by

��0

1

1 11B7 = 1 10

0 1 1

1

Matrix representing Fano matroid F�

or its dual F �� given by �B��t� We claim that F�� and hence

F �� are not regular� Indeed� if B� is signed so that all � � �

submatrices have f��g real determinants� then up to columnand row scaling� B� is already that matrix� But the �rst threecolumns of B� de�ne a � � � matrix with real determinant equalto �� The name is based on the fact that the matroid is theFano plane� which is the projective geometry PG�� The sevenelements of the matroid are the points of the geometry�


n � �� If M is regular� then M is the graphic matroid of M�W�� whereW� is the wheel with four spokes� Up to pivots� B is the matrix

��0

1

001 1

1

1 1 000 0 1

10

Matrix representing graphic matroid M�W��

If M is nonregular� then M is one of two matroids given by

��

0101 1

0

1 1B8.1 = 00

11 1 1

10

Matrix B�� representing nonregular matroidwith eight elements� case �

and

��

11

1 0 1 1B8.2 =

0 1 1 1

01101 1

Matrix B�� representing nonregular matroidwith eight elements� case �

The matroid represented by B�� is the ane geometry AG��which has eight points corresponding to the eight binary vectorswith three entries each� A subset of the points is �anely� GF��mindependent if the corresponding subset of vectors has even car�dinality and is linearly GF��dependent� and if it is minimal withrespect to these two conditions� For that matroid� contraction�resp� deletion� of any element produces the nonregular matroidF� �resp� F �

� � as a minor� Thus� that matroid is highly nonregular�In contrast� deletion of any row or column from B�� produces amatrix that represents a regular matroid�

So far� we have stressed what one might call the matrix viewpoint ofbinary matroids� As we shall see in Section �� that notion can be extendedto general matroids using matrices termed abstract� The main advantage


of that notion is the fact that a binary matrix �or in general� an abstractmatrix� displays numerous bases� circuits� cobases� and cocircuits of thematroid simultaneously� For this reason� we view matrices �binary� or overother �elds� or abstract� to be an important tool of matroid theory�

There are� however� other and equally important viewpoints� One ofthem considers the rank function r�� of M as a main tool� Recall thatthis function was de�ned via B of �� as follows� For any subsetZ � E� let X� � Z � X and Y� � Z � Y � By �� X� � X � X�

and Y� index the submatrix B� of B� Then declare the rank of Z to ber�Z� � jX�j�GF��rank B�� The utility of the rank function largely restsupon a property called submodularity� Section �� includes a de�nition ofsubmodularity for functions de�ned on matrices� There it is shown thatthe matrix rank function is submodular� Shortly� we de�ne submodularityfor functions that map the subsets of a set E to the nonnegative integers�We then prove the matroid rank function to be submodular�

The matrix viewpoint� the rank function viewpoint� as well as sev�eral others not mentioned so far �e�g�� the geometric viewpoint� the latticeviewpoint # see the books cited in Chapter �� have advantages and dis�advantages� Intuitively speaking� matrices very conveniently display bases�circuits� cobases� and cocircuits that di er just by a few elements� On theother hand� the relationships between radically di erent bases� circuits�etc�� are not well exhibited� The rank function approach� as well as sev�eral others� treats all such cases evenly� Indeed� for the solution of severalproblems involving radically di erent bases� circuits� etc�� the rank functionseems particularly suitable�

The results described in this book rely largely on the matrix viewpoint�But the reader should not be misled by this fact� It just turns out that theresults of this book are nicely treatable by matrices� But there are a numberof problems of matroid theory where other approaches� in particular onesrelying on the rank function� are superior to the matrix technique employedhere� Below� we describe two simple instances that exhibit the power ofthe rank function approach� But �rst we express the de�ning k�separationconditions of �� in terms of r��

k�Separation Condition for Rank Function

�� Lemma� Let M be a binary matroid on a set E and with rankfunction r�� Suppose E� and E� partition E� Then �a� and �b� belowhold�

�a� �E�� E�� is a k�separation of M if and only if

��jE�j� jE�j � k

r�E�� r�E�� r�E� � k � �


�b� �E�� E�� is an exact k�separation of M if and only if


r�E�� r�E�� r�E� � k � �

Proof� �a� To establish the �only if� part� take any representation matrixB of M � say indexed by X and Y as before� De�ne for i � �� Xi �Ei �X and Yi � Ei � Y � Let these sets partition B as in �� SincejE�j� jE�j � k� we have jX��Y�j� jX��Y�j � k� By the de�nition of r�� wehave for i � �� r�Ei� � jXij � GF��rank Di� and r�E� � jX�j � jX�j�Since r�E��r�E�� r�E��k�� we have jX�j�GF��rank D� � jX�j�GF��rank D� jX�j�jX�j�k�� or GF��rank D��GF��rank D� k� �� Thus� by �� X� � Y��X� � Y�� E�� E�� is a k�separation ofM � For the proof of the �if� part� one reverses the above arguments�

�b� This follows from the proof of �a� by suitable replacement of someinequalities by equations�

We now de�ne the submodularity property and prove that the rankfunction r�� has that property�

Submodularity of Rank Function

A function f�� from the set of subsets of a �nite set E to the nonnegativeintegers is submodular if for any subsets S� T � E�

�� f�S� � f�T � � f�S � T � � f�S � T �

�� Lemma� r�� is a submodular function�

Proof� Take any binary representation matrix of M � Let A�� A�� A�� andA� be the column submatrices of A � I j B! corresponding to the columnindex sets S� T � S�T � and S�T � respectively� Evidently� A� is a submatrixof A�� A�� and A�� and both A� and A� are submatrices of A�� Let B� be abasis of A�� For i � �� extend B� to a basis of Ai� say by adjoining Bi�Evidently� the number of columns of B� j B�!� B� j B�!� B� j B�!� andB� is the matroid rank of S� T � S � T � and S � T � respectively� Thus� thesubmodularity inequality �� holds if and only if B� and B� togetherhave at least as many columns as B�� The latter condition holds since bythe construction� B� j B�! is a basis of A�� while B� j B� j B�! spans allcolumns of A��

We use submodularity of r�� in the proof of the next result� De�neM c�z to be the matroid obtained from M�z by deletion of the elementsof each parallel class except for one representative of each class� Similarly�derive M d�z from Mnz by contracting the elements of each series classexcept for one representative of each class�


�� Lemma� Let M be a ��connected binary matroid on a set E�Take z to be any element of E� Then M c�z or M d�z is ��connected�

Proof� The arguments below rely repeatedly on three observations thatfollow directly from �� and �� First� if a set is not a circuit �resp�cocircuit� of M � then it cannot be a circuit �resp� cocircuit� in any minorderived from M by deletions only �resp� contractions only�� In particular�the ��connected matroid M has no parallel or series elements� and thus� forany element z� the minor Mnz �resp� M�z� has no parallel �resp� series�elements� Second� if an element z is not a coloop �resp� loop� of M � thenMnz �resp� M�z� has the same rank as M � Third� the rank of a set Z � Edrops at most by � when an element z �� Z is contracted� and stays thesame when z �� Z is deleted�

Assume the lemma fails� i�e�� for some ��connected M and z � E� bothM c�z and M d�z are ��separable� Let �P �� Q�� be a ��separation of M c�z�We know that no two elements of M c�z are in parallel� Thus� jP �j � �or P � contains two series elements� The obvious assignment of the parallelelements by which M�z and M c�z di er converts �P �� Q�� to a ��separation�P�Q� of M�z where P � P � and Q � Q�� Since M is ��connected� M�zhas no series elements� If jP �j � � and P � � P � then P must be a set oftwo series elements in M�z� which is impossible� Thus� jP �j � � impliesjP j � �� We conclude jP j � � in general� and jQj � � by symmetry� Usingduality� M d�z must have a ��separation �R�S� with jRj� jSj � ��

Assume jP �Rj �� Then jP � Sj� jQ �Rj � �� The same conclusionholds if jQ � Sj �� Thus� we may assume jP � Rj� jQ � Sj � � orjP � Sj� jQ � Rj � �� By symmetry� we may suppose that the formersituation holds�

Denote by r�� rM�z�� and rMnz�� the rank functions of M � M�z�and Mnz� respectively� Since �P�Q� and �R�S� are ��separations of M�zand Mnz� respectively� we have

��rM�z�P � � rM�z�Q� rM�z�E � fzg� ��rMnz�R� � rMnz�S� rMnz�E � fzg� ��

Now rM�z�P � � r�P � fzg�� rM�z�Q� � r�Q � fzg�� and rM�z�E �fzg� � r�E� � �� Also� rMnz�R� � r�R�� rMnz�S� � r�S�� and rMnz�E �fzg� � r�E�� We use these relationships to deduce from �� the in�equalities

��r�P � fzg� � r�Q � fzg� r�E� � �r�R� � r�S� r�E� � �

By submodularity� we also have

��r�P � R� � r�P �R � fzg� r�P � fzg� � r�R�r�Q � S� � r�Q � S � fzg� r�Q � fzg� � r�S�


Add the two inequalities of �� Similarly� add the two inequalities of�� The resulting two inequalities imply

�� r�P �R� � r�Q � S � fzg�!

� r�Q � S� � r�P � R � fzg�! �r�E� � �

But then at least one of the pairs �P �R�Q�S �fzg� and �Q�S�P �R � fzg� must be a ��separation of M � which is not possible�

Along the same lines� but with much simpler arguments� one can provethe following lemma� We leave the proof to the reader�

�� Lemma� Let M be a connected binary matroid on a set E�Take z to be any element of M � Then M�z or Mnz is connected�

We conclude this section by proving that submodularity of the r��function and submodularity of the GF��rank function are equivalent� Toshow this� we assume S and T to be subsets of E for a binary matroidM with a binary representation matrix B� Let the customary index setsX and Y of B be partitioned into X�� X�� X�� X�� and Y�� Y�� Y�� Y��respectively� so that

��

X� � X � �S � T �X� � X � �S � T �X� � X � �S � T �X� � X � �T � S�Y� � Y � �S � T �Y� � Y � �S � T �Y� � Y � �S � T �Y� � Y � �T � S�

Let the partitions of X and Y induce the following partition of B�

��B11 B12 B13

B21 B23B22

B31 B32 B33

Y1 Y0Y3Y2

Y

X2

X1

X0

X3

X

01


� � Abstract Matrices Produce All Matroids ��

De�ne submatrices D�� D�� D�� D� of B as in �� i�e��

��

B11 B12

B21 B22 ;D1 =B23B22

B32 B33D2 =

;B21 B23B22D3 =B12

B22

B32

D4 =

Submatrices D�� D�� D�� D� of B

By the discussion following �� the equations below relate the GF��rank of the Di to the matroid rank of S� T � S � T � and S � T �

��

r�S� � jX�j� jX�j� GF��rank D�

r�T � � jX�j� jX�j� GF��rank D�

r�S � T � � jX�j� jX�j� jX�j� GF��rank D�

r�S � T � � jX�j� GF��rank D�

Then clearly the submodularity inequality for r�� which is r�S� � r�T � �r�S � T � � r�S � T �� holds if and only if this is so for the submodularityinequality for the GF��rank function� which is GF��rank D� � GF��rank D� � GF��rank D� � GF��rank D��

In the next section� we move from binary matrices to abstract ones�Correspondingly� we obtain all matroids instead of just the binary ones�

�� Abstract Matrices Produce All

Matroids

Binary matrices produce the binary matroids� In a natural extension� wecould consider matrices over �elds other than GF�� and the matroids pro�duced by them� We skip that step� Instead� we move in this section directlyto abstract concepts of independence� bases� circuits� and rank� and in theprocess create the entire class of matroids� We also introduce abstract ma�trices as a generalization of matrices over �elds� and we describe some oftheir features� Routine arguments prove that the abstract matrices gen�erate all matroids� and that the matroids produce all abstract matrices�Indeed� one may view abstract matrices as one way of encoding matroids�Abstract matrices exhibit many features of linear algebra� They also dis�play several properties of matroids rather conveniently� and we have found


them to be very useful� In particular� they often help one to detect andprove new structural results that are hidden from view when� instead� onethinks of a matroid as a construction via certain sets� functions� geometries�or operators�

Abstract matrices behave to quite an extent like binary matrices� Thisfact explains why so many seemingly special results for binary matroidshold for the general case� To support that claim� toward the end of thissection we list results of Section �� for binary matroids that� sometimesafter an elementary modi�cation� hold for general matroids� We are nowready for the detailed discussion�

De�nition of General Matroid

Let E be a set of vectors over some �eld F � A central result of linearalgebra says that for any given subset of vectors of E� the bases of thatsubset have the same cardinality� We abstract from this fact the axiomsfor general matroids as follows� A matroid M on a ground set E is a pair�E�I�� where I is a certain subset of the power set of E� The set I is theset of independent subsets of M � A subset of E that is not in I is calleddependent� The set I must observe the following axioms�

��

�i� The null set is in I��ii� Every subset of any set in I is also in I��iii� For any subset E � E� the maximal subsets of E

that are in I have the same cardinality�

The cardinality of any maximal independent subset of any E � E iscalled the rank of E� A base of M is a maximal independent subset of E�A circuit is a minimal dependent subset of E� A cobase is the set E �Xfor some base X� Let I� be the set of cobases and their subsets� Thepair M� � �E�I�� is a matroid� as is easily checked� It is called the dualmatroid of M � A cocircuit of M is a circuit of M��

One can axiomatize matroids in terms of bases� circuits� and othersubsets of E� or by certain functions� geometries� and operators� It isusually a simple� though at times tedious� exercise to prove equivalence ofthese systems� Here we just include the axioms that rely on bases� circuits�and the rank function�

Axioms Using Bases� Circuits� Rank Function

For bases� the axioms are as follows� Let B be a set of subsets of E� Suppose


B observes the following axioms�

��

�i� B is nonempty��ii� For any sets B�� B� � B and any x � �B� �B��

there is a y � �B��B�� such that �B��fxg��fygis in B�

Then B is the set of bases of a matroid on E�Via circuits� we may de�ne matroids as follows� Let C be the empty

set� or be a set of nonempty subsets of E observing the following axioms�

��

�i� For any C�� C� � C� C� is not a proper subset ofC��

�ii� For any two C�� C� � C and any z � �C� � C��there is a set C� � C where C� � �C��C��fzg�

Then C is the set of circuits of a matroid on E�With the rank function� we specify a matroid as follows� Let r�� be a

function from the power set of E to the nonnegative integers� Assume r��satis�es the following axioms for any subsets S and T of E�

��i� r�S� jSj��ii� S � T implies r�S� r�T ��iii� r�S� � r�T � � r�S � T � � r�S � T ��

Then r�� is the rank function of a matroid on E�We omit the proofs of equivalence of the systems� It is instructive�

though� to express each one of I� B� C� and r�� in terms of the other ones�We do this next�

Suppose I is given� Then B is the set of Z � I with maximum car�dinality� C is the set of the minimal C � E that are not in I� For anyE � E� r�E� is the cardinality of a maximal set Z � E that is in I�

Suppose B is given� Then I is the set of all X � B plus their subsets�C is the set of the minimal C � E that are not contained in any X � B�For any E � E� r�E� is the cardinality of any maximal set X � E whereX � B�

Suppose r�� is given� Then I is the set of E � E where r�E� � jEj�B is the set of Z � E for which jZj � r�E�� C is the set of the minimalC � E for which r�C� � jCj � ��

Abstract Matrix

We take a detour to introduce abstract matrices� We want to acquire a goodunderstanding of such matrices� since they not only represent matroids� but


also display a lot of structural information about matroids that other waysdo not�

An abstract matrix B is a f�� g matrix with row and column indicesplus a function called abstract determinant and denoted by det� The func�tion det associates with each square submatrix D of the f�� g matrix thevalue � or �� i�e�� detD is � or �� Note that numerically identical squaresubmatrices with di ering row or column index sets may have di erentdeterminants� The reader should not be misled by the symbols � and ��Indeed� for the moment� we do not view abstract matrices as part of somealgebraic structure� It turns out� though� that � and � allow a rather ap�pealing use of linear algebra terms� For example� we call D nonsingular ifdetD � �� and singular otherwise�

The function det must obey several conditions� First� if D is the �� matrix � ! �resp� � !�� then detD � � �resp� detD � ��

Second� for any nonempty submatrix B� of B� the maximal nonsingu�lar submatrices must have the same size� This condition may be rephrasedas follows� Start with some nonsingular submatrix of B�� Iteratively add arow and a column such that each time another nonsingular submatrix re�sults� Stop when no further enlargement is possible� The above maximalitycondition demands that the order of the �nal nonsingular submatrix is thesame regardless of the choice of the initial nonsingular submatrix and ofthe rows and columns added to it� The order of any such �nal nonsingularsubmatrix is called the rank of B�� For the case where B� is trivial orempty� we declare rank B� to be �� Upon deletion of a column or row� wedemand that the rank drop at most by the rank of that row or column�

Third� the rank function of B must behave much like the rank functionof matrices over �elds� In particular� for any partition of any submatrix ofB of the form

��B11 B12 B13

B21 B23B22

B31 B32 B33

Partitioned submatrix of B

the submatrices

��

B11 B12

B21 B22 ;D1 =B23B22

B32 B33D2 =

;B21 B23B22D3 =B12

B22

B32

D4 =

Submatrices D�� D�� D�� D�

� � Abstract Matrices Produce All Matroids �

must observe

�� rank D� � rank D� � rank D� � rank D��

We call this property the submodularity of the rank function�We summarize the above requirements as follows�

Axioms for Abstract Matrix

��

�i� If D � Bxy!� then detD � Bxy��ii� For all submatrices B� of B� The maximal non�

singular submatrices of B� have the same size�called the rank of B�� When a row or column isdeleted from B�� the rank drops at most by therank of that row or column�

�iii� The rank function is submodular�

The transpose of an abstract matrix B is Bt with determinants de�ned asfollows� For any square submatrix B� of B� detB� is the determinant valuefor the submatrix �B��t of B� By symmetry of the conditions of ��Bt with its determinants is an abstract matrix� as expected�

We may create abstract matrices in several ways� In the simplest case�we start with a matrix A over some �eld F � Then we declare B to bethe support matrix of A� Thus� B is a f�� g matrix� We turn B into anabstract matrix as follows� We declare any square submatrix D of B tobe nonsingular if the corresponding submatrix of A is F�nonsingular� andto be singular otherwise� Well�known linear algebra results plus Lemma�� imply that the axioms of �� are satis�ed�

Representation of Abstract Matrix

Suppose an abstract matrix B can be derived by the above constructionfrom a matrix A over some �eld F � We say that B is represented by A overF � As an example� let A be the matrix

��

11d

10

f g h

1

0

eab

1

c 1 111 1

10 1

0

Matrix A producing an abstract matrix B


over GF�� Then B is numerically identical to A� and the determinantsfor the submatrices of B are given by the GF��determinants of A� Forexample� the submatrix D of B given by

��g

a11

h

1b1

D =

Submatrix D of abstract matrix B

has detD � �� since the related submatrix of A has GF��determinant ��The example may be modi�ed to produce an abstract matrix that is

not representable over any �eld� Let B and its determinants be as justde�ned� Then change the determinant of D of �� from � to �� Onemay check by enumeration that the new B observes the axioms of ��For a proof that the new B is not representable� suppose A over some �eldF represents B� Then one readily shows that the rows and columns of Acan be scaled so that the matrix of �� results� For D of �� assubmatrix of the scaled A� we have detF D � �� But for D as submatrixof B� we have detD � �� a contradiction�

For certain abstract matrices� the axioms of �� completely deter�mine the rank� For example� by axiom �ii� of �� zero matrices haverank �� A more interesting instance is given in the next result�

�� Lemma� Let B be an m�m triangular abstract matrix� ThendetB � � if and only if the diagonal of B contains only �s�

Proof� The case m � � is immediate by �i� of �� Thus� consider thecase m � ��

Assume that the diagonal of B has only �s� For an inductive proof�we partition B as

�� 01

y Y

B =BX

x

01

Triangular B with �s on the diagonal

where B is triangular and has only �s on the diagonal� Apply the sub�modularity condition �� as follows� Declare B�� resp� B�� B��B�� to be the submatrix of B indexed by x and Y �resp� x and y� Xand Y � X and y�� all other Bij are trivial or empty� Then by ��rank D� � rank B�� rank D� � rank B� rank D� � �� and by induc�tion� rank D� � m� �� By submodularity� rank B � � � �m� �� m� sodetB � ��


Assume now that the diagonal of B contains a �� Thus� we maypartition B as

��B1X1

x

X2

yY1 Y2

00

1

B =

B2

0

Triangular B with a zero on the diagonal

where both B� and B� are square� By axiom �ii� of �� the rows of Bindexed by X� � fxg have rank of at most jY�j� The remaining rows of Bhave rank of at most jX�j� Thus� again by axiom �ii� of �� the rankof B is at most jY�j� jX�j � jX�j� jX�j m� �� Hence� detB � ��

�� Corollary� The abstract matrices

�� ;00

0 ;00

;001

0;

011

0 001

1

Small singular abstract matrices

are singular� The matrices

��10

1 ; ;10

011

1

Small nonsingular abstract matrices

are nonsingular� The matrix

��11 1

1

Abstract matrix with determinant � or �

may be singular or nonsingular�

Proof� Lemma �� handles the cases of �� and �� A ma�trix over GF�� with support given by �� may be GF��singular orGF��nonsingular� This fact validates the claim about ��

Note that the GF��determinants of the matrices of Corollary ��agree with the abstract determinants� except possibly for the matrix of��


Abstract Matrices Encode Matroids

We claim that abstract matrices are nothing but one way of encoding ma�troids� For a proof� we �rst assume that an abstract matrix B with rowindex set X and column index set Y is at hand� We want to show thatB encodes some matroid� De�ne a rank function r�� on E � X � Y asfollows� Given Z � E� the sets X� � Z � X and Y� � Z � Y induce apartition of B of the form

�� X1B =

Y1

B1

D1X2

Y2

B2

D2


Then de�ne

�� r�Z� � jX�j� rank B��

�� Lemma� r�� is the rank function of a matroid�

Proof� We verify the rank axioms �� Let S� T � E� Clearly r�S� jSj� A simple case analysis and axiom �ii� of �� con�rm that S � Timplies r�S� r�T �� Finally� r�S� � r�T � � r�S � T � � r�S � T � is arguedas at the end of Section �� except that we use the abstract rank functioninstead of the GF��rank function�

Matroids Produce Abstract Matrices

Given a matroid M on E with rank function r�� we want to show thatM may be encoded by some abstract matrix� De�ne X to be a base of Mand Y � E � X� For any y � Y � the set X � fyg must contain at leastone circuit� Indeed� there must be precisely one circuit in X �fyg� say Cy�since otherwise there is a base X � contained in X � fyg with cardinalityjX �j jXj � �� The circuit Cy is called the fundamental circuit thaty creates with X� Declare the characteristic vectors of the fundamentalcircuits Cy� y � Y � to be the columns of a matrix B� Thus� we have

�� B

Y

X


where for x � X and y � Y � we have Bxy � � if and only if x � Cy�We endow B with determinants as follows� Let B� be a square submatrixof B� say indexed by X� � X and Y� � Y as in �� Then declaredetB� � � if X� � Y� is a base of M � and to be � otherwise�


�� Lemma� B and its determinants constitute an abstract matrix�

Proof� First� det Bxy! � Bxy for all x � X and y � Y � due to the equiv�alence of the following statements separated by semicolons� Bxy � �� x isin the fundamental cycle Cy� �X � fyg� � fxg does not contain a circuit��X � fyg�� fxg is a base of M � det Bxy! � ��

Next� let B� be an arbitrary submatrix of B indexed by some X� � Xand Y� � Y � De�ne X� � X � X� and Y� � Y � Y�� We prove thatthe maximal square nonsingular submatrices D of B� have same order�Any such matrix D corresponds to a base of M that contains X�� thatis contained in X� � X� � Y�� and that� subject to these two conditions�intersects Y� as much as possible� By the independence axioms �� thecardinality of any such maximal intersection is the same regardless of whichindependent subset of Y� one starts with� We conclude that the maximalnonsingular submatrices of B� have same order� That order is the rank ofB�� denoted by rank B��

The rank function r�� of M is thus related to rank B� by r�X��Y�� jX�j � rank B�� We apply the independence axioms of �� to M andverify that removal of a column or row from B� reduces the rank at mostby the rank of the removed column or row� Finally� the arguments at theend of Section �� linking submodularity of r�� with that of the GF��rank function are easily adapted to prove that the rank function for B issubmodular�

Thus� the axioms of �� are satis�ed� and B is an abstract matrixas claimed�

The constructions of the proof of Lemmas �� and �� areinverses of each other in the following sense� Let M be a matroid with abase X� De�ne B to be the abstract matrix constructed from M in theproof of Lemma �� The matroid deduced from B in the proof ofLemma �� is M again� For this reason� we say that B represents M �

At this point� we have established an axiomatic link between abstractmatrices and matroids� We could explore the ways in which matroid con�cepts manifest themselves in abstract matrices� But that discussion wouldlargely duplicate the material of Section �� about binary matroids� So wejust sketch the de�nitions and relationships� and omit all proofs� Through�out� B is an abstract matrix with row index set X and column index setY � The related matroid M on E � X � Y has X as a base�

Column y � Y of B is the characteristic vector of Cy �fyg� where Cy

is the fundamental circuit that y forms with X� Row x � X of B is thecharacteristic vector of C�

x � fxg� where C�x is the fundamental cocircuit

that x forms with Y � Two parallel �resp� coparallel or series� elements ofM manifest themselves in two nonzero columns �resp� rows� of B of rank�� or in a column �resp� row� unit vector� A loop �resp� coloop� of M isindicated by a column �resp� row� of �s� The transpose of B represents the


dual matroid M� of M � Let U and W be disjoint subsets of E� Assume Ucontains no circuit and W contains no cocircuit� Then there is a B whereX � U and Y � W � Delete from B the rows of U and the columns ofW � getting an abstract matrix B� Correspondingly� contraction of U anddeletion of W reduce M to the minor M � M�UnW � which is representedby B� Furthermore� expansion by U and addition of W extend M to M �

Partition B as in �� If for some k � �� jX� � Y�j� jX� � Y�j � kand rank D� � rank D� k � �� then �X� � Y��X� � Y�� is a k�separationof B and M � Let k � �� If B and M have no l�separation� � l � k� thenthey are k�connected� M is connected if it is ��connected� Consistent withSection �� B is connected if the graph BG�B� is connected�

Abstract Pivot

We introduce pivots in abstract matrices� For comparison purposes� werewrite the GF��pivot of �� as follows� Given is the pivot elementBxy � ��

��

�� We replace each Buw� u � �X � fxg�� w � �Y �fyg�� by det�Duw� where Duw is the submatrixof B given by

Buy

Bxy

y wx

u

Bxw

BuwDu =w

�� We exchange the indices x and y�

Let B be the abstract matrix for M as before� A pivot on Bxy is toproduce the abstract matrix B� corresponding to the base �X �fxg�� fygof M � We claim that the following procedure generates the desired B��

��

�� We replace each Buw� u � �X � fxg�� w � �Y �fyg�� by detDuw� where Duw is the submatrix ofB as given above�

�� We exchange the indices x and y� Let B� be theresulting matrix�

�� We endow the square submatrices D� of B� withdeterminants� Let U � be the row index set of onesuch D�� and W � be the column index set� ThendetD� � detD for the submatrix D of B indexedby U and W as speci�ed below�

If y � U �� x �W �� U � U ��fyg� W � W ��fxg�If y � U �� x ��W �� U � �U ��fyg��fxg� W � � W �If y �� U �� x �W �� U � U �� W � �W ��fxg��fyg�If y �� U �� x ��W �� U � U � � fxg� W � W � � fyg�


Validity of the procedure is argued as follows� By step ��X � � �X�fxg��fyg and Y � � �Y �fyg��fxg index the rows and columns�respectively� of B�� By step �� for u � �X�fxg� and w � �Y �fyg��B�uw � detBuw� Now detDuw � � if and only if �X � fx� ug� � fy�wg is

a base of M � The latter set is �X � � fug� � fwg� so B�uw is correctly

computed� The entries B�yw �� Bxw�� w � �Y � fyg�� are correct since

�X � fxg� � fwg � �X � � fyg� � fwg� Similarly� the entries B�ux �� Buy��

u � �X � fxg�� are correct since �X � fug� � fyg � �X � � fug� � fxg�Finally� B�

yx �� Byx � �� since �X � � fyg� � fxg � X� the assumed baseof M � Analogous arguments involving the sets U and W � instead of theelements u and w� validate �� We conclude that B� is the abstractmatrix corresponding to the base X � � �X � fxg� � fyg of M as claimed�

Except for the computationally tedious step �� the pivot isalmost identical to the GF��pivot of �� Indeed� for � � � matri�ces� Corollary �� establishes an almost complete agreement betweenGF��determinants and abstract determinants� Thus� the step ��of an abstract pivot looks very much like the step �� of a GF��pivot� Informally� one is tempted to say that general matroids behavelocally to quite an extent like binary matroids�

There remains� of course� the cumbersome step �� But we canalways imagine that this step is implicitly carried out� without our actuallyhaving to write down the list of determinant values� When we take thatattitude� the pivot operation becomes simple and useful�

Some Matroid Results

The similarity of local behavior of binary and general matroids is easilydemonstrated� We do this here by listing a number of results for generalmatroids that by trivial modi�cations may be obtained from the resultsfor binary matroids proved in Section �� In each case� the proof as givenin Section �� suces� or in that proof one simply substitutes an abstractmatrix whenever a binary one is employed� With each result� we cite inparentheses the related result of Section ��

�� Lemma �see Lemma �� Let C �resp� C�� be a circuit�resp� cocircuit� of a matroid M � Then jC � C�j ��

�� Lemma �see Lemma �� Let M be a matroid with aminor M � and B be an abstract representation matrix of M � Then M hasan abstract representation matrix B that displaysM via B and thus makesthe minor M visible�

�� Lemma �see Lemma �� The following statements areequivalent for any matroidM and any minorM ofM � Let B be an abstractrepresentation matrix of M �


�i� M is a contraction minor of M ��ii� M has an abstract representation matrix B displayingM via B� where

B is of the form

��X

U

B 0

WY

B =0 1

X

Y

Matrix B displaying contraction minor M

�iii� Every abstract representation matrix B of M displaying M via B isof the form given by ��

�� Lemma �see Lemma �� The following statements areequivalent for any matroidM and any minorM ofM � Let B be an abstractrepresentation matrix of M �

�i� M is a deletion minor of M ��ii� M has an abstract representation matrix B displayingM via B� where

B is of the form

��

Y

B =

Y

U

X B

0

W

X 01

Matrix B displaying deletion minor M

�iii� Every abstract representation matrix B of M displaying M via B isof the form given by ��

�� Lemma �see Lemma �� Let M be a matroid with anabstract representation matrix B� Then M is connected if and only if thisis so for B�

�� Lemma �see Lemma �� The following statements areequivalent for a matroid M with set E and an abstract representationmatrix B�

�i� M is ��connected��ii� B is connected� has no parallel or unit vector rows or columns� and

has no partition as in �� with rank D� � �� D� � �� and jX� �Y�j� jX� � Y�j � ��


�� Lemma �see Lemma �� Let M be a matroid on a setE and with rank function r�� Suppose E� and E� partition E� Then �a�and �b� below hold�

�a� �E�� E�� is a k�separation of M if and only if


r�E�� r�E�� r�E� � k � �

�b� �E�� E�� is an exact k�separation of M if and only if


r�E�� r�E�� r�E� � k � �

�� Lemma �see Lemma �� LetM be a ��connected matroidon a set E� Take z to be any element of E� Then M c�z or M d�z is ��connected�

�� Lemma �see Lemma �� Let M be a connected matroidon a set E� Take z to be any element of E� ThenM�z orMnz is connected�

Aspects of Representability

In the remainder of this section and in the next one� we examine severalaspects of the representability of abstract matrices� Recall that an abstractmatrix B is represented by a matrix A of the same size and over some �eldF if the following holds� For every square submatrix of B� the determinantof that submatrix is � if and only if the F�determinant of the correspond�ing submatrix of A is nonzero� We want to establish a direct connectionbetween the matroid M � de�ned by B� and any matrix A representing Bover some �eld F � To this end� we partition such A in agreement with thepartition of B of �� Thus�

��

Y1

C1

Y2

X1A =

A1

X2 A2

C2

Partitioned version of matrix A over �eld F

Evidently� the submatrix A� of A corresponds to B� of B� Assume thesesubmatrices to be square� We know that detF A� �� if and only if detB� �� We also know that detB� � � if and only if X��Y� is a base of M � Thus�


detA� �� if and only if X� � Y� is a base of M � When this relationshipholds� we also say that A over F represents M �

We have seen that some abstract matrices are not representable overany �eld� By de�nition� the same holds for some matroids� In particular�the abstract matrix deduced from �� by modifying the determinant ofD of �� produces a matroid that is not representable over any �eld�

Deciding whether or not a matroid is representable over some �eldis a nontrivial problem� Usually� one assumes that the matroid is givenby a black box or oracle that in unit time tells whether or not a subsetof E is independent in the matroid� No additional information about thematroid is available� Under these assumptions� even representability overGF�� cannot be tested in polynomial time� Indeed� the same conclusioncan be drawn for a great many representability questions� There is oneextraordinary exception� One can test in polynomial time whether or nota matroid is representable over every �eld� That representability problemis intimately linked to the problem of deciding whether a matrix is totallyunimodular� Details are covered in Chapter ��

Suppose an abstract matrix B and the related matroid M are repre�sentable over some �eld F � Let A be a matrix over F proving this fact�Then every matrix B� derivable from B by one or more pivots is also repre�sentable over F � For a proof� one carries out a pivot in B� say on Bxy � ��and a second F�pivot in A on Axy� Let B� and A� result� It is easily seenthat A� establishes B� to be representable over F � Combine this result withthe trivial observation that every proper submatrix of B and the transposeof B are representable over F � We conclude that every minor of M andthe dual M� of M are representable over F �

It is easy to check that all matroids with at most three elements arerepresentable over every �eld� Couple this observation with the fact thatthe taking of minors maintains representability� Evidently� a matroid Mnot representable over a given �eld F must have a minor that also is notrepresentable over F � but all of whose proper minors are representable overF � We call such a minor a minimal violator of representability over F � Notmuch is known about the minimal violators for the various �elds F � Com�plete lists of the minimal violators exist only for the �elds GF�� and GF��There is also a complete description for the case when representability overevery �eld is demanded� Beyond these cases� only incomplete results areknown�

For the case of GF�� there is just one minimal violator� It is amatroid on four elements called U�

� � In the next section� we de�ne thatmatroid and prove the claim we just made� In Chapter �� we determinethe minimal violators for GF�� and for the case of representability overevery �eld� In both cases� there are just two minimal violators plus theirduals� As we shall see� abstract matrices and abstract pivots are useful forthe proof of the results for GF�� and GF��

� � Characterization of Binary Matroids �

�� Characterization of Binary Matroids

In this section� we characterize the binary matroids in several ways� Inparticular� we prove that a certain matroid on four elements called U�

� isthe only minimal violator of representability over GF�� Accordingly� amatroid is binary if and only if it has no U�

� minors� To begin� we de�ne�for � m n� Um

n to be the matroid on n elements where every subset ofcardinality m is a base� It is easily checked that Um

n is indeed a matroid�It is the uniform matroid of rank m on n elements�

Let M be a matroid on a set E� For an arbitrary base X of M �compute the abstract matrix B� Suppose M is representable over F � Byde�nition� there is a matrix A over F with B as support matrix� such thatfor all corresponding submatrices D of B and D� of A� detD � � if andonly if detF D� �� If F is GF�� evidently A is numerically identical toB�

Consider Umn � for � m n � �� Since every set of cardinality

m is a base of Umn � every abstract matrix B for that matroid is of size

m� n� contains only �s� and has only nonsingular square submatrices� Inthe related binary matrix A� all square submatrices of order at least �are GF��singular� Thus� Um

n is nonbinary� The smallest nonbinary casehas � � m � n � �� i�e�� the matroid is U�

� � Representability over any�eld F is maintained under minor�taking� Thus� a binary matroid cannotpossibly have U�

� minors� We now prove that absence of U�� minors implies

representability over GF��Let M be a nonbinary matroid all of whose proper minors are binary�

Select any abstract representation B for M � and let A be the associatedbinary matrix� Then there are minimal submatrices D of B and D� of Ain the same position such that exactly one of detD and det�D� is �� Sinceevery proper minor of M is binary� we must have D � B and D� � A�Since the entries of B agree with those of A� the order of B must be atleast ��

If B is a zero matrix� then both detB and det�A are zero� a contradic�tion� Thus� B contains a �� say Bxy � �� If the order of B is greater than�� we perform a pivot in B and the corresponding GF��pivot in A� Inboth cases� we delete the pivot row and pivot column� Let B� and A� result�By the minimality assumptions on B� and A� and the rules �� and�� the two matrices must agree numerically� By �� and theanalogous rule of linear algebra for A� exactly one of detB� and det�A�

is zero� Thus� we have proved that a proper minor of M is not binary� acontradiction� Hence� B is a � � � matrix� By Corollary �� there isonly one choice for B and A� That is� B is the abstract matrix of ��and A is also that matrix when viewed as binary� We include the two ma�trices below in �� Since exactly one of the determinants of B and Ais nonzero and det�A � �� we must have detB � ��


��11

B =11

;11

A =11

Abstract matrix B of minimal nonbinary matroidand related binary matrix A

Evidently� M has four elements� and every ��element subset is a base� Thus�M is isomorphic to U�

� � We record this conclusion and state and prove arelated corollary for future reference�

�� Theorem� A matroid M is binary if and only if M does nothave U�

� minors�

�� Corollary� Let an abstract matrix B represent a nonbinarymatroidM � Let N be the binary matroid represented by the binary matrixA that is numerically identical to B� ThenM has a base that is not a baseof N �

Proof� Carry out the earlier proof� except that the proper minors of Mare not assumed to be binary� Accordingly� D and D� may be propersubmatrices of B and A� respectively� By the pivot argument� we know thatdetD � � and det�D� � �� This implies that the base of M correspondingto D is not a base of N �

The proof of Theorem �� implies the following statement� A ma�troid M is nonbinary if and only if M has an abstract representation matrixB with a �� submatrix D that is nonsingular and that contains only �s�This fact leads to an elementary proof of the following theorem�

�� Theorem� The following statements are equivalent for a matroidM on a set E�

�i� M is binary��ii� For any circuit C and cocircuit C�� jC � C�j is even��iii� The symmetric di�erence of two circuits is a disjoint union of circuits��iv� The symmetric di�erence of two disjoint unions of circuits is a disjoint

union of circuits��v� The symmetric di�erence of two distinct circuits contains a circuit��vi� Given any distinct circuits C� and C�� and any two elements e� f �

�C� � C�� there is a circuit C� � �C� � C�� fe� fg!��vii� For any base X and Y � E � X� any circuit C is the symmetric

di�erence of the fundamental circuits Cy corresponding to X and withy � �C � Y ��

Proof� Statement �i� plus Lemmas �� imply �ii��vi��Statement �vii� follows from �i� by the previous characterization of circuits

� � References ��

in terms of column submatrices of the representation matrix B� We provethe converse implications by contradiction� Thus� we assume M to bea nonbinary matroid� For M � we select an abstract matrix B with the�� submatrix D as described above and show by inspection and routinearguments that none of �ii��vii� holds�

�� References

The basic material on graphic� binary� and general matroids relies on Whit�ney �� and Tutte �� b�� Theorem �� is provedin Truemper ��b�� Corollary �� was �rst established in Tutte��b�� a short proof appears in Cunningham �� together with otherconnectivity concepts �see also Inukai and Weinberg �� Oxley ��a��and Wagner ��b�� Corollary �� and its generalization to generalmatroids are included in Cunningham �� the results also appear inDuchamp �� and Krogdahl ��

The ��isomorphism results �Lemma �� and Theorem �� aredue to Whitney ��a�� Shorter proofs are given in Truemper ��a��Wagner ��a�� and Kelmans �� together with an upper bound ofn�� switchings for ��connected graphs with n vertices� The generalizationto directed graphs is covered in Thomassen �� For a variation of the��isomorphism problem� de�ne a matroid from the node�edge incidencematrix of a graph as in Section �� except that the matrix is viewed to beover IR instead of GF�� Just as in the GF�� case� several graphs mayproduce the same matroid� A partial analysis of the graphs generating thesame matroid is given in Wagner �� Another variation� called vertex��isomorphism� is treated in Swaminathan and Wagner ��

The graphicness testing subroutine is due to L�ofgren �� Well im�plemented� it has led to the presently most ecient algorithms for thatproblem �see Fujishige �� and Bixby and Wagner �� Other rel�evant references are Gould �� Auslander and Trent �� Tutte �� Iri �� Tomizawa ��a�� Bixby and Cunning�ham �� Wagner �� and Bixby ��a�� The �rst polynomial testfor graphicness of binary matroids was given by Tutte �� An e�cient graphicness test for matroids not known to be binary is described inSeymour ��c�� see also Bixby ��a�� and Truemper ��a��

The notion of submodularity of the rank function as one of the cen�tral tools of matroid theory is due to Edmonds �see� e�g� Edmonds ��In this book� we use the submodularity concept rather infrequently� Anexcellent survey of the many facets and applications is given in Fujishi�ge �� For optimization and decomposition results� see� e�g�� Fujishi�ge �� and Frank ��b�� Seymour �� analyses


an unusual way of specifying a matroid� De�ne the connectivity functionof a matroid on a set E and with rank function r�� to be the functionc�X� � r�X� � r�E �X� � r�E�� X � E� Clearly� the connectivity func�tion determines the matroid at most up to duality� In Seymour �� itis shown that the connectivity function generally does not determine thematroid up to duality� but that this is so when the matroid is binary�

Lemma �� and its generalization �Lemma �� have also beenindependently proved by others �Seymour ��b�� Bixby ��b�� Thespecial case of graphs was �rst treated in Seymour ��b��

The concept of abstract matrices is based on that of partial represen�tation matrices of Truemper �� The example of a nonrepresentableabstract matrix is taken from that reference� Early examples of nonrepre�sentable matroids are in MacLane �� Lazarson �� Ingleton �� and Vamos �� Additional references about the representabilityproblem are included in Section ��

Theorem �� is one of the key contributions of Tutte �� Thattheorem set the stage for and motivated a number of subsequent results onrepresentability� The proof and Corollary �� are taken from Truem�per ��b�� The conditions of Theorem �� are from Whitney ��Rado �� Lehman �� Tutte �� Minty �� and Fournier�� Additional characterizations of the binary matroids may be foundin Bixby �� Duchamp �� and White ��

For additional material on binary matroids or matroids representableover �elds with characteristic �� see Gerards �� Hassin �� a��b�� Oxley ��b�� Lemos �� and Ziegler ��

Chapter �

Series�Parallel and Delta�Wye

Constructions

�� Overview

This chapter is the �rst of three onmatroid tools� Here� we construct graphsand binary matroids with elementary procedures� For graphs� the construc�tions involve addition of a parallel edge� or subdivision of an edge into twoseries edges� or substitution of a triangle by a ��star� or substitution of a��star by a triangle� We call the �rst two operations series�parallel exten�sion steps� for short SP extension steps� Either one of the triangle��starsubstitution steps is a delta�wye step� for short �Y step� These operationshave a natural translation to operations on binary matroids�

The power of SP extension steps is quite limited� Suppose in the graphcase one starts with a cycle with just two edges and applies SP extensionsteps� Then rather simple graphs are produced� They are usually calledseries�parallel graphs� for short SP graphs� In the binary matroid case�let us start with a circuit containing just two parallel elements� Then weproduce nothing else but the graphic matroids of the SP graphs� Theseresults and some related material are described in Section ��

The situation changes dramatically when we mix SP extension stepswith �Y steps� In the graph case� suppose we start again with a cyclewith two edges� Then we produce all �connected planar graphs and more�How much more is a di�cult open question� Similarly� suppose that inthe binary matroid case we start with a circuit with two edges� Thenwe produce the graphic matroids of the just�described graphs� as well asnongraphic binary matroids� Here� too� the class of matroids so obtained

��

�� Chapter � Series�Parallel and Delta�Wye Constructions

is not well understood� In Chapter �� it is proved that every matroid ofthat class is regular�

One may� of course� start a sequence of SP extension steps and �Ysteps with any collection of graphs� not just from the cycle with two edges�Little is known about the classes of graphs so obtained� The same goesfor binary matroids� with one exception� The class of almost regular ma�troids� which we de�ne later� is generated when one starts with two binarymatroids on seven and eleven elements� respectively�

The cited material on SP extension steps and �Y steps in graphs andbinary matroids is covered in Sections �� and �� respectively� The �nalSection � contains applications� extensions of some of the matroid resultsto general matroids� and relevant references�

The material of this chapter builds upon Chapters and ��

�� Series�Parallel Construction

Start with the cycle with just two edges� In that small graph� replace oneedge by two parallel edges or by two series edges� To the resulting graphapply either one of these two operations to get a third graph� and so on�Here is an example with three such extension steps�

��

Series�parallel extension steps

Each iteration is a series�parallel extension step� for short SP extension

step� The class of graphs producible this way are the series�parallel graphs�for short SP graphs� The inverse of an SP extension step is an SP reduction

step�In this section� we analyze the structure of SP graphs� In particular�

we characterize them in terms of forbidden minors� We begin with someelementary lemmas�

�� Lemma� Every SP graph is �connected and planar� Any minorof an SP graph is also an SP graph� provided the minor has at least twoedges and is �connected�

Proof� The cycle with two edges is �connected and planar� An SP exten�sion step in a graph with at least two edges cannot introduce a ��separationor destroy planarity� By induction� the SP graphs are �connected and pla�nar�

� � Series�Parallel Construction ��

For the proof of the second part� paint in a given SP graph the edgesof a given �connected minor red� Reduce the SP graph by SP reductionsteps until the cycle with two edges is obtained� We examine a singlereduction step and apply induction� We must consider two cases for thatstep� deletion of a parallel edge� and contraction of one of two edges witha common degree endpoint� Consider the deletion case� If both edgesare red� then the reduction is also an SP reduction in the minor� If exactlyone edge is not red� then that edge is deleted� The minor must still bepresent� since contraction of that edge would turn the red edge into a loop�contrary to the assumption that the minor is �connected� If both edgesare not red� then the minor is still present after deletion of one of theseedges� For if both edges must be contracted to produce the minor� then thesecond contraction involves a loop� and thus is a deletion� The contractioncase is handled analogously�

Recall that K� is the complete graph on four vertices�

�� Lemma� Every ��connected graph G with at least six edges hasa K� minor�

Proof� Take any cycle C of G of minimal length� Since G is ��connectedand has at least six edges� it must have a node that does not lie on C�By Menger�s Theorem� there are three internally node�disjoint paths fromthat additional node to three distinct nodes of C� Suitable deletions andcontractions eliminate all other edges and reduce the cycle and three pathsto a K� minor�

�� Lemma� K� is not an SP graph�

Proof� K� does not have series or parallel edges�

�� Lemma� No SP graph has a K� minor�

Proof� Presence of a K� minor would contradict Lemmas �� and��

We are ready to characterize the SP graphs in terms of excluded mi�nors�

�� Theorem� A �connected graph is an SP graph if and only if ithas no K� minor�

Proof� The �only if� part is handled by Lemma �� We thus provethe converse� Let G be a �connected graph without K� minors� Simplechecking validates the small cases with up to �ve edges� So assume G hasat least six edges� By Lemma �� G must be �separable� Choose a�separation so that for the two corresponding graphs G� and G�� we haveG� with minimal number of edges�


Suppose G� has exactly two edges� These edges must be parallel orincident at a degree node of G� Thus� we can reduce and apply induction�Suppose G� has at least three edges� Let k and l be the nodes of G� thatmust be identi�ed with two nodes of G� to produce G� Suppose G� has anedge z connecting k and l� That edge can be shifted from G� to G�� Thecorresponding new �separation contradicts the minimality assumption onthe edge set of G�� Similarly� the nodes k and l cannot have degree � inG��

Add an edge e to G� connecting nodes k and l� The new graph G�

�is

isomorphic to a proper minor of G� By induction� G�

�is an SP graph� By

the above discussion� in G�

�the edge e is not parallel to another edge� and

it does not have an endpoint of degree � Thus� any SP reduction step inG�

�can be carried out in G as well� We perform one such step in G� and

invoke induction for the reduced graph�

One may reformulate the construction of SP graphs as follows� Startwith some cycle� Iteratively enlarge the graph on hand as follows� Selecta path in the graph where all internal nodes have degree � Let k and l

be the endpoints of the path� Then add to the graph a path from k to l�Evidently� this construction creates precisely all SP graphs� It also allowsa short proof of the following result�

�� Lemma� An SP graph without parallel edges either is a cyclewith at least three edges� or has two internally node�disjoint paths with thefollowing properties� Each path has at least two edges� Each intermediatenode of the two paths has degree in the graph� while the endpoints havedegree of at least ��

Proof� Consider the alternate construction� The initial cycle must have atleast three edges� since otherwise the �nal graph has parallel edges� Whenthe �rst path is adjoined to the initial cycle� the lemma is satis�ed� or the�nal graph has parallel edges� The same conclusion holds by inductionafter each additional path augmentation�

Lemma �� has the following corollary�

�� Corollary� An SP graph with at least four edges and withoutparallel edges has at least two nonadjacent nodes with degree �

Proof� If the SP graph is a cycle� then the conclusion is immediate� Soassume that the SP graph is not a cycle� Then each one of the two pathspostulated in Lemma �� has at least one intermediate degree node�Thus� the graph has two nonadjacent degree nodes�

We introduce two interesting subclasses of the class of SP graphs byexcluding certain graphs as minors� One of the excluded graphs we alreadyknow� It is K�� the complete bipartite graph with two vertices on oneside and three on the other one� The second excluded graph is the double


triangle� obtained from the triangle by replacing each edge by two paralleledges� We denote that graph by C�

�� We want to characterize �rst the SP

graphs without K�� minors� and then those without C�

�minors� To this

end� de�ne a graph to be outerplanar if it can be drawn in the plane sothat all vertices lie on the in�nite face�

�� Theorem� The following statements are equivalent for a �connected graph G with at least two edges�

�i� G has no K� or K�� minors��ii� G is an SP graph without K�� minors��iii� G is outerplanar�

Proof� By Theorem �� G is an SP graph if and only if it has no K�

minors� Thus� �i� �� ii�� To show �ii� �� iii�� let G be an SP graphwithout K�� minors� De�ne C to be a cycle of G of maximum length�Suppose G has a node v that does not lie on C� Since G is �connected�there exist two paths from node v to distinct nodes i and j on C so thatthese paths have only the node v in common� If i and j are connected byan edge of C� then C can be extended to a longer cycle using the two paths�a contradiction� If i and j are not joined by an edge of C� then C and thetwo paths are easily reduced to a K�� minor of G� another contradiction�Thus� all nodes of G occur on C� For the proof of outerplanarity� we mayassume that G has no parallel edges� Draw C in the plane� say using acircle� Then draw the remaining edges� each time placing the edge insidethe circle as a straight line segment� If any two such edges cross� thenthese edges plus C can be reduced to a K� minor of G� a contradiction�Thus� no edges cross� and we have produced an outerplanar drawing of G�For �iii� �� i�� we note that K� and K�� are not outerplanar and thatouterplanarity is maintained under minor�taking�

For the second subclass of the class of SP graphs� we de�ne a suspended

tree to be any graph generated by the following process� Start with a tree�Create an additional vertex called the non�tree vertex� From that vertex�add one arc to each tip node of the tree� plus at most one arc to any othernode of the tree� An example is given below� The initial tree is drawn withbold lines�

��

Suspended tree

We permit the degenerate case where the initial tree is just one node�


We have the following theorem� which turns out to be nothing but thedual version of Theorem ��

�� Theorem� The following statements are equivalent for a �connected graph G with at least two edges�

�i� G has no K� or C�

�minors�

�ii� G is an SP graph without C�

�minors�

�iii� Except for parallel edges� G is �isomorphic to a suspended tree�

Proof� We use graphic matroids and duality to link Theorem �� andthe one at hand� Speci�cally� for each of the statements �i��iii� of The�orem �� we carry out the following process� From that statement�we deduce the matroid version� then dualize that matroid statement� and�nally show that the latter statement� when expressed in terms of graphs�yields the statement of Theorem �� with the same number� Then onereverses the sequence of arguments� going from each statement of Theo�rem �� to the corresponding one of Theorem �� The proof iscomplete once the following two observations are made� First� the just�described dualization process takes any K� �resp� K�� minor of one graphto a K� �resp� C�

�� minor of the other one� Second� the dualization process

takes the property of being an SP graph �resp� of being outerplanar� to theproperty of being an SP graph �resp� of being �isomorphic to a suspendedtree up to parallel edges�� and vice versa�

We relate the above material to binary matroids� We begin with thefollowing construction� Start with the � � � binary matrix B � � � �� It�eratively enlarge that matrix by adjoining a binary column or row vectorparallel to an existing column or row� or by adjoining a column or row unitvector� We call each such iteration a matroid SP extension step� De�nethe SP matroids to be the binary matroids represented by the matricesthat can be so produced� The inverse of a matroid SP extension step is amatroid SP reduction step� The similarity of terminology with the graphcase is no accident� as we shall see next�

�� Lemma� Every SP matroid is the graphic matroid of an SPgraph� and vice versa�

Proof� The key relationships are provided in Section �� The matrixB � � � � represents the graphic matroid of a cycle with two edges� Supposeafter some iterations of the construction process� a matrix B is on hand�Let B represent the graphic matroid of an SP graph G� Then the adjoiningof a column z parallel to a given column y of B can be translated to addingan edge z parallel to edge y in G� Similarly� any other extension of B can betranslated in G to an addition of parallel edges or to a subdivision of edgesinto series edges� Thus� every matrix produced by matroid SP extensionsteps represents the graphic matroid of some SP graph� With similar ease�


one proves that the graphic matroid of any SP graph is represented bysome B generated by SP extension steps�

With the aid of Lemma �� we translate Theorem �� to thefollowing result� Recall that by Corollary �� connectedness in agraph G is equivalent to connectedness in the graphic matroid M�G��

�� Theorem� A connected binary matroid is an SP matroid if andonly if it has no M�K�� minor�

Theorems �� and �� also have an interesting translation�Since these results are dual to each other� it su�ces that we translate The�orem �� To simplify matters� we rule out parallel elements� Observethat the matrices

��1

00

01

11

1

11

1 01

11 0

1;

Representation matrices of M�K�� and M�C�

��

represent the graphic matroids M�K�� and M�C�

�� respectively� In the

�rst case� the corresponding base of M�K�� is a star of K�� In the secondcase� the corresponding base of M�C�

�� contains any two nonparallel edges

of C�

��

�� Theorem� The following statements are equivalent for a con�nected binary matroidM with at least three elements� no two of which areparallel�

�i� M has no M�K�� or M�C�

�� minors�

�ii� M is an SP matroid without M�C�

�� minors�

�iii� No representation matrix of M has any one of the matrices of ��as submatrix�

�iv� M is the graphic matroid of a suspended tree��v� M is a minor of the matroid M � represented by a binary matrix B�

that is the node�edge incidence matrix of a tree�

Proof� Equivalence of �i�� ii�� iii�� and �iv� follows directly from Theorem�� To show that �iv� implies �v�� let M be the graphic matroid ofa suspended tree G� Add edges� if necessary� so that every tree node isconnected by exactly one arc with the extra node� The graphic matroidM � of that enlarged graph G� has M as a minor� The edges of G� incidentat the extra node form a tree X� Thus� X is a base of M � It is easy tosee that the representation matrix B� of M � corresponding to X is nothingbut the node�edge incidence matrix of the tree� Hence� �v� holds� Thearguments are easily reversed to prove that �v� implies �iv��

For subsequent reference� we include the following lemma�

� Chapter � Series�Parallel and Delta�Wye Constructions

�� Lemma� Let M be a connected binary matroid� Then any bi�nary matroid obtained by an SP extension step from M is connected�

Proof� By Lemma �� connectedness of a binary matroid is equiv�alent to connectedness of any one of its representation matrices� Clearly�any SP extension step maintains connectedness of representation matrices�Thus� the resulting matroid is connected�

�� Delta�Wye Construction for Graphs

The simplicity of SP graphs gives way to far more complicated graphs whenwe permit two operations in addition to SP extensions� One of them is thereplacement of a triangle by a ��star� and the second one is the inverseof that step� Either operation we call a �Y exchange� De�ne a sequenceof SP extensions and �Y exchanges to be a �Y extension sequence� Theinverse sequence is a �Y reduction sequence� A �connected graph is �Yreducible if there is a �Y reduction sequence that converts the graph toa cycle with just two edges� In this section� we show that �Y extensionsequences applied to such a cycle create all �connected planar graphs andmore� Any graph so producible is a �Y graph�

As an example for �Y reduction sequences� we reduce K�� the com�plete graph on �ve vertices� to the cycle with two parallel edges�

��

replacetriangle

replacetriangle

replace3-star

deletethreeparalleledges

contractseriesedge,done

contractseriesedge

replace3-star

replace3-star

deletethreeparalleledges

�Y reduction sequence for K�

In each graph of the reduction sequence� the triangle or ��star involvedin a �Y exchange is indicated by bold lines� Similarly� we emphasize the

� � Delta�Wye Construction for Graphs �

series or parallel edges of SP reductions� We know from Lemma ��that M�K�� is not cographic� Thus� K� is nonplanar� and the exampledemonstrates that �Y graphs may be nonplanar�

A �Y exchange may not preserve �connectedness� For example� whena vertex of a triangle in a �connected graph has degree � then replacementof that triangle by a ��star produces a ��separable graph� The next lemmagives the conditions under which �connectedness is maintained�

�� Lemma� Let G be a �connected graph� Then a triangle to ��starexchange �resp� ��star to triangle exchange� in G produces a �connectedgraph G� if and only if the triangle �resp� ��star� does not contain two edgesin series �resp� in parallel��

Proof� Consider the triangle to ��star exchange� The �only if� part hasbeen argued above� For proof of the �if� part� suppose a triangle doesnot contain two series edges� Equivalently� the triangle does not contain acocycle� Thus� G has a tree that does not include any edges of the triangle�With the aid of this tree� it is easy to see that the graph G� derived fromthe �connected graph G has no ��separation�

If a ��star has two edges in parallel� then a ��star to triangle exchangeis not possible� Thus� the �only if� part is trivially satis�ed� The �if�part is easily checked� analogously to the case of a triangle to ��star ex�change�

The asymmetry of arguments in the proof of Lemma �� is dueto the fact that a triangle is always a cycle of a graph� while a ��staris not always a cocycle� Note that the conditions of Lemma �� areautomatically satis�ed in �Y reduction sequences where �Y exchangesare done only when an SP reduction is not possible�

Our goal is to show that the class of �Y graphs includes all �connectedplanar graphs� That goal is restated in the next theorem�

�� Theorem� Every �connected planar graph is �Y reducible�

The proof relies on three lemmas� They show� in fact� that any �connected plane graph� i�e�� an embedding of a planar graph in the plane�is �Y reducible under the following restriction� We permit a triangle to��star exchange only if the triangle bounds a face�

The �rst auxiliary lemma is the analogue of Lemma ��

�� Lemma� If a �connected graph or plane graph G is �Y re�ducible� then every �connected minor H of G is �Y reducible as well�

Proof� We con�ne ourselves to the plane graph case� By omitting ref�erences to the embeddings of G� one obtains the proof for the generalsituation� We induct on the number of SP reductions and �Y exchangesthat reduce the planar graph G to the cycle with just two edges� We maysuppose that H has no series or parallel edges� H is then a minor of G


as well as of any graph derived from G by any SP reductions� Thus� byinduction� we only need to examine the case where the �rst step of a givenreduction sequence for G is a �Y exchange�

Consider a triangle to ��star exchange� By assumption� the triangle�say fe� f� gg� bounds a face� Let G� be the plane graph resulting fromthe exchange� Suppose e� f � and g occur in H� Since H is �connected�these edges must form a triangle in H that also bounds a face of H� Byassumption�H has no series or parallel edges� so we may replace the triangleof H by a ��star� getting a �connected graphH �� Then H � is a �connectedminor of G�� and the conclusion follows by induction� Suppose at least oneof the edges of the triangle fe� f� gg does not occur in H� It is easilyseen that we may delete e� f � or g from G while retaining H as a minor�Regardless of the speci�c situation� H is isomorphic to a minor of G�� andonce more induction can be invoked�

The case of a ��star to triangle exchange is handled analogously� In�deed� in the plane graph case� we may invoke duality�

The next two lemmas involve grid graphs� which are plane graphs ofthe form

��

1

2

3

4

m-1

m

1 2 3 4 n-1 n

; m , n ≥ 2

Grid graph

�� Lemma� Every plane graph is a minor of some grid graph�

Proof� �Sketch� We may assume that the given plane graph is �connected�since this can be achieved by the addition of edges� Split each vertex ofthat plane graph so that a �connected plane graph G results where thedegree of each vertex is at most �� By a suitable subdivision of edges� Gcan be embedded into a grid graph as follows� First embed any one face ofG� but not the outer one� Then embed one face at a time so that each oneof the successive subgraphs of G so embedded is �connected�

�� Lemma� Every grid graph is �Y reducible�

Proof� Two special �Y reduction subsequences will be used repeatedly�For the �rst case� suppose we have a grid graph plus one edge e so thate and two edges of a degree vertex of the grid graph form a triangle�

� � Delta�Wye Construction for Graphs ��

Consider the two �Y exchanges depicted in �� below�

��e

replacetriangle

replace3-star

e

Moving edge e toward lower left�hand corner

E�ectively� the two �Y exchanges have moved the edge e one step closertoward the lower left�hand corner of the grid graph�

The second situation is even simpler� Suppose an edge e forms atriangle with two edges of a ��star� Then that edge can be e�ectivelyeliminated via one �Y exchange plus a series reduction as follows�

��contractseriesedge

replacetriangle

e

Removal of edge e

We are ready to describe a �Y reduction sequence for grid graphs� By�� the graph is obviously an SP graph if m or n � � Hence� supposem�n � �� First we reduce the two pairs of series edges in the upper rightcorner and lower left corner of the grid graph to one edge each� Thus� theupper right�hand corner has become

��

e

Upper right�hand corner after series reduction

By repeated application of �� we move the edge e toward the left orbottom boundary of the graph� When that boundary is reached� e can beeliminated either by �� or by the fact that it has become parallel tothe lower left edge produced in one of the two initial series reductions�

We now have the upper right�hand portion as

��

f

g

Upper right�hand corner after removal of e


The edge g is in series with f and can be contracted� Then f can beeliminated analogously to the edge e� By repetition of this process andsuitable adjustment in the last iteration� we e�ectively eliminate all nodesof the right�hand boundary of the grid graph� By induction� the lemmafollows�

Proof of Theorem �� A given plane graph is by Lemma ��a minor of some grid graph� By Lemmas �� and �� both graphsare �Y reducible�

By Lemma �� the class of �Y graphs is closed under the takingof �connected minors� Thus� one is tempted to look for a characterizationof the �Y graphs by exclusion of minimal minors that are �connected andnot �Y reducible� As a �rst step toward �nding these minors� we introducethe following equivalence relation on the class of �connected graphs thatare not �Y reducible and that are minimal with respect to that property�Two such graphs are de�ned to be related if one can be obtained fromthe other one by a sequence of �Y exchanges� The above characterizationproblem is solved once one �nds one member of each equivalence class� Itis easy to see that the graph K� is one of the desired minimal graphs� Bystraightforward enumeration� the equivalence class represented by K� canbe shown to consist of the following graphs�

��

K6

Petersen Graph

Equivalence class of minimal nonreducible graphsrepresented by K�

One might conjecture� and would not be the �rst person to do so� thatthe graphs of �� constitute all minimal nonreducible graphs� Theconjecture is appealing but false� A counterexample due to Robertson isthe graph G constructed as follows� We start with the planar graph of�� below� We add to that graph one vertex v� which is connected to

� � Delta�Wye Construction for Binary Matroids ��

the circled nodes of the planar graph by one edge each� The graph G isevidently ��connected and has no triangles or ��stars� Thus� it is not �Yreducible� We claim that G does not have any one of the graphs of ��as a minor� By the construction� G can be reduced to a planar graph byremoval of the vertex v� But none of the graphs of �� becomes planarwhen any one its vertices is removed� as is easily veri�ed� Thus� G is acounterexample to the conjecture�

��

Planar graph for counterexample G

In the next section� we adapt the concept of �Y extension and reduc�tion sequences to binary matroids�

�� Delta�Wye Construction for Binary

Matroids

In this section� we translate the de�nitions and conditions for �Y graphsgiven in Section �� to statements about binary matroids� Thus� we obtain�Y extension steps� �Y reduction steps� �Y matroids� etc� Recall fromLemma �� that every SP matroid is the graphic matroid of an SPgraph� and vice versa� In contrast� the yet�to�be�de�ned �Y matroids turnout to be not just the graphic matroids of �Y graphs�

We start with the de�nitions� From Section �� we already know howSP extensions are performed in binary matroids� So let us translate the �Yexchange from graphs to binary matroids� To gain some intuitive insight�we perform a triangle to ��star step in a �connected graph G with edge setE� then translate that step into matroid language for the graphic matroidM �M�G� of G� Let the triangle of G be C � fe� f� gg� As in Section ��we assume that C does not contain a cocycle�

To carry out the triangle to ��star exchange in G� we �rst add the��star� say C� � fx� y� zg� getting a graph G�� A partial drawing of G� with


C and C� is shown below�

��e g

f

x

y z

Triangle and ��star

Then we delete the triangle C� getting the desired graph G�� Since C

contains no cocycle� the graph G has a tree X and a cotree Y � E �X sothat C � Y � The representation matrix B of M for this X is assumed tobe

��XB =

Ye f g

a b cB

Matrix B for M �M�G�

Clearly� X �fyg is a tree of the graph G�� We claim that the representationmatrix B� for M � �M�G�� corresponding to that tree must be

�� XB' =

Ye f g x z

0y 1 1

a b c a bB

Matrix B� for M � �M�G��

The proof of this claim is as follows� First� the edges x and z of G� producethe only fundamental circuits withX�fyg that contain y� This fact justi�esthe last row of B�� Second� contraction of y in G� makes the edge x �resp�z� parallel to e �resp� f�� Correspondingly� upon deletion of row z from B��the columns x and e �resp� z and f� must be parallel� These facts uniquelydetermine B� as shown in ��

Deletion of the columns e� f � and g from B� yields the desired repre�sentation matrix B�� for M �� M�G�� i�e��

�� a bB'' = ; Y−{e,f,g}0 1 1

zx Y~

y

X B Y =~

Matrix B�� for M �� M�G��


Instead of X �fyg� we could also have chosen X �fxg or X �fzg as tree ofG�� Correspondingly� the column vectors a and b explicitly shown in ��would have been a and c� or c and b� Each one of the latter matrices mayalso be obtained by a GF��pivot in row y of B��

The above translation of a triangle to ��star exchange in G to a triangleto triad exchange inM may be rephrased as follows� The latter exchange inM is allowed only if the triangle does not contain a cocyle� In the exchange�we �rst �nd a representation matrix B where the triangle elements arenonbasic� Next we delete one of the columns corresponding to the triangle�Then we add a row that has �s in the remaining two columns of the triangle�and �s elsewhere� Finally� we re�index the two columns formerly indexedby triangle elements� The diagram below summarizes this particular �Yexchange process� For clarity� the diagram omits indices other than e� f � gand x� y� z�

�� c

g

a b

e f

0 1 1

x z

a b

yB

B

�Y exchange� case �

Note that a � b � c � � �in GF�� There are other ways to display thetriangle to triad exchange� Speci�cally� when we select a B with exactlyone element of the triangle� say f � basic� we get a case of the form

��

1a

a

b b

e g

0f

b

x

a 1z0y

B B

�Y exchange� case

When we select a B with two elements of the triangle basic� say e and f �we get the third possible case

��

cbz

y

ax

b

0

g

a 1e1f

BB

�Y exchange� case �

where a�b�c � � �in GF�� Simple checking� analogous to that proving�� con�rms these claims� We emphasize that in each situation� the


symbols B� a� b� and c refer to di�erent matrices and vectors� However�the relationships between the triangle elements e� f � and g and the triadelements x� y� and z are displayed in agreement with ��

The de�nition of a triangle to triad exchange of a binary but notnecessarily graphic matroid M is as follows� We require that the triangle�say fe� f� gg� does not contain a cocycle� Let B be any representation matrixofM � Then B is one of the left�hand matrices of �� or ��The matroid resulting from the exchange is given by the correspondingright�hand matrix� Suppose we select B of �� to carry out the triangleto triad exchange� as we always may� Then the cocycle condition on thetriangle is equivalent to the requirement that the row vectors a and b ofB be nonzero and distinct� Hence� the row vectors a� b� and c � a � b

�in GF�� in the resulting matrix of �� are distinct� and fx� y� zg isindeed a triad in the corresponding matroid� The triad trivially does notcontain a cycle since the resulting matrix indicates x� y� and z to be partof a basis�

For the de�nition of a triad to triangle exchange� we invert the aboveprocess� We demand that the triad not contain a cycle� The exchangeis speci�ed by �� and �� when we start with the matrixon the right hand side and derive the one on the left hand side� By tak�ing transposes of the matrices involved in �� and �� wesee that a triangle to triad exchange in M is precisely a triad to triangleexchange in M�� Thus� these two operations are dual to each other�

We de�ne �Y extension sequence� �Y reduction sequence� and �Yreducibility for binary matroids by the obvious adaptation of the same termsfor graphs� Later� we need the following lemma about �Y exchanges�

�� Lemma� Let M be a connected binary matroid� Then any �Yexchange in M produces a connected binary matroid�

Proof� By duality� we only need to consider the triangle to triad exchangedepicted in �� By Lemma �� M is connected if and only if anyrepresentation matrix B ofM is connected� The left�hand matrix of �� is thus connected� Since the vectors a� b� c of that matrix are nonzero anda � b � c � � �in GF�� the right�hand matrix is easily veri�ed to beconnected as well� Thus� the related matroid is connected�

Suppose we start with the connected� binary� and graphic matroidhaving just two parallel elements� with representation matrix B � � � �� Letthe �Y matroids be the binary matroids that may be produced from thatmatroid by �Y extension sequences� Due to Lemmas �� and ��SP extensions and �Y exchanges maintain connectedness� Thus� the �Ymatroids are connected� By de�nition� the smallest �Y matroid is graphic�Furthermore� all graphic matroids of �Y graphs are �Y matroids� as arethe duals of these matroids� By �� the nonplanar graph K� is �Yreducible� as is� evidently� K�� Thus� the graphic matroids M�K�� and


M�K�� are �Y�reducible� as well as M�K�� and M�K�� By Lemma�� the latter matroids are not graphic� Thus� �Y matroids need notbe graphic� Indeed� there are �Y matroids that are not graphic and notcographic� For example� the nongraphic and noncographic matroid R��

introduced later in Chapter �� via the representation matrix

��

00

0

11

0

0101

110

1

1010

1

0101

1 11 110 1

1 0 01 0 0

B12 =

Matrix B�� for R��

is a �Y matroid� It would take up too much space to describe the SPreduction and �Y exchange steps that prove this claim� But with themachinery introduced in Chapter �� we have deduced B � � � � from B��

of �� using eight �Y exchange steps and ten SP reduction steps�Analogously to Lemma �� we now show that the class of �Y�

matroids is closed under the taking of connected minors� The proof mimicsthat of Lemma ��

�� Lemma� If a connected binary matroid M is �Y reducible�then every connected minor N of M is �Y reducible as well�

Proof� Select a representation matrix B for M that displays the minor N

by a submatrix B� Thus�

��Y

B = X

Y

01

BX

Matrix B for M displaying minor N by B

We induct on the number of SP reductions and �Y exchanges that reduceM to the matroid with just two parallel elements� We may assume that Nhas no series or parallel elements� N is then a minor of M as well as of anyminor derived from M by SP reductions� Thus� by induction and duality�we only need to examine the case where the �rst step of a given reductionsequence for M is a triangle to triad exchange� Let the triangle of M befe� f� gg�

Three cases are possible� depending on the number of elements offe� f� gg that index rows of B� In the �rst case� that number is �� Thus�e� f� g � Y � Suppose at most two of the columns e� f � g of B intersect


B� Then according to �� we drop from B one column that does not

intersect B� and add a row with two �s� The matrix B is still a submatrixof the matrix so derived from B� and we are done by induction� Suppose

all three columns e� f � g of B intersect B� Then the change of B given

by �� may also be viewed as a triangle to triad change involving B�provided we can prove that fe� f� gg is a triangle of N that does not containa cocycle� But the assumption that N is connected and has no series orparallel elements implies these two conditions� and once more we are doneby induction� With similar ease one argues the case where one or twoelements of fe� f� gg index rows of B�

In Chapter �� it is shown that the class of �Y matroids is a subset ofthe class of regular matroids� which we de�ne in a moment� But otherwisethat class is not well understood� Consider a slightly more complicatedsituation� This time� a class of binary matroids is produced by �Y exten�sion steps from a given connected binary matroid� or possibly from severalsuch matroids� In the remainder of this section� we cover an interestinginstance�

We need a few de�nitions concerning regular and almost regular ma�troids� These matroids are treated in depth in Chapters �� We con�neourselves here to a rather terse introduction� A real f��g matrix is to�

tally unimodular if every square submatrix has real determinant equal to� or �� A binary matrix is regular if its �s can be signed to become �sso that the resulting real matrix is totally unimodular� A binary matroidis regular if M has a regular representation matrix� It is not so di�cult tosee� and is also proved in Chapter �� that for a given binary matroid eithernone or all representation matrices are regular�

A binary matroid M is almost regular if it is not regular� and if forany element z of the matroid� at least one of the minors M�z and Mnz isregular� In addition� we demand the existence of a label for each element zso that we can identify at least one of the minors M�z or Mnz as regular�The label must be �con� or �del�� If the label for element z is �con� �resp��del�� then M�z �resp� Mnz� must be regular� We still have a rathertechnical condition that must be satis�ed by the labels� They must beso chosen that every circuit �resp� cocircuit� of M has an even number ofelements with �con� �resp� �del�� labels� Finally� there must be at leastone �con� element and at least one �del� element�

The reader is likely to be puzzled by the strange parity condition andthe existence condition� In Chapter �� it is shown how they come about�and why one might want to de�ne and investigate almost regular matroids�So for the time being� we suggest that the reader simply accept or at leasttolerate these seemingly strange requirements�

One way to construct some almost regular matroids is as follows� Wetake any square f��g real matrix A that is not totally unimodular but


all of whose proper submatrices do have that property� Call any suchmatrix minimal non�totally unimodular� There is a rather simple subclassof these matrices where every row and every column has exactly two �s�We assume that A is not of this variety� Let B be the support matrix ofA� View B as the binary representation matrix of a binary matroidM � Tothe elements of M corresponding to the rows �resp� columns� of B� assign�con� �resp� �del�� labels� Then M is an almost regular matroid� a factproved in Chapter �� An example is the minimal non�totally unimodularmatrix

�� 11

1

01

1

00

110

1

01

10A =

Minimal non�totally unimodular matrix

The binary support matrix B of A represents an almost regular matroidwhen labels are assigned as described above� The class of minimal non�totally unimodular matrices is quite rich� Thus� the class of almost regularmatroids is interesting as well�

Here are the representation matrices of two almost regular matroidsthat cannot be produced by the above process� Instead of row and columnindices� we record for each row and column the label of the correspondingelement of the matroid�

��

del11

1

0condeldel

del

100

0 1 1 1 1 11

1

del

con

con

con

con

con

110

011

0 10 11 1

0 1 1B11 =

1 01 0

condeldel

0 1 1 11

del

con

con

con

1 11B7 = ;

Labeled Matrices B� and B��

The matrix B� should be familiar� It is� up to column�row permutationsand labels� the representation matrix of �� of the nonregular Fanomatroid� The reader can easily check that the Fano matroid with the givenlabels does satisfy the above�mentioned parity and existence conditions onlabels� The origin of B�� is explained in Chapter �� At any rate� veri�ca�tion of the conditions on labels for B�� requires a moderate computationale�ort�

Due to the labels� we want to restrict SP extension steps and �Yexchanges a bit� Speci�cally� we allow a parallel �resp� series� extension ofM only if the involved element z of M has a �con� �resp� �del�� label� Thenew element receives the same label as z� A triangle to triad exchange is


permitted only if the triangle� say fe� f� gg� has exactly two �con� elements�The labels of the resulting triad� say fx� y� zg� can be deduced from thedrawing below� which should be interpreted the same way the drawing of�� is linked to �� and ��

�� xcon

econ

gcon

f del

y del z del

Triangle and ��star with labels

Note that the resulting triad has two �del� labels� as demanded by theparity condition� A triad to triangle exchange is just the reverse of theabove step� It is permitted only if the triad has exactly two �del� labels� Arestricted �Y extension sequence is a sequence of restricted SP extensionsand of restricted �Y exchanges� It is not di�cult to prove that restricted�Y extension sequences convert almost regular matroids to matroids withthe same property� We show this in Chapter ��

We include a short restricted �Y extension sequence that starts withB� of �� and that generates� with appropriate labels� the matrix B

deduced earlier from the minimal non�totally unimodular A of ��

��

1 01 0

condeldel

abc

0 1 1 11

dd e f g

el

con

con

con

1 11

1 01 0

condeldel

abc

0 1 11

dd e f

el

con

con

gcon

hcon

1

1

11

1

00

replacetriangle{d,g,h}

add h parallel

l

db c y z

ede

de

de

l l l

fe

1 1con

1

nx

0 1concon

a 1 1 1 1

0 1 1 00 0co

0GF(2)-pivot on two circled 1s

del

1 1del

nx

0 1cona

b1 1 1 1

ce f y z

oco

de

de

nn l l

0c 1 00 0co

01

1

to a

Example of restricted �Y extension sequence

The next theorem states a rather surprising fact about the power of re�stricted �Y extension sequences�

�� Theorem� The class of almost regular matroids has a partitioninto two subclasses� One of the subclasses consists of the almost regular

� � Applications� Extensions� and References ��

matroids producible by �Y extension sequences from the matroid repre�sented by B� of �� The other subclass is analogously generated byB�� of �� There is a polynomial algorithm that obtains an appropri�ate �Y extension sequence for creating any almost regular matroid fromthe matroid of B� or B�� whichever applies�

Unfortunately� the existing proof of Theorem �� is so long thatwe cannot include it here� In this book� it is one of the few results aboutbinary matroids that we state but do not prove� We outline a proof inChapter �� when we restate Theorem �� as Theorem �� Atthat time� we use the theorem to establish several matrix constructions�

In the �nal section� we cover applications� extensions� and references�

�� Applications� Extensions� and

References

The series�parallel construction is a basic idea of electrical network theory�The characterization of SP graphs in Theorem �� in terms of excludedK� minors is given in Dirac �� Another proof and basic results aboutSP graphs are provided in Du�n �� Theorem �� is due to Char�trand and Harary �� The dual of that result� Theorem �� isproved directly in Truemper and Soun �� and Soun and Truemper�� Decomposition results for SP graphs and outerplanar graphs aredescribed in Wagner ��

Onemay attempt to generalize SP matroids by dropping the restrictionthat the matroids be binary� One still starts with the matroid havingjust two parallel elements� The SP extensions are de�ned via addition ofparallel elements and expansion by series elements� These steps can benicely displayed by abstract matrices� With that approach� one very easilyproves that the supposedly more general procedure generates nothing butthe graphic matroids of SP graphs� By Theorems �� and �� aconnected matroid with at least two elements is thus an SP matroid if andonly if it has no U�

�orM�K�� minors� Additional material on combinatorial

aspects of series�parallel networks may be found in Brylawski ��Akers �� and Lehman �� contain the following conjecture� Let

G be a �connected graph that is a plane graph plus one edge called thereturn edge� Then G is conjectured to be �Y reducible to a cycle with justtwo edges� one of which must be the return edge� Note that the returnedge is not allowed to participate in any �Y exchange� The conjecturewas �rst proved in Epifanov �� by fairly complicated arguments� Asimple proof is given in Truemper ��a�� which also contains the proofof Theorem �� given here� An interesting but not simple proof of


Theorem �� is provided in Gr�unbaum �� That reference relies onTheorem �� to derive a very elementary proof of Steinitz�s Theoremlinking ��connected graphs and ��dimensional polytopes� Computationalaspects of �Y graphs are treated in Feo �� and Feo and Provan ��Gitler �� characterizes �Y reducible planar graphs where k speci�ednodes� k � �� may not be removed� That is� none of the k nodes may beeliminated by Y�to�� exchanges or series reductions� The return edge casediscussed above is equivalent to the situation with k � �

The counterexample graph G de�ned via the planar graph of ��is due to Robertson �� The class of �Y graphs may be specialized bydemanding that all �Y exchanges are either ��to�Y exchanges or Y�to��exchanges� The �Y graphs so created� we call ��to�Y graphs and Y�to��graphs� The two classes are completely characterized in Politof ��a��b��

The special structure of SP graphs� �Y graphs� ��to�Y graphs� and Y�to�� graphs has been exploited for numerous applications� in particular forcombinatorial optimization and reliability problems� Relevant referencesare Moore and Shannon �� Akers �� Lehman �� Nishizekiand Saito �� Rosenthal and Frisque �� Monma and Sid�ney �� Farley �� Farley and Proskurowski �� Takamizawa�Nishizeki� and Saito �� Wald and Colbourn ��a�� b�� Agrawaland Satyanarayana �� Satyanarayana and Wood �� Arn�borg and Proskurowski �� Politof and Satyanarayana �� Colbourn �� and El�Mallah and Colbourn ��

The class of almost regular matroids is de�ned in Truemper ��a��Theorem �� is taken from that reference�

Chapter �

Path Shortening Technique

�� Overview

In this chapter� we introduce a matroid tool called the path shorteningtechnique� It is an adaptation of an elementary graph method to matroids�In Section �� we �rst motivate that technique� then use it to prove severalresults concerning the existence of certain separations and of certain minorsin binary matroids� In subsequent chapters� we rely a number of times onthese results�

In Section �� we employ the technique to devise a polynomial al�gorithm that solves the following problem� Given are two matroids M�

and M� on a common set E� One must �nd a maximum cardinality setZ � E that is independent in both M� and M�� This problem is calledthe cardinality intersection problem� for short� intersection problem� Corre�spondingly� the algorithm is called the intersection algorithm� That schemealso solves the following problem� which is called the partitioning problem�As before� two matroids M� and M� on a common set E are given� saywith rank functions r�� and r�� One must partition E into two sets�say E� and E�� such that r�E� � r�E� is minimized� The intersectionalgorithm also provides a constructive proof of a max�min theorem thatlinks the intersection problem with the partitioning problem� The resultsof Section �� and many more on the intersection and partitioning of ma�troids are almost entirely due to Edmonds� They constitute some of themost profound results of matroid theory�

The results of Section �� are related to the remaining chapters asfollows� There we frequently must �nd certain separations of matroids�

��

�� Chapter � Path Shortening Technique

The intersection algorithm may be employed to do this� But one may alsolocate the desired separations with the separation algorithm of the nextchapter� Thus� in principle� we do not require the intersection algorithm forthe remainder of the book� As a consequence� the material of Section ��may be just scanned or even skipped on a �rst reading without loss ofcontinuity�

The �nal section� �� contains extensions and references� The chapterutilizes the material of Chapters � and ��

�� Shortening of Paths

Consider the following simple graph problem� Given is a ��connectedgraph G with at least two edges� among them edges e and f � We are askedto prove that G has a ��connected minor consisting of just e and f � Thesolution is straightforward� As stated in Section �� any two edges of a��connected graph lie on some cycle� In particular� the edges e and f lieon some cycle C of G� Evidently� C consists of e� f � and two node�disjointpaths� say P� and P�� We obtain the desired minor by deleting all edgesnot in C and contracting all edges of P� and P�� One could call the secondstep a shortening of the paths P� and P��

The path shortening operation has numerous uses in graph theoryinvolving far more complicated situations than the trivial example treatedabove� The method also has an interesting translation to matroid theory�where we will refer to it as the path shortening technique� In this section�we describe that technique while proving the matroid analogue of the just�used fact that in a ��connected graph any two edges lie on some cycle�Subsequently� we rely on the path shortening technique to establish othermatroid results that will be repeatedly invoked in later chapters� We beginwith the matroid version of the cited graph result�

�� Lemma� A binary matroid M is connected if and only if for

every two elements x and z of M � there is a circuit containing both x and

z�

Proof� Let B be a binary representation matrix of M � First we prove the if� part by contradiction� If M is not connected� then by Lemma ��B is not connected� Evidently� M then has elements x and z that cannotboth be in any circuit�

We turn to the only if� part� Thus� we assume M � and hence B�to be connected� If the rank of M is �� then due to the connectednessassumption� M has at most one element� and the desired conclusion holdsvacuously� So assume that the rank of M is at least �� Correspondingly� Bhas at least one row� Let x and z be two elements of M � We must showthat some circuit of M contains both x and z�

� � Shortening of Paths ��

Since B is connected� by at most one GF��pivot we can assure that xindexes a row of B� Suppose that z indexes a column of B� If the elementBxz of B is a �� then x is in the fundamental circuit of M given by thecolumn z of B� and we are done� If Bxz � �� then any GF��pivot incolumn z produces the case where z also indexes a row� Thus� from nowon we may assume that both x and z index rows of B�

At this point� we switch from the connected matrix B to the connectedbipartite graph BGB� The analysis of BGB consists of the followingelementary step� We examine a shortest path from row node x to row nodez� Suppose that the row nodes of the path are x� r�� r�� rm� z� forsome m � �� and that the column nodes are y� s�� s�� sm� The nodesare encountered in the given order as one moves along the path from x toz� We claim that B can be partitioned as

�� .. .

. ..

s1. . .

1

1x

z

r1

rm

y sm

1B =

0 1

11

11

Shortest path from x to z displayed by B

In the submatrix B of B indexed by x� r�� r�� rm� z and y� s�� s�� sm� the �s� circled or not� correspond to the edges of the path� Recall aconvention of Section �� about the display of matrices� If a submatrix orregion of a matrix contains explicitly shown �s while leaving the remainingentries unspeci�ed� then the latter entries are to be taken as �s� Accord�ingly� the display of the submatrix B implies that B does not contain any�s beyond those corresponding to the edges of the path� We must prove� ofcourse� that this is the case� So suppose there are additional �s in B� Thenone readily con�rms that the path has a chord� and thus is not shortest� acontradiction�

By GF��pivots on the circled �s of B of �� we derive from Bthe following matrix B��

�� . . . .

. . .

.

. 1s1

1y

zsm

x r1 rm

all1s

B' =

0 111

Matrix B� obtained by path shortening pivots


The matrix B� proves that z is contained in the fundamental circuit thatx forms with the base producing B�� and we have proved x and z to be insome circuit of M as desired�

Note that in BGB� the path connecting x and z has at least twoedges� In contrast� the graph BGB� has an edge connecting x and z�Thus� the GF��pivots transforming B to B� have reduced the path ofBGB to a shorter path in BGB�� For this reason� we call the abovemethod the path shortening technique� Evidently� the proof procedure ofLemma �� can be implemented in a polynomial algorithm that de�termines a circuit containing two given elements in any connected binarymatroid�

We now prove additional matroid results using the path shorteningtechnique� They concern the existence of certain separations or of certainminors� In each case� the proof procedure has an obvious translation toa polynomial algorithm that very e�ectively locates one of the claimedseparations or minors�

The �rst case concerns the existence of connected ��element extensionsof a given minor in a binary matroid�

�� Lemma� Let N be a connected minor of a connected binary

matroid M � De�ne z to be an element of M not present in N � Then Mhas a connected minor N � that is a ��element extension of N by z�

Proof� By Lemma �� M has a representation matrix B that displaysN via a submatrix B� Let the rows of B be indexed by X and the columnsby Y � Since M and N are connected matroids� both B and B are connectedmatrices� Suppose the element z indexes a column in B� We thus have Bgiven by �� below� If the subvector e of column z of B is nonzero� thenthe connected submatrix �B j e� represents the desired connected minor N �

with z�

��

z

e

B =

0 1

BX

Y

Matrix B displaying minor N by B

So suppose e � �� We know that the bipartite graph BGB is connected�Thus� that graph contains a path from X � Y to z� Take a shortest path�say with row nodes r�� r�� rm and column nodes s�� s�� sn� z� for


some m�n � �� The increasing indices indicate the order in which thenodes are encountered as one moves along the path from X � Y to z� Theendpoint of the path in X � Y is r� or s�� whichever applies� Below� wedisplay B� The explicitly shown �s correspond to the edges of the path�Two cases are possible� depending on whether r� or s� is the endpoint ofthe path in X � Y �

��

...

s1. . .

or ...

......

s1. . Y

1 11

sn z

B =

B

Y

X

0 1

01

111

r1

rm

r1

1111

01 11 1

1

sn z

BX

0 1

rm

.

Case � of B� Case � of B�r� � endpoint of path s� � endpoint of path

In each matrix of �� the unspeci�ed entries� which by our conventionare �s� are justi�ed as follows� Suppose such an entry is instead a �� forexample in a row x � X and column sj � j � �� Then the path does nothave minimal length since there is a path starting at node x that is shorter�All other unspeci�ed entries are argued similarly�

Assume case � of �� applies� Then pivots on the circled �s of Bproduce the already�discussed instance with e �� For case � of ��pivots on the circled �s produce a B� with a submatrix �B�d�� where thevector d is nonzero and indexed by z� That submatrix represents the desiredN �� The case where z indexes a row of B is handled by duality�

So far� we have explained each step of the path shortening techniquein detail� In the proofs to follow� we skip such details since the argumentsare identical or at least very similar to those above�

The next result concerns the presence of minors that are isomorphicto the graphic matroids of wheels� Recall from Section �� that for n � ��the wheel Wn is constructed from a cycle with n nodes as follows� Add oneextra node and link it with one edge each to the nodes of the cycle� Smallwheels are as follows�

��

W� W� W� W�

Small wheels


The spokes of each wheel Wn form a tree� With that tree as base ofMWn� we obtain from �� the following representation matrices forMW�� MW��

��0

1

001 1

1

1 1 000 0 1

10

01 1 01 0 1

11011 1

1

MW� MW� MW� MW�Representation matrices

Indeed� for n � �� MWn is represented by the n� n matrix

�� 1 .. ..

11 1

1

1

1 1

Representation matrix for MWn� n � �

Clearly for n � �� Wn is ��connected� Thus� by Theorem �� Wn isthe unique graph producing MWn�

The graphic matroids of wheels are cited very often in the remainderof this book� We adopt an abbreviated terminology and simply call themwheels� The double meaning of wheel� should not cause a problem� Thecontext invariably makes it clear whether we mean a matroid or a graph�At any rate� we use wheel graph and wheel matroid if there is even a slightchance of confusion�

A basic result about wheels is as follows�

�� Lemma� Let M be a binary matroid with a binary represen�

tation matrix B� Suppose the graph BGB contains at least one cycle�

Then M has an MW� minor�

Proof� Since the graph BGB has at least one cycle� it has a cycle Cwithout chords� Now BGB is bipartite� so C has at least four edges� Thesubmatrix B of B corresponding to C evidently is up to indices either the�� matrix for MW� of �� or for some n � �� the n� n matrix of�� for MWn� Path shortening pivots on �s of B convert the lattercase to the former one�

Lemma �� and another application of the path shortening tech�nique lead to a proof of the following result�


�� Lemma� Let M be a connected binary matroid with at least

four elements� Then M has a ��separation or an MW� minor�

Proof� Let B be a binary representation matrix of M � Since M is con�nected� the bipartite graph BGB is connected� Suppose BGB does notcontain a cycle� Thus� that graph is a tree� Any tip node of the tree corre�sponds to a row or column unit vector in B� Thus� M has parallel or serieselements� Since M has at least four elements� M has a ��separation�

We turn to the remaining case where BGB has a cycle� By Lemma�� M has an MW� minor� We may assume that B displays thatminor via the � � � matrix of �� with four �s� Enlarge that � � �matrix to a maximal submatrix of B containing only �s� Let D be thatsubmatrix� say with rows indexed by a set R and columns indexed by a setS� We use these sets to partition B as shown in �� below� In B� eachrow of the submatrix U and each column of the submatrix V is assumedto be nonzero� By the maximality of D� at least one � must be containedin each row of U and in each column of V �

��

D V 0

U

0

R

PB =

S Q

01

Partitioned matrix B

Let F be the graph obtained from BGB by deletion of the edges corre�sponding to the �s of the submatrix D� Suppose no path of F connectsa node of R with one of S� Let X� resp� Y� be the row resp� columnnodes of F that are connected by some path with some node of R� SinceX� � R� we have jX� � Y�j � �� De�ne X� � X X� and Y� � Y Y��Since Y� � S� we have jX� � Y�j � �� By derivation of F from BGB�any arc of BGB connecting a node of X� � Y� with one of X� � Y� mustcorrespond to a � of D� But X� � R and Y� � S� so the partitioning of Baccording to X�� X�� Y�� and Y� must result in

��D

0

0

R

X2

B =

S Y1

01

01

Y2

X1

Matrix B with ��separation X� � Y��X� � Y�


Since GF��rank D � �� X� � Y��X� � Y� is a ��separation of B� and weare done� So suppose a path in F does connect a node of R with one of S�Choose a path of minimal length� Evidently� the length must be odd andat least ��

Assume the length to be �� Then the center edge of the path corre�sponds in B of �� to a � in some row p � P and some column q � Q�Now row p of U contains a � and a �� say in columns s� � S and s� � S�respectively� Similarly� column q of V has a � and a �� say in rows r� � Rand r� � R� respectively� Then in B of �� the submatrix indexed byr�� r�� p and s�� s�� q is

��r1r2p

s1 s2 q

01 1 11 1 0

11

Submatrix of B representing MW� minor

A GF��pivot on the � in row p and column q produces up to indices thematrix for MW� of �� Thus� M has an MW� minor�

Finally� suppose the shortest path in F from R to S has length greaterthan �� With the path shortening technique� we reduce that path to oneof length �� The related GF��pivots in B of �� do not a�ect thesubmatrices D� U � or V � Thus� the pivots produce the earlier situationwith length ��

Lemma �� supports the following conclusion about ��connectedbinary matroids�

�� Corollary� Every ��connected binary matroid M with at least

six elements has an MW� minor�

Proof� By Lemma �� M has a ��separation or an MW� minor�Since M is ��connected� the former case is not possible�

At times� one would like to claim that a given matroid M has a certainminor containing a speci�ed element� A simple case is given in the followingresult�

�� Lemma� LetM be a connected binary matroid with anMW�minor� Then for every element z of M � there is an MW� minor of Mthat contains z�

Proof� Let N be the assumed MW� minor of M � and z be any elementof M � If z is an element of N � we are done� Otherwise� by Lemma ��M has a connected minor of the form N�z or N�z� Consider the �rstcase� We may assume N�z to be represented by the matrix of �� forMW�� plus one nonzero column indexed by z� A straightforward case

� � Intersection and Partitioning of Matroids ��

analysis proves that for each possible column z� the matroid N�z has anMW� minor with z� Thus� M has an MW� minor containing z as well�The situation for N�z is handled analogously� or by duality�

In the next section� we investigate the intersection and partitioningproblems� The subsequent chapters do not rely on the material of thesection� so it may be skipped without loss of continuity�

�� Intersection and Partitioning of

Matroids

Let M� and M� be two arbitrary matroids de�ned on a common set E andwith rank functions r�� and r�� respectively� The cardinality intersec�

tion problem demands that we �nd a maximum cardinality set Z � E thatis independent in M� and M�� The partitioning problem requires that welocate a partition of E into E� and E� such that r�E� � r�E� is mini�mized� In this section� we present an algorithm that simultaneously solvesboth problems� We call it the intersection algorithm� The method consistsof repeated applications of the path shortening technique� though carriedout in a rather unusual fashion� The algorithm also provides a constructiveproof of a max�min theorem that links the two problems in an unexpectedway�

We have elected to describe the intersection algorithm for general ma�troids instead of just binary ones� since in many� if not most� applications�at least one of the two matroids M�� M� is nonbinary� Correspondingly�the algorithm makes use of abstract matrices� Thus� the reader should befamiliar with the material on abstract matrices in Section �� in particularwith Lemma �� That result says that any triangular submatrix ofan abstract matrix is nonsingular if and only if the submatrix has only �son its diagonal�

We begin with the intersection problem� As stated above� we havetwo matroids M� and M� on a common set E� The algorithm must �nd amaximum cardinality set Z that is independent in both M� and M�� Thescheme begins with any set Z that is independent in both matroids� Forexample� Z � will do� The method iteratively replaces the given set Zby a larger one that is also independent in both matroids� until a set ofmaximum cardinality is found� It su�ces that we describe one iteration�It consists of three steps� which we summarize next�

In step �� we deduce two matrices from certain abstract matrices forM� and M�� We combine the two matrices to a new matrix C that has astrange form� but that actually is a handy encoding of the initial abstractmatrices of M� and M��


In step �� we derive a graph from C and search in that graph fora certain path� Suppose a path of the desired kind can be located� Weinterpret that path in terms of the abstract matrices for M� and M� anddeduce a set Z � that is larger than Z and independent in M� and M�� Withthe set Z � in hand� we terminate the iteration� If a path of the desired kindcannot be found� we go to step ��

In step �� we conclude that the absence of paths of the desired kindimplies a certain partition of the matrix C� We interpret that partitionin terms of the abstract matrices for M� and M� and conclude that Z isoptimal� Thus� we stop� The proof of optimality for Z also shows that thepartition of C implies a partition of the set E into two sets� say E� andE�� that solve the partitioning problem� In addition� the proof establishesthe previously mentioned max�min theorem that connects the intersectionproblem with the partitioning problem�

We begin the detailed description� In step �� we �rst �nd for i � �� abase Xi of Mi that contains the set Z� This is possible since Z is indepen�dent in M� and M�� Let Bi be the abstract matrix of Mi correspondingto the base Xi� Thus� Bi has row index set Xi and column index setE Xi� We adjoin an identity to Bi� getting �I j Bi�� In agreement withthe indexing rules introduced in Section �� we index the columns of thesubmatrix I of �I j Bi� by Xi� Next we permute the columns of �I j B��and �I j B�� such that the columns of the two matrices in same positionhave the same column index� Furthermore� the columns indexed by Z areto become the leftmost columns� Finally� we add zero rows if necessary� sothat both matrices have the same number of rows� For i � �� let Ai bethe matrix so obtained from �I j Bi�� and de�ne Y � E Z� Then thematrix Ai is of the form

�� Ai = Z

Z Y

0

11

1 0 1 ; i = 1, 2

Matrix Ai obtained from �I j Bi�

The rows of Ai without index either are rows of �I j Bi� indexed by XiZ�or are added zero rows�

Before going on� we would like to establish a simple lemma about thematrix Ai of �� The result will allow easy veri�cation that certainsubsets of Mi are independent�

�� Lemma� For some k � �� let Z be a subset of E with k elements�

If the column submatrix of Ai indexed by Z contains a k � k triangular


submatrix that has only �s on the diagonal� then Z is independent in

Mi�

Proof� Except possibly for column permutations and additional zero rows�Ai is the matrix �I j Bi�� Thus� the postulated k � k triangular submatrixof Ai has block triangular form� where one of the blocks is an identitysubmatrix of I� and where the other block is a triangular submatrix of Bi

with only �s on the diagonal� By Lemma �� the set Z must be asubset of a base of Mi� and thus is independent in Mi�

We continue with step �� In the matrix A� resp� A�� we replace each� by � resp� �� getting a matrix �A� resp� �A�� We compute a matrixC � �A� � �A� by adding entries termwise according to the following rule�� and � � � � �� By �� the matrix Cis of the form

�� C = Z

Z

P

Y

0

γγ

γ α β γ0

Matrix C � �A� � �A�

Note the row index subset P in �� The rows of P arise from the rowsof A� and A� shown in �� without index� We consider P to be a newindex set that is disjoint from the index sets of A� and A�� This concludesstep ��

In step �� we �rst examine C of �� for a trivial way of augmentingthe set Z to a larger set Z � that is independent in M� and M�� Speci�cally�assume that C contains a column z that in rows indexed by P has both an� and a �� or a �� Then in the matrices A� and A�� the column submatrices

A�

and A�

indexed by Z � � Z � fzg have triangular submatrices that viaLemma �� prove Z � to be independent in M� and M�� Thus� we canstop the iteration� So from now on� we assume that C has no such columnz� Thus� each column of C contains in the rows indexed by P either just �sand �s� or just �s and �s� or just �s� Let Q� resp� Q� index the columnsof the �rst resp� second kind� Using these two index sets� we partitionC of �� further as shown in �� below� From the submatrix Cde�ned by the row index set Z and the column index set Y of C of ��we construct the following directed bipartite graph G� We start with theundirected bipartite graph BGC� Let i� j be an edge of BGC where iis a row node of Z and j is a column node of Y � If the entry of C producingthat edge is an � resp� �� then we direct that edge from i to j resp� j


to i� If that edge corresponds to a �� then we replace it by two directededges of opposite direction�

��C =

Q1 Q2

Z

Z

P

Y

0 0

γγ

γeachcolumnhas α

eachcolumnhas β

C

Matrix C partitioned by Q� and Q�

Using any convenient shortest route algorithm� we locate a shortest pathfrom Q� to Q�� or determine that no such path exists� A case of the lattervariety is on hand� for example� if Q� or Q� is empty�

If a shortest path does not exist� we go to step �� to be covered shortly�Otherwise� let such a path connect a node q� � Q� with a node q� � Q��

Let C be the submatrix of C de�ned by the nodes of that path� We claim

that C either is the matrix

��

q2q1

U

W

ββα

α

β αno β, γ

no α, γ

C =

Matrix C de�ned by the nodes of the path

or is obtained from the matrix of �� by replacing any number of the

explicitly shown �s and �s by �s� In C of �� the statements no �� and no �� are valid since any violating entry would permit a shorterpath from Q� to Q�� a contradiction� For the same reason� we have for

i � �� Qi �W � fqig� We use the index sets U and W of C to de�neZ � � Z U �W � By �� jW j � jU j� �� so jZ �j � jZj� �� We claimthat Z � is independent in M� and M�� For a proof� we examine the columnsubmatrix C � of C indexed by Z �� By �� and �� the correspondingcolumn submatrices of A� and A� contain triangular matrices that con�rmthe claim� Thus� we have completed the iteration�

As an aside� consider one pivot in B� and B�� each time on a particular� in column q�� Speci�cally� in B� the � corresponds to an � entry of C incolumn q� and in a row of P � and in B� to the � entry of C explicitly shown

in column q� and in the �rst row� say x� of C� Correspondingly� we drop


the element x from Z and add the element q�� getting� say� a set �Z� Thesets Z and �Z have the same cardinality� but if we repeat the above processfor �Z instead of Z� we discover a shorter path� By suitable repetition ofthe above procedure� the path becomes ever shorter until we �nally justadd an element to obtain the set Z ��

Finally� we discuss step �� We enter that step when a path from Q�

to Q� does not exist� Let Z� � Z and Y� � Y be the nodes reachable fromthe nodes of Q�� and de�ne Z� � Z Z� and Y� � Y Y�� The sets Z��Z�� Y�� Y� induce the following partition in C of ��

��

Y2Z2

Z2

C =

Y1Z1

Z1

P

YZ

Z

0

0

0

0 0 0

γγ

γγ α β γ0

eachcolumnhas α

eachcolumnhas β

no β, γ

α β γ0 no α, γ

Matrix C when a path does not exist

In particular� the statements no �� and no �� in the submatricesindexed by Z�� Y� and Z�� Y�� respectively� are correct since otherwise atleast one additional node could be reached from Q�� For i � �� de�neEi � Zi � Yi� By de�nition� E� and E� partition E� From C of ��it is obvious that Ei as subset of the matroid Mi has rank equal to jZij�Furthermore� since Z � Z� � Z�� we have

�� jZj � r�E� � r�E�

Now for any set Z independent in M� and M�� and for any partition of Einto any sets E� and E�� we must have for i � �� jZ �Eij � riZ �Ei �riEi� Adding over i � �� we obtain

�� jZj � jZ �E�j� jZ � E�j � r�E� � r�E�

By �� and �� the set Z on hand in step � solves the intersec�tion problem� and the sets E� and E� found at that time solve the partitionproblem� Thus� the algorithm has solved both problems simultaneously�Clearly� the algorithm is polynomial � indeed� very e�cient�

The preceding arguments also prove the following theorem due to Ed�monds�

�� Theorem Matroid Intersection Theorem� Let M� and M� be

two matroids on a set E and with rank functions r�� and r�� Then

�� max jZj � minfr�E� � r�E�g


where the maximization is over all sets Z that are independent in both M�

and M�� and where the minimization is over all partitions of E into sets

E� and E��

We describe two representative example applications� The �rst exam�ple involves an undirected bipartite graphG� say with edge set E connectingthe nodes of a set V� with those of a set V�� De�ne a matching to be asubset Z of E such that every node of G has at most one edge of Z incident�For i � �� let Mi be the matroid on E where a subset Z is independentif the nodes of Vi have at most one edge of Z incident� It is easily checkedthat Mi is the disjoint union of jVij uniform matroids of rank �� and thus isa matroid� Clearly� the matchings of G are precisely the subsets of E thatare independent in both M� and M�� De�ne a node cover to be a nodesubset such that every edge has at least one endpoint in that subset� Wenow reformulate the matroid intersection Theorem �� to a basic resultof graph theory due to K�onig�

�� Theorem� The cardinality of a maximum matching of a bipar�

tite graph is equal to the cardinality of a minimum node cover�

We leave it to the reader to prove that Theorem �� follows fromTheorem ��

The second application concerns separations in matroids� Suppose wewant to know whether a given matroid M on a set E has a k�separationwith at least k � l elements on each side� for some nonnegative integersk and l� How can we �nd such a separation or prove that none exists�Suppose we can e�ciently solve the following� more restricted� problem�We are given two disjoint subsets F� and F� of E� each of cardinality k� l�We must decide whether M has a k�separation E�� E� such that E� � F�

and E� � F�� If we can solve the restricted problem� then we can solvethe original one by enumerating all possible choices of the sets F� and F��The overall algorithm is polynomial if the possible values of k � l can beuniformly bounded by some constant�

We analyze the restricted problem� De�ne r� to be the rank functionof M � We need to �nd a pair E�� E� such that

��E� � F� E� � F�

rE� � rE� � rE � k �

or prove that such a pair does not exist� An answer to the following prob�lem�

�� min�E��E��

E��F��E��F�

frE� � rE�g

obviously su�ces� We want to eliminate the conditions E� � F� and E� �F� from �� by some matroid construction� Let M� � M!F�nF� and

� � Extensions and References ��

M� � M!F�nF�� with respective rank functions r�� and r�� De�neE � E F� � F�� and for i � �� Ei � Ei �E� Then for i � �� rEiis nothing but rFi � riEi� Thus� rE� � rE� � rF� � r�E� �rF��r�E� � r�E��r�E�� a constant� Thus� equivalent to ��is the problem

�� min�E��E��

fr�E� � r�E�g

which can be solved by the intersection algorithm� For uniformly boundedk � l� the above scheme is polynomial� But from a practical standpoint�even small bounds on k � l� such as � or �� are likely to make the schemepractically unusable� So one might want to explore other avenues to �nd k�separations� In the next chapter� we learn about a method for a particularcase that turns out to be very important for several settings� Speci�cally�we will have k � � and l � ��

In the next section� we point out additional material about intersectionand partitioning problems� and list references�

�� Extensions and References

The path shortening technique is introduced in Truemper �� As wehave seen in Section �� the technique fully applies to abstract representa�tion matrices of general matroids� Thus� virtually all results of Section ��can be translated to almost identical ones for general matroids� One ad�ditional class of nonbinary matroids must be introduced� though� It is theclass of whirls Wn� We de�ne these matroids next� A whirl of rank n � �is derived from the wheel matroid with same rank by declaring the circuitcontaining the elements of the rim edges to be independent� Small whirlsare represented over GF� by the matrices of �� below� In general� forn � �� Wn is represented over GF� by the matrix of �� below� where� � f��g is so chosen that the GF��determinant of the matrix is ��Equivalently� the real sum of the entries must be �mod �� Note that W�

is also U�� the uniform matroid of rank � on four elements�

��0

1

001 1

1

1 1 000 0 -1

10

01 1 01 0 1

11111 1

-1

W� W� W� W�

Representation matrices over GF� for whirls W�"W�


��11 .. ..

11

1

α

1 1

Representation matrix over GF� for whirl Wn� n � �

Lemmas �� and �� are valid for general matroids� Lemmas ��and �� and Corollary �� are readily extended to the general caseby allowing occurrence of the whirl W� as possible minor besides the givenwheel matroids� All these results� including the extensions to nonbinarymatroids� are implicit in Whitney �� and Tutte ��

We have no reference for Lemma �� but that result is well known�The nonbinary version of that lemma is proved in Bixby �� It involvesU�� instead of MW�� and claims the existence of a U�

� minor with z� asfollows�

�� Lemma� Let M be a connected matroid with a U�� minor� Then

for every element z of M � there is a U�� minor of M that contains z�

The proof is virtually identical to that of Lemma �� except that ab�stract matrices are used here� The result motivated a long series of papersconcerning the presence of speci�ed elements in given minors Seymour��e� ��b� ��a� ��b� Oxley �� a� ��a� Kahn�� Coullard �� Coullard and Reid �� Oxley and Row ��Oxley and Reid �� and Reid �� a� ��b� ��

The matroid intersection and partitioning results of Section �� arejust a small sampling of a wealth of material� The roots of these problemscan be traced back to several matching results of which Theorem ��due to K�onig �� is an example� Lov#asz and Plummer �� give avery complete account of these developments� Other early results relatedto matroid intersection and partitioning are the solution of the problem ofpartitioning a graph into forests in Nash�Williams �� and Tutte�� and the solution of the so�called optimum branching problem� �rstin Chu and Liu �� and later in Edmonds ��a� Bock �� andKarp ��

Edmonds ��a� �� introduced and solved the matroidintersection and partitioning problems� as well as generalizations to ma�troids with weighted elements and to so�called polymatroids� He provedthat these problems can be converted to structurally simple linear programssince certain polytopes have only integer vertices� In the terminology oflinear programming� the intersection problem is then the linear program�ming dual of the partitioning problem� This work and Edmonds$s profoundresults for matching problems for a complete coverage� see Lov#asz andPlummer �� establish Edmonds as the founder of polyhedral combi�


natorics� For applications� generalizations� and other algorithms� see Leh�man �� Edmonds ��b� ��b� Edmonds and Fulkerson ��Lawler �� Frank �� Hassin �� Lawler and Martel��a� ��b� Gr�otschel� Lov#asz� and Schrijver �� and Fujishige�� Schrijver �� houses many of the problems treated inthose references under one roof�

As far as we know� the graph formulation of the intersection algorithmdescribed in Section �� is due to Krogdahl see Lawler �� exceptfor our use of abstract matrices to simplify the arguments� In Cunning�ham �� a considerably faster version of the algorithm is given� Theimprovement rests on Menger$s Theorem and the observation that the pathused to derive the set Z � from the given set Z should be chordless� butneed not be shortest� Incidentally� Menger$s Theorem may also be used todetermine� for the given set Z� a set R � EZ of minimum cardinality sothat Z becomes an independent set of maximum cardinality in the minorsM�nR and M�nR of M� and M�� The graph approach may also be usedto improve the already very appealing algorithm of Frank �� for theintersection problem with weighted elements�

The second application cited at the end of Section �� which involvescertain k�separations of a matroid� is due to Cunningham �� and Cun�ningham and Edmonds ��

Finally� we should mention that the intersection case involving at leastthree matroids� is in general NP�hard since it includes the NP�completeHamiltonian cycle problem see Garey and Johnson ��

Chapter �

Separation Algorithm

�� Overview

So far� we have described two simple matroid tools� the series�parallel anddelta�wye constructions of Chapter �� and the path shortening technique ofChapter �� In this chapter� we introduce a third tool called the separationalgorithm� Before we summarize that method and some of its uses� letus recall from Section �� some denitions and results concerning matroidseparations� Let M be a binary matroid on a set E and with rank functionr�� Furthermore� let B be a binary representation matrix of M with rowindex set X and column index set Y � Suppose two sets E� and E� partitionE� For i � � �� dene Xi � Ei �X and Yi � Ei � Y � Then E�� E�� is ak�separation of M � provided the matrix B when partitioned as

�� X1B =

Y1

B1

D1X2

Y2

B2

D2


satises

�� jX� � Y�j� jX� � Y�j � k

GF��rank D� �GF��rank D� � k �

��

� � Separation Algorithm ��

The separation is exact if the inequality of �� involving the GF��rankof D� and D� holds with equality� In terms of E�� E�� E� and the rankfunction r�� the conditions of �� are

�� jE�j� jE�j � k

rE�� rE�� rE� � k �

In the case of an exact separation� the inequality involving the rank functionholds with equality� Finally� for k � �� the matroid M is k�connected if itdoes not have an l�separation for any � l � k�

We are ready to summarize the material of this chapter� In Section ��we describe and validate the just�mentioned separation algorithm� Thescheme solves the following problem� Given is a binary matroid M with aminor N � For some k � � an exact k�separation F�� F�� is known for N �We want to decide whether or not M has a k�separation E�� E�� wherefor i � � �� Ei � Fi� In the a�rmative case� we say that F�� F�� induces

the k�separation E�� E��The problem of nding induced separations may seem rather technical�

But several important matroid results can be derived from its solution�Two such results are included in Sections �� and �� The rst resultprovides su�cient conditions for the existence of induced separations� Inlater chapters� we rely upon these conditions to prove the existence of anumber of decompositions� The second result builds upon the rst one�It concerns the existence of certain extensions of ��connected minors in��connected binary matroids� We use that result in the next chapter toestablish the so�called splitter theorem and the existence of some sequencesof minors� Finally� in Section �� we sketch extensions of the results to thenonbinary case and provide references�

The chapter relies on the material of Chapters �� and ��

�� Separation Algorithm

Suppose we are given a binary matroid M on a set E� Let N be a minorof M on a set F � E� Assume that N has� for some k � � an exact k�separation F�� F�� We want to know whether or notM has a k�separationE�� E�� where for i � � �� Ei � Fi� If such E�� E�� exists� we declareit to be induced by F�� F�� In this section� we describe a simple methodcalled the separation algorithm for deciding the existence of E�� E��

We begin with an informal discussion that relies on a particular binaryrepresentation matrix BN of N � Let X� be a maximal independent subsetof N contained in F�� Then select a subset X� from F� so that X� �X�

is a basis of N � For i � � �� let Yi � Fi �Xi� The desired representation

�� Chapter � Separation Algorithm

matrix BN of N corresponds to the base X� �X� of N � By the derivationof that base� BN is of the form

��

Y1 Y2

X1BN =

A1

DX2 A2

0

Partitioned version of BN

for some A�� A�� and D� Indeed� the zero submatrix indexed by X� andY� is present since X� is a maximal independent subset of F�� and sinceY� � F� �X�� By assumption� F�� F�� is an exact k�separation of N � sowe have by ��

��jX� � Y�j� jX� � Y�j � k

GF��rank D � k �

We embed BN into a representation matrix ofM � and thus make the minorN visible� as follows� Since X� �X� is independent in N � that set is alsoindependent inM � Thus� we can nd a set X� � E�F so that X��X��X�

is a base of M � Let Y� � E�F ��X�� The representation matrix B of Mfor this base contains BN as submatrix� We depict B below� For reasonsto become clear shortly� we have placed A�� A�� D� and the � submatrix ofBN into the corners of B�

��

Y1

A2

A1

D

Y2Y3

X3B =

X2

X1 0

0 1

Matrix B for M displaying partitioned BN

Recall that we want to nd a k�separation E�� E�� of M where for i � � ��Ei � Fi� or to prove that no such k�separation exists� In terms of the indexsets of �� we want to partition the set X� into X�� X�� and the setY� into Y�� Y�� so that the correspondingly rened matrix B of �� is


of the form

�� A1~

Y1 Y31 Y32

A2

B =

A1

D

Y2

Y3

X3

X31

X32

X2

X1

0

0

D~

A2~

Partition of B induced by that of BN

with GF��rank �D � k � � or we are to prove that such partitions of X�

and Y� do not exist� In the a�rmative case� D is a submatrix of �D� andsince GF��rank D � k � � we must have GF��rank �D � k � � Putdi�erently� if an induced k�separation of M exists at all� then it must bean exact induced k�separation�

We employ a recursive scheme to decide whether or not an inducedk�separation exists� As the measure of problem size for the recursion� weuse jX��Y�j� If jX��Y�j � �� then M � N � and for i � � �� Ei � Fi givesthe desired induced k�separation of M � Suppose jX� � Y�j � �� Redraw Bof �� so that an arbitrary row x X� and an arbitrary column y Y�are displayed as follows�

��

Y2Y1 y

A2

B =

g

α

h

A1

e

D

Y3

X3

x

X2

X1 0

f

Matrix B for M with partitioned BN �row x X�� and column y Y�

The recursive method relies on the analysis of the following three cases ofB of �� Collectively� these cases cover all situations�

In the rst case� we suppose that for some row x X�� the subvector eis not spanned by the rows ofD� We claim that in any induced k�separation�we must have x X�� For a proof� take any such separation as depictedby �� If x X�� then the subvector e of row x occurs in �D� Since eis not spanned by the rows of D� we have GF��rank �D � GF��rank D�


which contradicts the condition GF��rank �D � GF��rank D� Thus� xmust be in X�� as claimed� We now examine the subvector f of row x�Suppose that subvector is nonzero� Using �� once more� we see thatin any induced k�separation� the nonzero f forces x to be in X�� But thelatter requirement con�icts with the one determined earlier for x� Thus� aninduced k�separation cannot exist� and we stop with that conclusion� Sowe now assume the subvector f of row x to be zero� We know already thatx must be in X�� in any induced k�separation� Suppose in B of �� weadjoin e to A� and f to the explicitly shown � submatrix� getting a new A�

and a new � submatrix� Correspondingly� we extend N by x to N�x andredene N to be the extended matroid� Evidently� X� �fxg�Y��X� �Y��is a k�separation of the new N � and that k�separation induces one in M ifand only if this is so for the k�separation X� � Y��X� � Y�� of the originalN � Thus� we may replace the original problem by one involving the newN � By our measure of problem size� the new problem is smaller than theoriginal one� and we may apply recursion�

In the second case� we suppose that for some column y Y�� thesubvector g is nonzero� Arguing analogously to the rst case via �� weconclude that y must be in Y�� in any induced k�separation� Furthermore�suppose that the column subvector h of column y is not spanned by thecolumns of D� Using �� once more� we see that y must also be in Y�� inany induced k�separation� Thus� an induced k�separation cannot exist� andwe stop with that conclusion� So suppose that h is spanned by the columnsof D� Then we adjoin g to A�� h to D� and correspondingly redene N tobecome N�y� Then the k�separation X� � Y� � fyg�X� � Y�� of the newN induces a k�separation of M if and only if this is so for the k�separationX��Y��X��Y�� of the originalN � Once more� we may replace the inducedk�separation problem involving the originalN by one with the new N � Thelatter problem is smaller� and we may invoke recursion�

For the discussion of the third and nal case� we suppose that neitherof the above cases applies� Equivalently� for all x X�� the subvector e ofrow x is spanned by the rows of D� and for all y Y�� the subvector g ofcolumn y is zero� By �� X� � Y��X� �X� � Y� �Y�� is a k�separationof M induced by the one of N � and we stop with that conclusion�

We call the above recursive method the separation algorithm� It clearlyhas a polynomial implementation� For later reference� we summarize thealgorithm below�


� Suppose B of �� has a row x X� with the indicated row sub�vectors e and f such that GF��rank �e�D� � GF��rank D� Then xmust be in X�� Suppose� in addition� that f is nonzero� Then x mustalso be in X�� i�e�� B cannot be partitioned� and we stop with thatdeclaration� On the other hand� suppose f � �� Since x must be in


X�� we adjoin e to A�� and f to the explicitly shown � matrix� Thenwe start recursively again with the new B and the new BN �

�� Suppose B of �� has a column y Y� with the indicated columnsubvectors g and h such that g is nonzero� Then y must be in Y��Suppose� in addition� GF��rank �D j h� � GF��rank D� Then ymust also be in Y�� i�e�� B cannot be partitioned� and we stop withthat declaration� On the other hand� suppose GF��rank �D j h� �GF��rank D� Since y must be in Y�� we adjoin g to A�� and h toD� Then we start recursively again with the new B and the new BN �

�� Finally� suppose that for all rows x X�� the row subvector e satisesGF��rank �e�D� � GF��rank D� and that for all columns y Y��the column subvector g is �� Then X�� Y�� Y�� Y�� Y�� Y�gives the desired partition of B�

In the next two sections� we put the separation algorithm to good use�Preparatory to that discussion� we establish in the next lemma that certainextensions of binary matroids are ��connected�

�� Lemma� Let N be a ��connected binary matroid on at least sixelements� Suppose a �� or ��element binary extension of N � say M �has no loops� coloops� parallel elements� or series elements� Then M is��connected�

Proof� Let C be a binary representation matrix of M that displays arepresentation matrix� say B� for N � By assumption� B is ��connected�Suppose C is not connected� A straightforward case analysis proves C tocontain a zero vector or unit vector� Thus� M has a loop� coloop� parallelelements� or series elements� a contradiction� Hence� C is connected� If Cis not ��connected� then by Lemma �� there is a ��separation of Cwith at least ve rows�columns on each side� Then� necessarily� the matrixB has a ��separation with at least two rows�columns on each side� anothercontradiction� We conclude that C� and hence M � are ��connected�

From Lemma �� we deduce the following result for �edge exten�sions of ��connected graphs�

�� Lemma� Let H be a ��connected graph with at least six edges�Then a connected �edge extension of H is ��connected if and only if it isproducible as follows� Either two nonadjacent vertices of H are connectedby a new edge� or a vertex of degree at least � is partitioned into twovertices� each of degree at least �� and the two new vertices are connectedby a new edge�

Proof� The described extension steps are precisely the ways in which Hcan be extended by one edge to a larger connected graph that does not


have loops� coloops� parallel edges� or series edges� Thus� the �only if�part is obvious� The �if� part follows from Lemma ��

We are prepared for the next section� where we derive su�cient con�ditions for the existence of induced separations under various assumptions�

�� Su�cient Conditions for Induced

Separations

In this section� we employ the separation algorithm to establish su�cientconditions under which induced separations can be guaranteed� Beforewe begin with the detailed discussion� we describe the general setting inwhich these conditions will be invoked� To this end� let M be a class ofbinary matroids� The class is assumed to be closed under minor�taking andisomorphism�

We select a matroid N M� say on set F � Suppose by some methodwe nd� for some k � �� an exact k�separation F�� F�� for N � At thatpoint� we would like to claim the following�

��

��

Suppose an M M has an N minor� say N � � Let F �� F �

��

be a k�separation of N � that corresponds to F�� F�� underone of the isomorphisms between N � and N � Then the k�separation F �

�� F �

�� of N � induces a k�separation of M �

Results of type �� are valuable if an M M is known to have anN minor� and if the induced k�separation of M may be employed to e�ecta useful decomposition ofM � In Chapters �� we will see that these twoassumptions are satised in a number of cases� Thus� nontrivial instancesof �� are indeed useful�

The technique for proving �� is in principle straightforward� Withmachinery yet to be described� we compute all minimal binary matroidssatisfying the assumptions of �� but not its conclusion� If no suchmatroid is in M� then �� indeed holds�

Application of the technique entails two di�culties� First�M� N � andF�� F�� must be properly selected� Second� we need structural insight andcomputational tools to prove that M has no M for which �� fails�In this section� we ignore the rst aspect� It will be treated in depthin Chapters �� Instead� we concentrate on the development of thestructural insight and of the computational tools�

We break down that development into two phases� In the rst one�we accomplish the following task� where N is the previously mentionedmatroid with exact k�separation F�� F��

� � Sufficient Conditions for Induced Separations ��

��

��

Find computationally tractable properties of minimal bi�nary matroids M that have N as a minor� but that donot have a k�separation induced by the exact k�separationF�� F�� of N �

An M satisfying the conditions of �� we simply call minimal� Inthe second phase� we expand the task of �� to the task �� below� Itessentially says that the requirements of �� are to hold for some minorisomorphicN � and that M is to be minimal with respect to that condition�The precise statement is as follows�

��

��

Find computationally tractable properties of binary ma�troids M satisfying the following conditions� M must haveat least one N minor� Some k�separation of at least onesuch minor corresponding to F�� F�� of N under one of theisomorphismsmust fail to induce a k�separation of M � Thematroid M is to be minimal with respect to those condi�tions�

We say that an M satisfying the conditions of �� is minimal under

isomorphism� Evidently� minimality under isomorphism demands morethan the previously dened minimality�

Answers to �� give su�cient conditions so that �� holds� Thatis� if none of the binary matroids M with the yet�to�be�determined prop�erties of �� is in M� then necessarily �� must hold� Thus� answersto �� e�ectively supply su�cient conditions under which �� is sat�ised�

We begin with the task �� We are given a binary matroid Non a set F � In the next lemma� we rely on rank functions instead ofrepresentation matrices� Thus� we let rN �� be the rank function of N � Forsome k � � we have an exact k�separation F�� F�� of N � Thus� by ��

��jF�j� jF�j � k

rN F�� rN F�� rN F � � k �

LetM be any binary matroid on a set E and with rank function rM ��Assume that M has N as a minor� and that F�� F�� does not induce a k�separation of M � Thus� the system

��E� � F�� E� � F�

rM E�� rM E�� rM E� � k �

has no solution� Assume M to be minimal as dened above� Thus� everyproper minorM � ofM withN as a minor has a solution for an appropriatelyadapted �� First� we show that there is a unique partition of the setE � F into Z� and Z� so that N � M�Z�nZ��


�� Lemma� Let M � E� N � F � and F�� F�� be as de�ned above�Suppose �� has no solution and that M is minimal� Then for all z E � F �� M�z or Mnz does not have N as a minor�

Proof� Suppose both M�z and Mnz have N as a minor� By the minimal�ity of M � they both have induced separations� say U�� U�� for M�z andW��W�� for Mnz� Let r

M�z�� and rMnz�� be the rank functions of M�z

and Mnz� According to ��

��

U��W� � F�� U��W� � F�

rM�zU�� r

M�zU�� rM�zE � fzg� � k �

rMnzW�� rMnzW�� rMnzE � fzg� � k �

By the minimality of M � z cannot be a loop or coloop of M � Thus� the twoinequalities of �� imply the inequalities

��rM U� � fzg� � rM U� � fzg� � rM E� � krM W�� rM W�� rM E� � k �

We add the two inequalities of �� and apply submodularity to get

�� rM U� � fzg �W�� rM U� � fzg� �W��

� rM U� � fzg �W�� rM U� � fzg� �W�� rM E� � �k �

But each one of U� � fzg � W�� U� � fzg� � W�� and U� � fzg �W�� U� � fzg� �W�� is a pair E�� E�� satisfying E� � F� and E� � F��Since �� cannot be satised� we have

�� rM U� � fzg �W�� rM U� � fzg� �W�� rM E� � k

rM U� � fzg �W�� rM U� � fzg� �W�� rM E� � k

Summing the latter two inequalities� we obtain a contradiction of��

For further insight into the structure of a minimal M � we employ theseparation algorithm of Section �� That is� we have the representation


matrix B of �� for M � repeated here for ease of reference�

��

Y2Y1 y

A2

B =

g

α

h

A1

e

D

Y3

X3

x

X2

X1 0

f

Matrix B for M with partitioned BN �row x X�� and column y Y�

The submatrix of B composed of A�� A�� D� and the explicitly shown �matrix is BN of �� which we also repeat here� The latter matrixrepresents N �

��

Y1 Y2

X1BN =

A1

DX2 A2

0

Partitioned version of BN

We apply the separation algorithm to search for a partition as given by�� Since �� cannot be satised� the algorithm terminates in step or � announcing that no partition with the desired properties exist� Below�we list that algorithm again� with references adapted to the just�denedmatrices�


� Suppose B of �� has a row x X� with the indicated row sub�vectors e and f such that GF��rank �e�D� � GF��rank D� Thenx must be in X�� Suppose� in addition� that f is nonzero� Then xmust also be in X�� i�e�� B cannot be partitioned� and we stop withthat declaration� On the other hand� suppose f � �� Since x must bein X�� we adjoin e to A�� and f to the explicitly shown zero matrix�Then we start recursively again with the new B and the new BN �

�� Suppose B of �� has a column y Y� with the indicated columnsubvectors g and h such that g is nonzero� Then y must be in Y��Suppose� in addition� GF��rank �D j h� � GF��rank D� Then ymust also be in Y�� i�e�� B cannot be partitioned� and we stop with


that declaration� On the other hand� suppose GF��rank �D j h� �GF��rank D� Since y must be in Y�� we adjoin g to A�� and h toD� Then we start recursively again with the new B and the new BN �

�� Finally� suppose that for all rows x X�� the row subvector e satisesGF��rank �e�D� � GF��rank D� and that for all columns y Y��the column subvector g is �� Then X�� Y�� Y�� Y�� Y�� Y�gives the desired partition of B�

By �� for any x X� and any y Y�� the minorsM�x andMnyof M have N as a minor� Thus� by the minimality of M � the separationalgorithm does nd a partition if we delete any row x X� or any columny Y� from B�

We now prove some results about the structure of a matrix B producedby a minimal M � We start with the special case where B of a minimal Mcontains just one row or column beyond that of BN � Consider the case of asingle additional row x� In the notation of �� X� � fxg and Y� � �By step of the separation algorithm� the row subvectors e and f of rowx satisfy

�� e is not spanned by the rows of D� and

f is nonzero

Similarly� we deduce for the case of a single additional column y� i�e�� whenX� � and Y� � fyg�

�� g is nonzero� and

h is not spanned by the columns of D

We now treat the remaining cases� where B has at least two additionalrows or columns beyond those of BN � Thus� jX� � Y�j � �� We want toshow that both X� and Y� are nonempty� and that there exist x X� andy Y� so that the subvectors e� f of row x� and g� h of column y� as wellas the scalar �� obey certain conditions� First we prove the following factabout the subvectors e of the rows x X� and about the subvectors g ofthe columns y Y��

�� Lemma� Exactly one of the two cases i� and ii� below applies�

i� There is exactly one row x X� such that the subvector e is notspanned by the rows of D� In that row x� the subvector f is zero�Furthermore� for all y Y�� the subvector g of column y is zero�

ii� There is exactly one column y Y� such that the subvector g isnonzero� In that column y� the subvector h is spanned by the columnsof D� Furthermore� for all x X�� the subvector e of row x is spannedby the rows of D�


Proof� If the condition about f or h does not hold� then we have thesmaller case of �� or �� a contradiction�

To prove the claims about e and g� we apply the separation algo�rithm to B� Consider each application of steps and � except for thelast application� when the algorithm stops� In each such application� arow x X� or column y Y� is moved to X�� or Y�� and the re�lated row subvector f satises f � �� or the column subvector h satisesGF��rank �D j h� � GF��rank D� Exactly one of these conditions is vi�olated in the last iteration� The rows and columns moved to X�� and Y��plus the row or column encountered in the last application� su�ce to provethat B has no induced partition� Thus� by the minimality of M � these rowsand columns comprise the rows and columns that B has beyond those ofBN � We conclude that in each row x X�� the row subvector f is zeroexcept for at most one such vector� and that in each column y Y�� thecolumn subvector h is spanned by D except for at most one such vector�Furthermore� if there is a nonzero f � then all vectors h are spanned by D�and if there is an h not spanned by D� then all vectors f are zero� Toprove the claims about e and g� we use duality� or equivalently� we applythe above arguments to Bt� Then g plays the role of f above� and e that ofh� The conditions just proved for f and h establish the statements aboute and g of the lemma�

We investigate the two cases of Lemma �� further� We beginwith the situation where a unique row x X� has a subvector e that isnot spanned by the rows of D� and where for all y Y�� the subvector g ofcolumn y is zero� The next lemma tells more about the columns y Y��

�� Lemma� Suppose case i� of Lemma �� applies� Thenthere exists a y Y� such that the scalar � of column y is and thesubvector h is nonzero�

Proof� As in the proof of Lemma �� we apply the separation algo�rithm to B� By the assumptions� in the rst iteration the row x is movedto X�� By Lemma �� in the next recursive application of the sep�aration algorithm� step � must apply� Thus� a column y Y� is movedto Y�� By Lemma �� the subvector g of that column is zero� Then� � since otherwise step � could not move column y to Y��

Suppose h is zero� Since g is also zero� we can pivot on � � with�out disturbing the submatrix BN � This implies that both Mny and M�yhave N as a minor� in violation of Lemma �� Thus� h must benonzero�

Consider now the second case of Lemma �� Thus� B has aunique column y Y� with nonzero subvector g� and the subvector h isspanned by the columns of D� Furthermore� for all x X�� the subvector


e of row x is spanned by the rows of D� Analogously to Lemma �� we have the following result�

�� Lemma� Suppose case ii� of Lemma �� applies� Thenthere exists an x X� such that the subvector e is spanned by the rows ofD� If the subvector f is zero� then the subvector e is nonzero� Furthermore�the subvector �e j �� is not spanned by the rows of �D j h��

Proof� Apply the separation algorithm to B� In the rst iteration� the col�umn y is moved to Y�� In the next recursive application of the separationalgorithm� step must apply� Thus� a row x X� is moved to X�� Thesubvector e must be spanned by the rows of D� but the subvector �e j �� ofrow x is not spanned by the rows of �D j h��

Suppose f is zero� If e is also zero� then necessarily � � � A pivot on� does not disturb the submatrix BN � Thus� both M�x and Mnx containN as a minor� a contradiction of Lemma �� Thus� e is nonzero�

With �� and Lemmas �� and �� we assemblethe following theorem�

�� Theorem� Let M be minimal� and let B be the representationmatrix of �� for M �

a� Suppose X� � fxg and Y� � � Then the subvector e of row x is notspanned by the rows of D� and f is nonzero�

b� Suppose X� � and Y� � fyg� Then the subvector g of column y isnonzero� and h is not spanned by the columns of D�

c� Suppose jX� � Y�j � �� Then either c� � or c�� below applies forsome x X� and y Y��

c� � The subvector e of row x is not spanned by the rows of D� andf is zero� The subvector g of column y is zero� � � � and thesubvector h is nonzero�

c�� The subvector g of column y is nonzero� and the subvector h isspanned by the columns of D� The subvector e is spanned bythe rows of D� If the subvector f is zero� then the subvector eis nonzero� The subvector �e j �� is not spanned by the rows of�D j h��

Proof� Statements �� and �� establish a� and b�� Lemmas�� and �� prove parts c� � and c��

For our purposes� Theorem �� su�ces as answer for the task�� Thus� we turn to the task �� That problem demands that wend computationally tractable properties of binary matroids M that areminimal under isomorphism� That is� any such M has a minor isomorphicto N � For at least one such minor the following holds� Some k�separationof that minor corresponds to F�� F�� of N under one of the isomorphisms�


and fails to induce a k�separation of M � We want M to be minimal withrespect to these conditions�

Let B of �� be the representation matrix of anM that is minimalunder isomorphism� Since minimality under isomorphism implies the min�imality dened for �� B observes the conditions of Theorem �� Evidently� minimality under isomorphism is a stronger requirement thanminimality� Thus� we expect e� f � g� h� and � of Theorem �� to obeyadditional conditions� The next lemma supplies computationally tractableones� The notation is that of Theorem ��

�� Lemma�

c� � If case c� � of Theorem �� applies� then the following holds�

c� � � The subvector e of row x of B is not parallel to a row of thesubmatrix A��

c� �� Suppose column z Y� of A� is nonzero� Then the subvector

e of row x of B is not a unit vector with in column z of B�

c�� If case c�� of Theorem �� applies� then the following holds�

c�� Suppose D� the matrix obtained from D by deletion of acolumn z Y� of D� has the same GF��rank as D� Thenthe subvector �g�h� of column y of B is not parallel to columnz of �A��D��

c�� Suppose the rows of D do not span a row z X� of A��Then �g�h� is not a unit vector with in row z�

Proof� c� � �� For a proof by contradiction� suppose the subvector e ofrow x of B is parallel to a row z X� of A�� In B� we exchange therows x and z� and appropriately adjust X� to X �

�� X� � fzg� � fxg and

X� to X �� X� � fxg� � fzg� By c� � of Theorem �� f is zero�

Thus� the submatrix of B indexed by X �� X�� Y�� and Y� is BN except

for the change of the index z to x� Let N � be the corresponding minorof M � We know that N�x does not induce a k�separation of M � Thus�N ��z � N�x does not induce one either� The same conclusion applies toN �� since row z contains e� Now M is minimal under isomorphism� so Mmust be minimal with respect to N �� We show that the latter conclusionleads to a contradiction� For the proof� let us examine the e�ect of theexchange of rows x and z on column y Y�� By that exchange� the role ofthe zero subvector g indexed by X� is taken on by a vector g� indexed byX �

�� By c� � of Theorem �� the entry � in row x is nonzero� Thus�

the vector g� is nonzero� Apply Lemma �� to N � and the subvectorse and g�� Since e is not spanned by the rows of D and since g� is nonzero�these subvectors violate the conclusions of that lemma� and thus providethe desired contradiction�

c� �� Suppose that column z Y� of A� is nonzero� and that e is a unitvector with in column z of B� Perform a pivot in column z Y� of A��


Then the subvector e becomes parallel to a row of A�� and the above casec� � � applies�

c�� Suppose D� the matrix obtained from D� by deletion of a columnz Y� of D� has the same GF��rank as D� Further assume that thesubvector �g�h� of column y of B is parallel to column z of �A��D�� Ex�change columns y and z of B� Adjust Y� to Y �

�� Y� � fzg� � fyg and

Y� to Y �� Y� � fyg� � fzg� The swap of columns e�ectively replaces N

by an isomorphic minor N � and modies D to D� and e to e�� A simplerank calculation conrms that under the assumption on D� the rows of D�

do not span e�� Arguing analogously to the case c� � �� M is not minimalunder isomorphism�

c�� By a pivot in row z of A�� this case becomes c�� as is readilychecked�

We summarize the preceding conclusions in the next theorem� whichnishes the task �� The statement of the theorem is rather detailedto simplify its application�

�� Theorem� Let M be minimal under isomorphism� Then oneof a�� b�� or c� below holds�

a� M is represented by

�� x

Y2Y1

X1

B =

A1

DX2 A2

e f

0

Matrix B for M minimalunder isomorphism� case a�

In row x� e is not spanned by the rows of D� and f is nonzero�

b� M is represented by

��B

Y1

X2

Y2

X1 A1

D A2

0=

y

g

h

Matrix B for M minimalunder isomorphism� case b�

In column y� g is nonzero� and h is not spanned by the columns of D�


c� M has a minor M with representation matrix

��

Y1

x

Y2

X1 A1

DX2 A2

e f

0

y

g

h

αB =

Matrix B for minor Mof M minimal under isomorphism

Either c� � or c�� below holds for e� f � g� h� and ��

c� � e is not spanned by the rows of D� f � �� g � �� h �� e isnot parallel to a row of A�� If column z Y� of A� is nonzero� thene is not a unit vector with in column z of B�

c�� g �� h is spanned by the columns of D� e is spanned by the rows ofD� f � � implies e �� e j �� is not spanned by the rows of �D j h��If D� the matrix obtained from D by deletion of a column z Y��has the same GF��rank as D� then �g�h� is not parallel to columnz of �A��D�� If the rows of D do not span a row z X� of A�� then�g�h� is not a unit vector with in row z�

Proof� The statements follow directly from Theorem �� and Lemma��

Recall our main goal for this section� We want to determine su�cientconditions for induced separations� In the next corollary� we deduce suchconditions from Theorem ��

�� Corollary� Let M be a class of binary matroids that is closedunder isomorphism and under the taking of minors� Suppose that N givenby BN of �� is in M� but that the � and ��element extensions of Ngiven by �� and by the accompanying conditionsare not in M� Assume that a matroid M M has an N minor� Then anyk�separation of any such minor that corresponds to X� � Y��X� � Y�� ofN under one of the isomorphisms induces a k�separation of M �

Proof� Take M M satisfying the assumptions� We know M to beclosed under isomorphism� Thus� we may suppose that the N minor ofM is N itself� Suppose the k�separation of N does not induce one in M �Then M � or a minor of M containing N � is minimal under isomorphism�By Theorem �� M has a minor represented by one of the matricesof �� Since M is closed under minor�taking� anysuch minor of M is in M� But presence of such a minor in M is ruled outby assumption� a contradiction�


Sometimes an abbreviated version of Corollary �� su�ces to pro�duce the desired conclusion of induced separations� The following result isone such version�

�� Corollary� Let M be a class of binary matroids that is closedunder isomorphism and under the taking of minors� Suppose a ��connectedN given by BN of �� is inM� Assume that N�X��Y�� has no loopsand that NnX� � Y�� has no coloops� Furthermore� assume for every ��connected �element extension of N in M� say by an element z� that thepair X� � Y��X� � Y� � fzg� is a k�separation of that extension� Then forany ��connected matroid M M with an N minor� the following holds�Any k�separation of any such minor that corresponds to X� �Y��X� �Y��of N under one of the isomorphisms induces a k�separation of M �

Proof� Suppose the conclusion is false� By Corollary �� M con�tains a matroid M represented by one of the matrices �� or �� We rst dispose of the cases �� and �� The as�sumed ��connectedness of BN and the conditions of Theorem �� onthe matrices of �� and �� imply that these matrices do not con�tain zero vectors� unit vectors� or parallel vectors� Then by Lemma ��these matrices represent ��connected �element extensions of N � By as�sumption� any ��connected �element extension of N does have an inducedk�separation� Hence� the cases �� and �� cannot occur�

Consider �� case c� �� Delete column y from that matrix� Thereduced matrix represents N�x� We claim that N�x is ��connected� Byc� �� the subvector e of row x is not spanned by the rows of D� It also isnot parallel to a row of A�� Now N�X� � Y�� has no loop� so A� has nozero columns� Then by c� �� e is not a unit vector� By Lemma ��N�x is ��connected as claimed� Since e is not spanned by the rows of D�X� � Y��X� � Y� � fxg� is not a k�separation of N�x� By assumption�N�x cannot be in M� Yet N�x is a minor of M M� a contradiction�

Consider �� case c�� We rst establish an auxiliary result�Suppose that for some z Y�� deletion of column z from D reduces theGF��rank� or that for some z X�� the rows of D span row z of A��We claim that z is a coloop of NnX� � Y�� contrary to assumption� Fora proof� we delete from �I j BN � the columns indexed by X� � Y�� By asimple rank calculation� every basis of the reduced matrix contains columnz� This establishes the claim�

By c�� and the auxiliary result� g and h of �� satisfy the fol�lowing conditions� g �� and �g�h� is not parallel to a column of �A��D�and is not a unit vector� Then by Lemma �� and �� N�y is��connected� and X� � Y��X� � Y� � fyg� is not a k�separation of N�y�Thus� N�y cannot be in M� Yet N�y is a minor of M M� a contradic�tion�

In Chapter �� we require the graph version of Corollary �� for


k � �� For that situation� we adapt the above matroid language as follows�Suppose we have ��connected graphs G and H� On hand is a ��separationF�� F�� for H� Then that ��separation of H induces one for G if the lattergraph has a ��separation E�� E�� where E� � F� and E� � F�� Here isthe special graph version of Corollary �� for k � ��

�� Corollary� Let G be a class of connected graphs that is closedunder isomorphism and under the taking of minors� Let a ��connectedgraph H G have a ��separation F�� F�� with jF�j� jF�j � �� Assumethat H�F� has no loops and HnF� has no coloops� Furthermore� assumethat for every ��connected �edge extension of H in G� say by an edge z�the pair F�� F� � fzg� is a ��separation of that extension� Then for any��connected graph G G with an H minor� the following holds� Any ��separation of any such minor that corresponds to F�� F�� of H under oneof the isomorphisms induces a ��separation of G�

Proof� By the assumptions and Corollary �� F�� F�� is a ��separ�ation of MH�� and that ��separation induces in the matroid sense� a��separation E�� E�� in MG�� For i � � �� Ei � Fi� and thus jEij �jFij � �� By Theorem �� part c�� E�� E�� is a ��separation of G asdesired�

We touch upon the complexity of nding a minor that prevents aninduced k�separation� We consider this problem in the following setting�We are given a binary matroid M � a minor N of M� and a k�separation ofN � We would like to obtain a k�separation of M induced by that of N � Ifthat is not possible� we want to nd a minor M represented up to indicesby one of the matrices of �� The next theorem says thatthis problem can be solved in polynomial time�

�� Theorem� There is a polynomial algorithm for the followingproblem� The input consists of a binary matroid M � a minor N of M �and a k�separation of N � The output is to be either a k�separation of Minduced by that of N � or a minor of M that is isomorphic to one of thematroids represented by the matrices of ��

Proof� If an induced k�separation does exist� then one such k�separation isfound by the separation algorithm� Suppose there is no such k�separation ofM � Then we use a polynomial implementation of the constructive proofs ofLemmas �� and �� to locate a minor of M representedup to indices by one of the matrices of ��

Sometimes a class M of matroids under investigation is only closedunder restricted isomorphism and under special minor�taking� We wantsu�cient conditions under which the above results for induced decomposi�tions remain valid in the new setting� We state one such instance followinga denition�


Let L be a set of elements� Assume that two binary matroids containthe set L� We say that the two matroids are L�isomorphic if there is anisomorphism that is an identity on the set L� The conditions on M are asfollows� Each matroid of M contains L� Furthermore� M is closed underL�isomorphism and under the taking of minors� provided the minors areconnected and contain L� Analogous denitions apply to graphs� or to theterm �minimal under L�isomorphism��

As before� let N be a binary matroid with a k�separation F�� F�� Wealso assume that N is inM� and that L � F�� The next theorem says thatCorollaries �� and �� remain valid under the additional condi�tions� Under a suitable change to graph terminology� the same conclusionapplies to Corollary �� Finally� Theorem �� remains valid whenL�isomorphisms replace isomorphisms�

�� Theorem� Corollaries �� and �� remain valid whenM and N satisfy the following two conditions for some set L contained inthe set F� of N � First� each matroid of M contains the set L� Second� Mis closed under L�isomorphism and under the taking of minors� providedthe minors are connected and contain L� Corollary �� remains validwhen the above conditions on M and N are applied to the class G and tothe graph H� Theorem �� remains valid when L�isomorphisms areclaimed instead of isomorphisms�

Proof� The cited results rely on Theorem �� which is nothing butTheorem �� plus Lemma �� Now Theorem �� is a state�ment about a minimalM � and thus does not involve any isomorphism� Butin the proof of Lemma �� the matroid N is replaced by an isomorphicmatroid N �� However� N � can be derived from N by a relabeling of someelements of F� � X� � Y�� Thus� the elements of F� are not a�ected� andN � is F��isomorphic to N � Since L � F�� N � is also L�isomorphic to N � Weapply these observations to rewrite Theorem �� so that it becomesa statement about a matroid minimal under L�isomorphism� Indeed� weonly need to change the claims about the matrices of �� and�� by allowing for a relabeling of indices other than those of L to getthe desired theorem� It is now an easy matter to verify that the theorem soderived from Theorem �� implies the claimed results for Corollaries�� and �� and Theorem ��

An example application of Theorem �� is covered in the nextsection� There we prove the existence of certain extensions of ��connectedbinary minors in ��connected binary matroids� In Chapters �� and ��we use Corollaries �� and Theorem ��

� � Extensions of ��Connected Minors ��

�� Extensions of ��Connected Minors

An important matroid problem is as follows� We are given a ��connectedbinary matroid M with a ��connected minor N � The minor has at leastsix elements� We want to obtain a ��connected minor N � of M that� forsome small k � � is a k�element extension of an N minor of M � In thissection� we show that an N � with k � or � can always be found� Ourmain tool for establishing this result is Theorem �� of the precedingsection� In Chapter �� we rene the conclusion proved here to obtain theso�called splitter theorem�

The precise statement of the above claim about N � and k is as follows�

�� Theorem� Let M be a ��connected binary matroid with a ��connected proper minor N � Suppose N has at least six elements� Then Mhas a ��connected minor N � that is a � or ��element extension of some Nminor of M � In the �element case� N � is derived from the N minor by oneaddition and one expansion�

Proof� Let z be any element of M that is not in N � Lemma ��says that the connected M has a connected minor N � that is a �elementextension of N by z� Now Theorem �� holds for M and N if and onlyif it holds for M� and N�� Hence� by duality� we may assume that theextension is an addition� Let N be represented by a matrix B� Thus� forsome vector a� the minor N � of M is represented by the matrix

��B a

zY

X

Matrix for �element extension N � of N

Since N � is connected� the vector a must be nonzero� If a is not a unitvector and is not parallel to a column of B� then by Lemma �� N � is��connected and we are done� Otherwise� due to at most one GF��pivotin B� we may assume a to be a unit vector� say with in row u Y � Let

d be the row vector of B indexed by u� Partition B into d and B� and also

partition X into fug and X � X � fug� We thus can rewrite �B j a� of�� as

�� 0

1

zY

du

BX

Partitioned version of matrix of �� for N �


The partition in �� corresponds to the ��separation X � Y � fu� zg�of N �� Since M is ��connected� that ��separation of N � does not induceone in M � There must be a minor M � of M that proves this fact andthat is minimal under isomorphism� The matroid M � has an N � minor� Ifnecessary� we change the element labels of M � so that N � itself is that N �

minor�We apply Theorem �� The just�dened M � plays the role of M

of the theorem� The submatrices B� d� and � � of �� correspond toA�� D� and A�� respectively� of the theorem� We list enough conditions ofparts a�� b�� and c� of Theorem �� to derive the desired conclusion�

At the same time� we substitute B� d� and � � for A�� D� and A�� On theother hand� the indices x and y� the subvectors e� f � g� h� and the scalar �employed below should be interpreted exactly as in Theorem ��

a� M � is represented by

�� 0

f1

zY

ed

xu

BX

Case a� of Theorem ��

In row x� the subvector e is not spanned by d� and f � �We evaluate this condition� Evidently� e is nonzero and not parallel to

d� Indeed� one easily veries that the matrix of �� has no zero vectors�unit vectors� or parallel vectors� By Lemma �� the matroid M � istherefore ��connected� and hence is a ��connected ��element extension ofN produced by one addition and one expansion�

b� M � is represented by

�� g 0

h 1

y zY

du

BX

Case b� of Theorem ��

In column y� the subvector h is not spanned by the columns of d�This condition is clearly incompatible with the fact that d is a nonzero

vector� Thus� this case cannot occur�

c� M � has a minor M with representation matrix

� � Extensions of ��Connected Minors ��

��g 0

αh

f1

y zY

ed

xu

BX

Case c� of Theorem ��

From the conditions c� � and c�� of Theorem �� we extract thefollowing�

c� � The vector e is not spanned by d and is not parallel to a row of B�Furthermore� e is not a unit vector with in a column t Y for which

column t of B is nonzero�We evaluate these conditions� Since the matrix B � �B�d� for N is

��connected� each column of B is nonzero� Thus� the above conditions on

e imply that the matrix composed B� d� and e has no zero vectors� unitvectors� or parallel vectors� By Lemma �� that matrix represents a��connected �element extension of N �

c�� The vector g is nonzero� If d� the subvector obtained from d bydeletion of an element t Y � has the same GF��rank as d� then �g�h� is

not parallel to the column t of �B�d�� If d does not span a row t X of B�then �g�h� is not a unit vector with in row t�

We interpret these conditions� By the ��connectedness of N � the vector

d has at least two s and does not span any row of B� Thus� the above

conclusions about �g�h� hold for all t Y and all t X� Put di�erently��g�h� must be nonzero� cannot be a unit vector� and cannot be parallel to a

column of �B�d�� By Lemma �� the matrix composed of B� d� g� andh represents a ��connected �element extension of N �

The minor N � of Theorem �� may be e�ciently found� Indeed�one only needs to implement the preceding constructive proof� The precisecomplexity claim is as follows�

�� Theorem� There is a polynomial algorithm for the followingproblem� The input is a connected binary matroid M � and a ��connectedproper minor N of M on at least six elements� The output is either a ��separation of M � or a minor N � of M that is a ��connected � or ��elementextension of an N minor of M � In the �element case� N � is derived fromthe N minor by one addition and one expansion�

Proof� We implement the proof of Theorem �� using as subrou�tine the polynomial algorithm of Theorem �� to produce either a


��separation of M or one of the matrices �� It is easy to seethat a polynomial algorithm can thus be assembled for the stated prob�lem�

In the next section� we discuss extensions of the results of this chapterand include references�


The results of Section �� overlap signicantly with results of Seymour ��b� for general matroids� even though the terminology is quite di�er�ent� The precise relationships are as follows� Lemma �� is the binaryversion of a result taken from Seymour ��b�� That reference proceedsto describe a number of properties of matroids called minimal here� Theseresults are then used to deduce a version of Corollary �� In contrast�the approach taken here is based on properties that can be e�ciently ver�ied for matroids that are minimal or minimal under isomorphism� Suchproperties are investigated via abstract matrices for general matroids inTruemper �� The su�ciency conditions and testing algorithms so ob�tained are substantially stronger than those relying on Corollary ��or on the even weaker Corollary �� The simpler results given heresu�ce for the proofs in the chapters to come� Truemper �� contains ad�ditional material about induced decompositions� For example� it is shownthat the number of non�isomorphic minimal matroids is nite for a givenN � provided all matroids under consideration are representable over a givennite eld�

Truemper �� treats in detail the case of graphs� which we haveskipped here entirely except for the specialized Corollary ��

Theorem �� may be viewed as a weak version of the splitter the�orem of Seymour ��b�� We use Theorem �� in the next chap�ter to prove that result� Upon slight modication� the approach of Sec�tion �� yields other important theorems about ��connected extensions of��connected matroids� We sketch the main ideas and provide related refer�ences in Section �� of the next chapter�

Chapter 7

Splitter Theorem and Sequences of

Nested Minors

7.1 Overview

Chapters 4, 5, and 6 cover three basic matroid tools: the series-paralleland delta-wye constructions, the path shortening technique, and the sep-aration algorithm. The chapters also include a number of basic matroidresults whose proofs rely on these tools. With this foundation, we derivein this chapter and the next one several fundamental results about thedecomposition and composition of matroids. Specifically in this chapter,we define matroid splitters, characterize them, and deduce consequences ofthat characterization.

The concept of splitters and their characterization is due to Seymour.The idea can be summarized as follows. Let M be a class of binary matroidsthat is closed under isomorphism and under the taking of minors. Thena 3-connected matroid N ∈ M on at least six elements is declared to bea splitter of M if every matroid M ∈ M with a proper N minor has a2-separation. Some researchers define graph or matroid 2-separations tobe splits. The term “splitter” is in agreement with that notion.

The concept of splitters may seem rather abstract. But in subsequentchapters, we rely on it a number of times, and without doubt it is one of thecentral ideas for the decomposition of matroids. We characterize splittersin Section 7.2 in the so-called splitter theorem.

Define two minors of a graph or matroid to be nested if one of them is aminor of the other one. In Section 7.3, we derive from the splitter theoremseveral existence theorems about sequences of nested minors, among them

151

152 Chapter 7. Splitter Theorem and Sequences of Nested Minors

Tutte’s wheel theorem for graphs. A special case of nested minor sequencesis used in Section 7.4 to prove Kuratowski’s characterization of planargraphs. According to that result, a graph is planar if and only if it does nothave K3,3 or K5 minors. In the final section, 7.5, we point out a numberof extensions and list references.

The chapter requires familiarity with the material of Chapters 2, 3, 5,and 6.

7.2 Splitter Theorem

Let M be a class of binary matroids that is closed under isomorphism andunder the taking of minors. Recall that a splitter of M is a 3-connectedmatroid N ∈ M such that every matroid M ∈ M with a proper N minoris 2-separable. We employ the same terminology for graphs. For example,if G is a class of graphs that is closed under isomorphism and under thetaking of minors, then a splitter of G is a 3-connected graph G ∈ G suchthat every graph of G with a proper G minor is 2-separable.

In this section, we derive surprisingly simple necessary and sufficientconditions for a given N ∈ M to be a splitter. The next theorem statingthese conditions is the splitter theorem due to Seymour.

(7.2.1) Theorem (Splitter Theorem). Let M be a class of binary ma-troids that is closed under isomorphism and under the taking of minors.Let N be a 3-connected matroid of M on at least six elements.

(a) If N is not a wheel, then N is splitter of M if and only if M does notcontain a 3-connected 1-element extension of N .

(b) If N is a wheel, then N is a splitter of M if and only if M does notcontain a 3-connected 1-element extension of N and does not containthe next larger wheel.

Proof. If N is a splitter of M, then the 3-connected extensions citedin (a) or (b) obviously cannot occur in M. We prove the converse bycontradiction. Thus, we suppose that M does not contain the 3-connectedextensions cited in (a) or (b), whichever applies, and that nevertheless Nis not a splitter of M. Thus, M contains a 3-connected matroid M witha proper N minor, and M is not one of the cases excluded under (a) or(b). Since M is closed under isomorphism, we may assume N itself tobe that N minor. To M and N we apply Theorem (6.4.1). According tothat theorem, M has a 3-connected minor N ′ that is a 3-connected 1- or2-element extension of an N minor. In the 2-element extension case, N ′ isderived from the N minor by one addition and one expansion. Again, sinceM is closed under isomorphism and minor taking, we may take N itself tobe that N minor. The 1-element extension case has been ruled out by (a)

7.2. Splitter Theorem 153

and (b). Thus, N ′ is derived from N by one addition and one expansion.Suppose a binary matrix B with row index set X and column index setY represents N . Then N ′ can be represented by a binary matrix C thatdisplays B, and thus N , as follows.

(7.2.2)

qY

αbp

C = aX B

Matrix C representing N ′

We now show either that N ′ contains a 3-connected 1-element extensionof an N minor, a case ruled out by both (a) and (b), or that N is a wheeland N ′ is the next larger wheel, a case ruled out by (b). We accomplishthis by the following investigation into the structure of C of (7.2.2).

Since N ′ is 3-connected, the matrix C does not contain zero vectors,unit vectors, or parallel vectors. In particular, the subvectors a and b ofC must be nonzero. Furthermore, the submatrices [B | a] and [B/b] of C,which represent 1-element extensions of N , cannot be 3-connected sinceotherwise we have an eliminated case of (a) and (b). Thus, the subvectora (resp. b) is a unit vector or is parallel to a column (resp. row) of B.

Because of pivots in B and row exchanges in C, we may assume thata is a unit vector with 1 in the topmost position, and that b is parallel toa row of B. Since C is 3-connected, we necessarily have α = 1. We thenmay partition C as follows.

(7.2.3)

q

C =

Y

c 1

0

1bp

0 1X

Initial partition of C with a = unit vector

In the subsequent processing of C of (7.2.3), we introduce a number of rowand column exchanges and pivots that affect the index sets substantially.Since M is closed under isomorphism, we do not have to keep track of suchindex changes. So, instead, we just make sure that the matrix obtainedfrom the current C by deletion of the rightmost column and bottom row


does represent N up to a relabeling of the elements. We always refer tothat matrix as the current B. With this convention, we can freely introducenew indices or reuse old ones.

Suppose during the subsequent processing of C of (7.2.3), we detectthat the current C contains the current B plus a nonzero row or columnthat is not parallel to a row or column of B and that is not a unit vector.Then by Lemma (6.2.6), B plus that row or column up to indices representsa 3-connected 1-element extension of N , which has been ruled out. Thus,we assume below that this case does not occur. The proof of the theoremis complete once we show N to be a wheel and N ′ to be the next largerwheel. We are now ready to process C of (7.2.3).

We know that the vector b of C of (7.2.3) is parallel to a row of B.That vector of B cannot be c, for otherwise, C is 2-separable. So assumeb is parallel to the second row of B, say row v. An exchange of rows v andp of C produces

(7.2.4)

c

p1

b 1

0

0bv

0 1

Matrix C after exchange of rows p and v

Except for the replacement of the row index p by v, that row exchangedoes not affect the submatrix B. The column vector to the right of thecurrent B must be parallel to, say, the first column of B. Exchange thefirst column and the last column of C. By the 3-connectedness of N ′, wemust have

(7.2.5)

111

00

0 1

0 1

1 b

b

c

Matrix C after exchange of first and last column

Pivot on the circled 1 of (7.2.5). That pivot produces the matrix of (7.2.6)below.

Inductively, assume that the current C is given by (7.2.7) below. Sup-pose that the subvector b of row x is a unit vector, say with 1 in col-umn z ∈ Y2, and that column z of the submatrix B is zero. Then we

7.2. Splitter Theorem 155

have (X1 ∪ Y1 ∪ {x, y, z}, (X2 ∪ Y2) − {z}) as a 2-separation of C unless|(X2 ∪ Y2)−{z}| ≤ 1. If |(X2 ∪ Y2)−{z}| = 1, then C contains a zero rowindexed by X2, or C has a zero or unit vector column indexed by Y2 −{z}.Either case is a contradiction of the 3-connectedness of N ′.

(7.2.6)

011 1

00

0 1

0 1

b

b

c ’

Matrix C after pivot

(7.2.7)1

.

.

.

.

.

.

Y1 Y2 y

X1

X2

x

1 0 0

0

1

00

b

0

c

0 11

10 b

B

Matrix C for inductive proof

Thus, |(X2 ∪ Y2) − {z}| = 0, i.e., X2 = ∅ and Y2 = {z}, which impliesb = [ 1 ]. Since the columns z and y of C must be distinct, we also havec = [ 1 ]. Then C is

(7.2.8) 1 ..........

Y1 yz

X1

x

1 11

11

11

1

01

10

Matrix C displaying wheel case

Evidently, the current B is a matrix of type (5.2.9). Accordingly, N iswheel. A pivot in C on the 1 in the bottom right corner confirms that N ′

is a wheel as well.Once more, assume that in the matrix C of (7.2.7), the row subvector

b is a unit vector with 1 in column z. But this time, suppose that column


z of B is nonzero. By a pivot in column z of B, we convert the situation tothe third possible case, where b is not a unit vector. In that third case, thevector b in row x of C must be parallel to a row of B, say row p. Exchangerows p and x of C. We get

(7.2.9)1

.

.....

y

p

x

1 0

0

11

0

0

11

00

0

bb

c

b0 1

Matrix C after exchange of rows p and x

The remaining arguments are analogous to those for (7.2.4)–(7.2.6). Theyproduce an instance of (7.2.7) where |Y1| has been increased by 1. Byinduction, the case already discussed must eventually be encountered whereN is a wheel and N ′ is the next larger wheel.

When specialized to graphs, the splitter Theorem (7.2.1) becomes thefollowing result.

(7.2.10) Corollary. Let G be a class of connected graphs that is closedunder isomorphism and under the taking of minors. Let H be a 3-connectedgraph of G with at least six edges.

(a) If H is not a wheel, then H is a splitter of G if and only if G doesnot contain any graph derived from H by one of the following twoextension steps:

(1) Connect two nonadjacent nodes by a new edge.

(2) Partition a vertex of degree at least 4 into two vertices, each ofdegree at least 2, then connect these two vertices by a new edge.

(b) If H is a wheel, then H is a splitter of G if and only if G does notcontain any of the extensions of H described under (a) and does notcontain the next larger wheel.

Proof. Let M be the collection of graphic matroids produced by thegraphs of G. Define N to be the graphic matroid of the graph H. Lemma(6.2.7) says that the extensions of H described under (a) are precisely the3-connected 1-edge extensions of H. Thus, these extensions correspond tothe 3-connected 1-element graphic extensions of N . The result then followsfrom the splitter Theorem (7.2.1).

Typically, we will specify M or G by exclusion of certain minors andof all their isomorphic copies. Clearly, any collection of matroids or graphsso specified is closed under isomorphism and under the taking of minors.

7.3. Sequences of Nested Minors and Wheel Theorem 157

Two graph examples of splitters are given in the next theorem. Recallthat Wn is the wheel graph with n spokes.

(7.2.11) Theorem. W3 is a splitter of the graphs without W4 minors,and K5 is a splitter of the graphs without K3,3 minors.

Proof. There is no 3-connected 1-edge extension of W3. Thus, by part (b)of Corollary (7.2.10), W3 is a splitter of the graphs without W4 minors.For the second part, we note that up to isomorphism there is just one 3-connected 1-edge extension of K5. To obtain it, one partitions one vertexof K5 into two vertices of degree 2 and connects the two vertices by a newedge. The resulting graph is readily seen to have a K3,3 minor. Thus, bypart (a) of Corollary (7.2.10), K5 is a splitter of the graphs without K3,3

minors.

We will see a number of other splitter examples in subsequent chapters.In the next section, we deduce from the splitter Theorem (7.2.1) certainsequences of nested minors and Tutte’s wheel theorem.

7.3 Sequences of Nested Minors and

Wheel Theorem

Recall from Section 7.1 that two matroids are nested if one of them is aminor of the other one. In this section, we prove the existence of certainsequences of nested minors of binary matroids. As a special case, we es-tablish Tutte’s wheel theorem. Main tools are the splitter Theorem (7.2.1)and results proved in Chapter 5 with the path shortening technique.

For a given binary matroid, an arbitrary sequence of nested minors iseasy to find. The task becomes difficult and interesting when one imposesconditions on the sequence. We need a few definitions to express suchconditions.

Suppose two matroids are nested. Define the rank gap between the twomatroids to be the absolute difference in rank between them. Analogously,define the corank gap. Finally, let the gap be the sum of the rank gap andthe corank gap. Evidently, the gap is the number of elements that occuronly in the larger matroid.

Consider a sequence of nested minors of a given binary matroid. Thenthe rank gap of the sequence is the maximum rank gap among the pairs ofsuccessive minors of the sequence. Analogously, define the corank gap andthe gap of the sequence.

We now state the conditions under which we want to find nested minorsequences. Suppose for some k ≥ 2, we have a k-connected binary matroidM with a k-connected minor N . In the typical situation, we want to find a


sequence of nested k-connected minors M0, M1, M2, . . . , Mt = M , whereM0 is demanded to be isomorphic to N . Furthermore, the rank gap, or thecorank gap, or the gap is to be bounded by some given constant. Othervariants are possible. For example, one may require that a given elementof M that does not occur in M0 be present in M1, that a given elementnot occurring in M1 be present in M2, and so on.

Sequences of the desired kind are readily determined when k = 2.The main ingredient for their construction is a recursive application ofLemma (5.2.4). We skip the details of that simple case, and instead turnimmediately to the much more complicated case with k = 3. Specifically,we prove the existence of three types of sequences for k = 3, then point outextensions in Section 7.5.

In each of the cases treated here, we are given a 3-connected binarymatroid M with a 3-connected proper minor N on at least six elements.In the first case, we desire a sequence of nested 3-connected minors M0,M1, . . . , Mt = M , where M0 is isomorphic to N and where the gap is small.The next theorem shows that the gap can be held to 1 or 2.

(7.3.1) Theorem. Let M be a 3-connected binary matroid having a3-connected proper minor N on at least six elements.

(a) Assume N is not a wheel. Then for some t ≥ 1, there is a sequenceof nested 3-connected minors M0, M1, . . . , Mt = M , where M0 isisomorphic to N and where the gap is 1.

(b) Assume N is a wheel. Then for some t ≥ 1, there is sequence of nested3-connected minors M0, M1, . . . , Mt = M with the following features.M0 is isomorphic to N . For some 0 ≤ s ≤ t, the subsequence M0,M1, . . . , Ms consists of wheels and has gap 2, and the subsequenceMs, Ms+1, . . . , Mt = M has gap 1.

Proof. We first establish part (a). Thus, we assume that N is not a wheel.Indeed, inductively we assume for some i ≥ 0, the existence of a sequence ofnested 3-connected minors M0, M1, . . . , Mi of M , where M0 is isomorphicto N , where Mi is not a wheel, and where the gap is 1. If Mi = M , we aredone. So assume that Mi is a proper minor of M .

We rely on the contrapositive statement of part (a) of the splitter The-orem (7.2.1) to find a larger sequence. To this end, we define M to be thematroid collection containing M , all minors of M , and all matroids isomor-phic to these matroids. By this definition, M is closed under isomorphismand under the taking of minors. Since Mi is a 3-connected proper minorof the 3-connected M ∈ M, it cannot be a splitter of M. Thus, by part(a) of Theorem (7.2.1), M contains a matroid Mi+1 that is a 3-connected1-element extension of a matroid isomorphic to Mi. Now every 1-elementreduction of a wheel with at least six elements is 2-separable. Thus, if Mi+1

is a wheel, then Mi is 2-separable, a contradiction. We conclude that Mi+1

is not a wheel.


If necessary, we relabel M0, M1, . . . , Mi so that they constitute asequence of nested minors of Mi+1. These matroids plus Mi+1 satisfy theinduction hypothesis for i + 1. By induction, the claimed sequence existsfor M .

The proof of part (b) is essentially the same, except that we establishMi+1 using part (b) of Theorem (7.2.1) when Mi is a wheel.

We have used the splitter Theorem (7.2.1) for a simple proof of The-orem (7.3.1). Indeed, the two theorems are essentially equivalent, sinceone may deduce the splitter Theorem (7.2.1) from Theorem (7.3.1) just aseasily. We sketch the proof.

Let M and N be as specified in the splitter Theorem (7.2.1). SupposeN is not a wheel. We must show that N is a splitter of M if and only if Mdoes not contain any 3-connected 1-element extension of N . We prove thenontrivial “if” part by contradiction. So let M be a 3-connected matroidof M with N as proper minor. By Theorem (7.3.1), there is a sequence ofnested 3-connected minors M0, M1, . . . , Mt = M , where M0 is isomorphicto N , and where the gap is 1. Since M is closed under isomorphism, wemay assume M to be so chosen that M0 is equal to N . Then M1 is a3-connected 1-element extension of N and M1 ∈ M, which contradicts theassumed absence of such extensions. The case where N is a wheel is treatedanalogously.

A direct translation of Theorem (7.3.1) into graph language results inthe following corollary.

(7.3.2) Corollary. Let G be a 3-connected graph having a 3-connectedproper minor H with at least six edges.

(a) Assume H is not a wheel. Then for some t ≥ 1, there is a sequence ofnested 3-connected minors G0, G1, . . . , Gt = G, where G0 is isomor-phic to H, and where each Gi+1 has exactly one edge beyond those ofGi.

(b) Assume H is a wheel. Then for some t ≥ 1, there is a sequence ofnested 3-connected minors G0, G1, . . . , Gt = G with the followingfeatures. G0 is isomorphic to H. For some 0 ≤ s ≤ t, the subsequenceG0, G1, . . . , Gs consists of wheels, where each Gi+1 has exactly oneadditional spoke beyond those of Gi. Furthermore, in the subsequenceGs, Gs+1, . . . , Gt = G, each Gi+1 has exactly one edge beyond thoseof Gi.

One may combine Corollary (7.3.2) with Corollary (5.2.15) to obtainTutte’s wheel theorem, which is listed next.

(7.3.3) Theorem (Wheel Theorem). Let G be a 3-connected graph onat least six edges. If G is not a wheel, then G has some edge z such thatat least one of the minors G/z and G\z is 3-connected.


Proof. Corollary (5.2.15) says that a 3-connected graph with at least sixedges, in particular G specified here, has a W3 minor. Thus, G has a largestwheel minor, say H. Since G is not a wheel, H is a proper minor of G.We apply Corollary (7.3.2) to G and H. Accordingly, G has a sequence ofnested 3-connected minors G0, G1, . . . , Gt = G, where G0 is isomorphic toH. Since H is a largest wheel minor of G and since G is not a wheel, theindex s of part (b) of Corollary (7.3.2) must be zero, and t ≥ 1. We alsoconclude from that part that G = Gt has exactly one edge beyond those ofGt−1. Put differently, the 3-connected minor Gt−1 is for some edge z equalto G/z or G\z, which proves the theorem.

Theorem (7.3.3) can obviously be rewritten so that it becomes a wheeltheorem for binary matroids instead of graphs. The proof relies on Theorem(7.3.1) instead of Corollary (7.3.2).

From the sequence of nested 3-connected minors M0, M1, . . . , Mt = Mof Theorem (7.3.1), one can derive a number of other interesting sequences.A particular construction utilizes the representation matrices of these mi-nors. We present details following some observations.

By Lemma (3.3.12), a binary matroid M with a given minor M hasa representation matrix that displays M . We apply this result inductivelyto the sequence of nested minors M0, M1, . . . , Mt = M , and concludethat M has a representation matrix B that simultaneously displays M0,M1, . . . , Mt = M , say by nested matrices B0, B1, . . . , Bt = B. Let Ci

be the column submatrix of B that has the same column index set as Bi.Let bi be a row vector of Ci, say with row index x. Assume that bi isnot a row of Bi. Indeed, assume that bi is nonzero, is not a unit vector,and is not parallel to a row of Bi. Lemma (6.2.6) shows that under theseassumptions, Bi plus bi represent a 3-connected minor Mi&x of M . Thereare other minors Mj without x for which Mj&x is 3-connected. Specifically,let k be the largest index, i ≤ k ≤ t, such that Mj does not contain x. Weclaim that for each i < j ≤ k, Mj&x is 3-connected. The proof consistsof the following observation. Let bj be the row vector of Cj indexed by x.Since j ≤ k, bj is not part of Bj . Evidently, Bi is a submatrix of Bj, and bi

is a subvector of bj. We know that bi is nonzero, is not a unit vector, and isnot parallel to a row of Bi. Then the latter statement must also hold whenwe use j as superscript instead of i. Accordingly, Mj&x is 3-connected.

We use these observations as follows. In the sequence M0, M1, . . . ,Mt = M , we redefine each Mj , i ≤ j ≤ k, to be Mj&x. Correspondingly,we redefine the Bj matrices by adjoining the bj row vectors. The result is anew sequence of nested 3-connected minors. The rank gap of that sequenceis larger than that for the original one. But the corank gap is still the same.Indeed, since the gap of the original sequence was 1, the corank gap of thenew sequence is 0 or 1.

We repeat the above process, using all possible indices i and x, until


no further changes are possible. This occurs when each final Bi cannotbe extended to a larger 3-connected matrix within Ci. The final sequencemay contain duplicate minors. In that case, we delete just enough minorsto eliminate all such duplicates. Then we redefine the indices so that M0,M1, . . . , Mt = M is now the sequence resulting from the above process.Correspondingly, we redefine the indices of the Bi and Ci matrices.

If M0 and M have same corank, then t = 0, and the outcome isM0 = Mt = M , an uninteresting case. If M0 and M have different corank,we must have t ≥ 1. Consider the latter case. Evidently, each Bi+1 isdeduced from Bi by first adjoining any number of row vectors bi to Bi,and then adjoining a column vector a. We claim that each one of thesevectors bi is a unit vector or is parallel to a row of Bi. Indeed, in any othercase of a nonzero bi, the previously described process would have adjoinedbi to Bi and would have redefined Mi prior to termination, a contradiction.In the case of a zero vector bi, Bi+1 would have a zero or unit vector, andthus would be 2-separable, again a contradiction.

In terms of the minors of the final sequence, we thus have proved thateach Mi+1 may be derived from Mi by extensions involving any number ofseries elements, possibly none, followed by a 1-element addition.

The above construction is the main ingredient in the proof of thefollowing variant of Theorem (7.3.1).

(7.3.4) Theorem. Let M be a 3-connected binary matroid with a 3-connected proper minor N on at least six elements. If M does not containa 3-connected 1-element expansion (resp. addition) of any N minor, then Mhas a sequence of nested 3-connected minors M0, M1, . . . , Mt = M , whereM0 is an N minor of M , and where each Mi+1 is obtained from Mi byexpansions (resp. additions) involving some series (resp. parallel) elements,possibly none, followed by a 1-element addition (resp. expansion).

Proof. Clearly, the parenthetic case is dual to the stated one. Thus, weonly consider the case where M does not contain any 3-connected 1-elementexpansion of any N minor. We prove the existence of the claimed sequenceof nested minors as follows.

Apply the above construction process to the sequence of nested 3-connected minors M0, M1, . . . , Mt = M of Theorem (7.3.1). The result is anew sequence M0, M1, . . . , Mt = M where each Mi+1 may be derived fromMi by expansions involving any number of series elements, possibly none,followed by a 1-element addition. Since M does not contain a 3-connected1-element expansion of any N minor, the construction must have left M0

unchanged. The sequence so produced is the desired one.

Theorem (7.3.4) may be specialized to graphs as follows.

(7.3.5) Corollary. Let G be a 3-connected graph with a 3-connectedminor H on at least six edges.


(a) Suppose no H minor of G can be extended to another 3-connectedminor of G by the following process: Some node of H of degree at least4 is partitioned into two nodes, each with degree at least 2, and thenthe two nodes are connected by a new edge. Then G has a sequenceof nested 3-connected minors G0, G1, . . . , Gt = G with the followingproperties. G0 is an H minor of G. Each Gi+1 may be obtained fromGi as follows. First, at most two edges of Gi are replaced by paths oflength 2. Next, an edge is added so that the new graph has no degree2 nodes and no parallel edges.

(b) Suppose no H minor of G can be extended to another 3-connected mi-nor of G by connecting two nonadjacent nodes by a new edge. ThenG has a sequence of nested 3-connected minors G0, G1, . . . , Gt = Gwith the following properties. G0 is an H minor of G. Each Gi+1 maybe obtained from Gi as follows. First, some edges, possibly none, inci-dent at some vertex are replaced by two parallel edges each. The newvertex must have degree of at least 4. Next, that vertex is partitionedinto two vertices, each with degree at least 2, such that no two edgesremain parallel. Finally, the two vertices just created are joined by anew edge.

Proof. Part (a) is a routine translation of the non-parenthetic part ofTheorem (7.3.4) into graph language, in the same sense that Corollary(7.3.2) is the graph version of Theorem (7.3.1). Part (b) is a translation ofthe parenthetic part of Theorem (7.3.4).

We conclude this section by proving that the sequences of nested 3-connected minors of the above theorems and corollaries can be readilyfound. Additional material about nested sequences is contained in Sec-tion 7.5.

(7.3.6) Theorem. There is a polynomial algorithm for the followingproblem. The input consists of M and N of Theorem (7.3.1) or (7.3.4), orof G and H of Corollary (7.3.2) or (7.3.5). The output is a 2-separationof M or G, or a sequence of nested 3-connected minors with properties asspecified in the respective theorem or corollary.

Proof. The arguments that prove the cited theorems and corollaries canbe summarized as follows. Theorem (6.4.1) implies the splitter Theorem(7.2.1), which in turn implies Theorem (7.3.1) and Corollary (7.3.2). Thelatter results imply Theorem (7.3.4) and Corollary (7.3.5). The polynomialalgorithm claimed for the stated problems is essentially an efficient imple-mentation of the constructive proofs of these implications. We first sketchthe details for the matroid case with given M and N . With the polynomialalgorithm of Theorem (6.4.7), we locate a 2-separation of M or producethe 3-connected 1- or 2-element extensions claimed by Theorem (6.4.1).We use these extensions plus the proof procedure of the splitter Theorem

7.4. Characterization of Planar Graphs 163

(7.2.1) to derive the sequence of nested minors of Theorem (7.3.1). Fromthat sequence we construct the sequence of Theorem (7.3.4).

The graph case may be handled by appropriate translation of eachstep of the above algorithm into graph language. Alternately, one couldrepresent the given G and H by graphic matroids M and N , apply thepolynomial algorithm already described, and extract from G the minorsG0, G1, . . . , Gt = G corresponding to the sequence M0, M1, . . . , Mt = Mfound by the algorithm.

The remainder of this book includes several demonstrations of thepower and utility of sequences of nested 3-connected minors. The discussionin the next section may be viewed to be one such demonstration.

7.4 Characterization of Planar Graphs

In this section, we prove Kuratowski’s characterization of graph planarityin terms of excluded K3,3 and K5 minors. We state that theorem anddescribe a proof of beautiful simplicity due to Thomassen.

(7.4.1) Theorem. A graph is planar if and only if it has no K3,3 or K5

minors.

Proof. The “only if” part follows from the fact that planarity is main-tained under the taking of minors, and that by Lemma (3.2.48) both K3,3

and K5 are not planar. For a proof of the nontrivial “if” part, let G bea connected nonplanar graph each of whose proper minors is planar. Weneed to show that G is isomorphic to K3,3 or K5.

We first prove that G cannot be 1- or 2-separable. The case of 1-separability is trivial. Suppose G is 2-separable. Let G be produced byidentifying nodes k and l of a graph G1 with nodes m and n, respectively,of a graph G2. Now G1 plus an edge connecting nodes k and l is planar,since that graph, say G′

1, is isomorphic to a proper minor of G. The graphG′

2 similarly defined from G2, is also planar. It is easy to combine planardrawings of G′

1 and G′2 to one of G.

Thus, G is 3-connected. According to Lemma (5.2.15), G must havea W3 minor, say H. Since W3 is also the complete graph K4, no H minorof G can be extended to another minor of G by addition of an edge thatconnects two nonadjacent nodes. Under the latter condition, part (b) ofCorollary (7.3.5) states that G has a sequence of nested 3-connected minorsG0, G1, . . . , Gt = G with the following properties. G0 is an H minor ofG. Each Gi+1 may be obtained from Gi as follows. First, some edges,possibly none, incident at some vertex are replaced by two parallel edgeseach. The new vertex must have degree of at least 4. Next, that vertex ispartitioned into two vertices, each with degree at least 2, such that no two


edges remain parallel. Finally, the two vertices just created are joined bya new edge.

By the minimality of G, the graph Gt−1 of the sequence is planar,while G = Gt is not. Consider a realization of Gt−1 in the plane. Notethat we do not rely on the fact, not proved here, that the drawing of Gt−1

is essentially unique. As just described, G may be derived from Gt−1 bythe replacement of some edges incident at a vertex by some parallel edges,followed by a partitioning of that new vertex, etc. Define G′

t−1 to be thegraph on hand when the first step has been carried out — that is, when inGt−1 some edges incident at a vertex have been replaced by two paralleledges each. Let v be the vertex of G′

t−1 to be partitioned, say into verticesv1 and v2. By the connectivity conditions, a partial drawing of G′

t−1 thatemphasizes vertex v is either

(7.4.2)

en

e2

e1

f2

fnf1

v

Vertex v of G′t−1

or is deduced from that drawing by deletion of any number of the paral-lel edges f1, f2, . . . , fn. The dashed segments represent internally node-disjoint paths. The vertex v is so partitioned into v1 and v2 that theresulting graph G is nonplanar. This implies that we cannot replace v ofthe assumed drawing for G′

t−1 by v1 and v2, and then join these two nodesby an edge, while retaining a planar drawing. There are two possible causesfor the nonplanarity. First, some of the edges e1, e2, . . . , en may cross whenthey are attached to v1 and v2. In fact, no matter which particular situa-tion occurs, just four edges can always be selected to produce the graph of(7.4.3) below.

(7.4.3) v1 v2

Partition of vertex v producing K3,3 minor

7.5. Extensions and References 165

The graph contains a subdivision of K3,3, so we are done. In the secondcase, the ei edges can be attached to v1 and v2 so that they do not cross.However, crossing edges are encountered when we attach the fi edges. Wemay suppose this to be so even when we relabel edges such that some ei

become fi, since otherwise we can produce the earlier situation.This leaves just one case, where n = 3, and where for i = 1, 2, 3, both

ei and fi are present. We then have

(7.4.4)

f2

f3

e1 v1 v2

f1

e2

e3

Partition of vertex v producing K5 minor

which contains a subdivision of K5.

Actually, Kuratowski proved for a nonplanar graph G the presence ofa subgraph that is a subdivision of K3,3 or K5 and not just a K3,3 or K5

minor. Now a K3,3 minor induces a subdivision of K3,3. For a proof, carryout the expansion steps that convert the K3,3 minor to a subgraph of G.The same argument applies to a K5 minor, except when an expansion stepsplits a vertex of degree 4 into two vertices each of which has degree 3 uponinsertion of the new edge. The graph so produced has a K3,3 minor. Thus,Theorem (7.4.1) is equivalent to Kuratowski’s original formulation.

In the final section, we cover extensions and references.

7.5 Extensions and References

The splitter Theorem (7.2.1) and its extension to nonbinary matroids isdue to Seymour (1980b); see also Tan (1981) and Truemper (1984). Theextension to the nonbinary case requires the introduction of the whirls givenby (5.4.1) and (5.4.2). The latter matroids constitute a second specialcase besides that of wheels. The graph version of Theorem (7.3.1), andthus effectively Corollary (7.2.10), were independently proved in Negami(1982). An early splitter example is implicit in Wagner (1937a). A numberof splitters are given in Seymour (1980b), Oxley (1987b), (1987c), (1989a),(1989b), (1990a), and Truemper (1988).


Tutte’s wheel Theorem (7.3.3) appeared in Tutte (1966a). The exten-sion of that result to general matroids is also due to Tutte, and is knownas the wheels and whirls theorem (Tutte (1966b)). Strengthened versionsare in Halin (1969), Oxley (1981b), and Coullard and Oxley (1992).

Theorem (7.3.4) is a binary and slightly weaker version of results inSeymour (1980b) and Truemper (1984). The related graph result appearedfor the first time in Barnette and Grunbaum (1969). That graph result hasbeen repeatedly rediscovered.

Kuratowski’s original characterization of planar graphs in terms offorbidden subgraphs appeared in Kuratowski (1930). The equivalence ofthat result to the excluded minor version given by Theorem (7.4.1) is dueto Wagner (1937b). The amazingly short proof of Theorem (7.4.1) is fromThomassen (1980). The matroid version of Theorem (7.4.1) is proved inBixby (1977).

Substantially stronger results about sequences of nested 3-connectedminors exist. Bixby and Coullard (1987) contains the most recent andstrongest one: Let N be a 3-connected proper minor of a 3-connectedmatroid M . Then for any element z of M that is not in N , there is a 3-connected minor N ′ of M that contains z, that has N as a minor, and thathas at most four elements beyond those of N . Note that isomorphisms arenot involved in this result. When the reference to the element z is dropped,then N ′ can be guaranteed to have at most three elements beyond those ofN instead of four (Truemper (1984)). The latter result can be strengthenedwhen N has no triangles and no triads (= dual triangles). In that situation,N ′ need to have at most two elements beyond those of N (Bixby and Coul-lard (1984)). The sequences of nested 3-connected minors that are impliedby these theorems may be modified by the construction given in Section 7.3prior to Theorem (7.3.4). Some examples are worked out in Truemper(1984). For a recursive characterization of 3-connectivity using so-calledseparating cocircuits, see Bixby and Cunningham (1979).

The original proofs of the results cited in the preceding paragraphare by no means simple. But there is a unified way in which they canbe obtained. We sketch the main idea. Recall that the splitter Theorem(7.2.1) is based on the notion of induced 2-separations and on Theorem(6.3.20), which deals with minors called minimal under isomorphism inSection 6.3. Similarly, the theorems cited in the preceding paragraph maybe derived using the notion of induced 2-separations plus the followingresult of Truemper (1986) about minors called minimal in Section 6.3:Suppose that a 2-separation of a matroid N on at least four elements doesnot induce a 2-separation of a matroid M containing N , and that M isminimal with respect to that condition. Then M has at most five elementsbeyond those of N .

For k ≥ 4, characterizations of sequences of nested k-connected minorsseem to become very complex. Rajan (1986) and Robertson (1984) contain

7.5. Extensions and References 167

results about the graph case for various kinds of 4-connectivity.Work cited in Section 5.4 for minors with specified elements (Bixby

(1974), Seymour (1981e), (1985b), (1986a), (1986b), Oxley (1984), (1987a),(1990a), Kahn (1985), Coullard (1986), Coullard and Reid (1988), Oxleyand Row (1989), Oxley and Reid (1990), and Reid (1990), (1991a), (1991b),(1993), (1996)) is closely related to the material of this chapter. Several ofthese results are readily proved with the earlier mentioned generalizationof Theorem (7.3.1) due to Bixby and Coullard (1987).

Related to the above results are theorems claiming the following pro-cess to be possible. First, a minor N of a matroid M is replaced by anisomorphic copy that still contains a specified subset Z of the elements ofN . Second, that isomorphic copy is extended to another minor of M withcertain attractive properties. A theorem of this type is given in Tseng andTruemper (1986).

Chapter �

Matroid Sums

�� Overview

In this chapter� we describe ways of decomposing or composing binarymatroids� using a class of constructs called k�sums� where k ranges overthe positive integers� Thus� there are ��sums� ��sums� ��sums� etc� Insubsequent chapters� we use k�sums frequently� in particular to analyze orconstruct certain graphs or matroids� for example the graphs without K�

minors� the regular matroids� and the max��ow min�cut matroids� Thesetting in which k�sums are then invoked is as follows� One wants to un�derstand or construct a given class of graphs or matroids� Already availableis some insight into several proper subclasses� One conjectures that eachgraph or matroid not in any one of those subclasses can be recursivelyconstructed by composition steps where the elementary building blocks aretaken from the subclasses� It turns out that the k�sums dened in thischapter are well suited for such a composition process� as well as for theinverse decomposition process�

Generally� the structural complexity of k�sums grows as k increases�Thus� ��sums represent the simplest� indeed trivial� case of decompositionor composition� We cover that case in Section �� In the same section� wealso discuss the more interesting but still elementary case of ��sums�

For ��sums� or generally for k�sums with k � �� the simplicity of ��and ��sums gives way to a setting of rich structure that permits manyinteresting conclusions� In Section �� we explore these k�sums� especially��sums� In Section �� we acquire an e�cient method for nding ��

��

� � �� and ��Sums ��

and ��sums� Two alternatives of ��sums� called �sum and Y �sum� arecovered in Section ��

In the nal section� �� we summarize applications of k�sums� listextensions of the k�sum concept to general matroids via abstract matri�ces� mention other ways to decompose or compose matroids� and providereferences�

We close this section with a review of the exact k�separations denedin Section �� They turn out to be the key ingredient for k�sums� Supposea binary matroidM on a set E has rank function r�� Then a pair �E�� E��partitioningE is an exact k�separation if jE�j� jE�j � k and r�E��r�E�� r�E� � k � �� Given such a separation� let X� be a maximal independentsubset of E�� then enlarge X� by a subset X� of E� to a base of M � Denefor i � �� Yi � Ei�Xi� The representation matrix B ofM correspondingto the base X� �X� must be of the form

�� X1B =

Y1

A1

DX2

Y2

A2

0

Matrix B with exact k�separation

where jX� � Y�j� jX� � Y�j � k and GF��rank D � k � �� Below� werepeatedly utilize B of �� when we work with exact k�separations�

The chapter requires knowledge of Chapters �� and �� We alsouse the process of Y exchanges of Chapter ��

�� and ��Sums

In this section� we learn how to deduce and manipulate �� and ��sum de�compositions and compositions� We start with the ��sum case� Let M bea binary matroid on a set E and with a representation matrix B� Lemma�� says that M has a ��separation if and only if B is not connected�Assume the latter case� Then B can clearly be partitioned as shown in�� below� with jX� �Y�j� jX� �Y�j � �� The latter condition is equiva�lent to demanding that the submatrices A� and A� of B are nonempty� i�e��they are not � � � matrices� We declare that the binary matroids repre�sented by A� and A�� say M� and M�� are the two components of a ��sumdecomposition of M � The decomposition is reversed in the obvious way�giving a ��sum composition of M� and M� to M � We mean either process

�� Chapter � Matroid Sums

when we say thatM is a ��sum ofM� andM�� denoted byM �M��M��

�� X1B =

Y1

A1

0X2

Y2

A2

0

Matrix B of ��separation

Under the assumption of graphicness� the ��sum has a straightforwardgraph interpretation given by the next lemma� We omit the elementaryproof via Theorem �� part �a��

�� Lemma� Let M be a binary matroid� Assume M to be a ��sumof two matroids M� and M��

�a� If M is graphic� then there exist graphs G� G�� and G� for M � M��andM�� respectively� such that identi�cation of a node of G� with oneof G� creates G�

�b� If M� and M� are graphic �resp� planar�� then M is graphic �resp�planar��

Analogously to the matroid case� we call the graphG of Lemma ��a ��sum of G� and G�� and denote this by G � G� �� G��

We move to the more interesting case of ��sums� We assume that thegiven binary matroidM is connected and has a ��separation �E�� E�� SinceM is connected� the ��separation must be exact� Thus� in any matrix Bof �� corresponding to that exact ��separation� the submatrix D musthave GF��rank �� Hence� B is the followingmatrix� where for the momentthe indices x � X� and y � Y� are to be ignored�

��

Y1y

A2

B =

A1

Y2

x

X1

X2

0

0

1all1s

Matrix B of exact ��separation

We refer to the submatrix of B of �� indexed by X� and Y� as D�in agreement with �� We want to extract from B two submatricesB� and B� that contain A� and A�� respectively� and that also contain

� � �� and ��Sums ��

enough information to reconstruct B� Evidently� the latter requirement isequivalent to the condition that we must be able to compute D from B�

and B�� Now D has GF��rank �� Thus� knowledge of one nonzero rowof D and of one nonzero column of D su�ces to compute D�

With this insight� we choose B� to be A� plus one nonzero row of D�say row x� and B� to be A� plus one nonzero column of D� say column y�The two indices x and y are shown in �� Below� we display B� andB� so selected�

��A2B2 =

y Y2x

X210

Y1y

B1 = A1

x

X1

0 1 1

1

Matrices B� and B� of ��sum

We reconstruct D� and thus implicitly B� from B� and B� by computing

�� D � �column y of B�� row x of B��

Let M� and M� be the minors of M represented by B� and B�� We callthese minors the components of a ��sum decomposition of M � The reverseprocess� which corresponds to a reconstruction of B from B� and B�� is a��sum composition of M� and M� to M � Both cases are handled by sayingthat M is a ��sum of M� and M�� denoted by M �M� ��M�� For futurereference� we record in the next lemma that ��separations of connectedbinary matroids produce ��sums with connected components�

�� Lemma� Any ��separation of a connected binary matroid Mproduces a ��sum with connected components M� and M�� Conversely�any ��sum of two connected binary matroids M� and M� is a connectedbinary matroid M �

Proof� The above denitions establish the lemma except for the connect�edness claims� It is easily veried that connectedness of B of �� impliesconnectedness of B� and B� of �� and vice versa� By Lemma ��connectedness of representation matrices is equivalent to connectedness ofthe corresponding matroids� Thus� connectedness of B� B�� and B� isequivalent to connectedness of M � M�� and M�� respectively�

The ��sum decomposition or composition has the following appealinggraph interpretation�

�� Lemma� Let M be a connected binary matroid that is a ��sumof M� and M�� as given via B� B�� and B� of �� and ��


�a� If M is graphic� then there exist ��connected graphs G� G�� and G� forM � M�� andM�� respectively� with the following feature� The graph Gis produced when one identi�es the edge x of G� with the edge y of G��and when subsequently the edge so created is deleted� The drawingbelow depicts the general case�

�� H1H1 H2 x H2y

Graph G Graph G� Graph G�

�b� If M� and M� are graphic �resp� planar�� then M is graphic �resp�planar��

Proof� �a� Given B of �� for M � let E� � X� � Y� and E� � X� �Y�� We know that the pair �E�� E�� is a ��separation of M � Select anygraph G for M � Since M is connected� G is ��connected� Let H� andH� be the subgraphs of G with edge sets E� and E�� respectively� ByTheorem �� part �b�� the ��separation �E�� E�� ofM implies that theconnected components of H� and H� are connected in cycle fashion� Bythe switching operation of Section �� we may rearrange G so that bothH� and H� are connected� We may assume that G already is that graph�Thus� G is as given by �� with connected H� and H�� By Lemma�� the matrices B� and B� of �� are connected� By �� andthe connectedness of B�� contraction in G of the edges of X� � fxg anddeletion of the edges of Y� must produce the graph G� of �� Similarly�contraction of the edges of X� and deletion of the edges of Y� � fyg mustproduce the graph G� of �� Thus� G� and G� are graphs for thegraphic matroids given by B� and B� of �� The two graphs are ��connected since the corresponding matrices are connected�

�b� Let G� and G� be ��connected graphs for M� and M�� For some H�

and H�� the drawings of �� correctly depict G� and G�� Identify theedge x of G� with the edge y of G�� then delete the so�created edge� Thereare two ways to accomplish the identication� but either way is acceptableand produces a graph G as depicted by �� Elementary checking offundamental cycles of G versus fundamental circuits of M conrms thatG represents the matroid M dened by B of �� Thus� M is graphic�If G� and G� are plane graphs� we can carry out the edge identicationso that G becomes a plane graph� Thus� M is planar if M� and M� areplanar�

We turn to the more challenging case of general k�sums� with k � ��

� � General k�Sums ��

�� General k�Sums

We are given a ��connected binary matroidM on a set E� For some k � ��we also know an exact k�separation �E�� E�� We want to decompose Min some useful way� We investigate this problem using the matrix B of�� Slightly enlarged� we repeat that matrix below�

��

A2

B =

A1

D

Y1 Y2

X1

X2

0

Matrix B with exact k�separation

Recall that the submatrix D of B has GF��rank k � �� We want todecompose M into two matroids M� and M� that correspond to two sub�matrices B� and B� of B� As in the ��sum case� we postulate that B�

and B� include A� and A�� respectively� Furthermore� B� and B� mustpermit a reconstruction of B� The latter requirement can be satised byincluding in B� �resp� B�� a row �resp� column� submatrix of D with thesame rank as D� i�e�� with GF��rank k � �� Indeed� the submatrix D ofB can be computed from these row and column submatrices� We providethe relevant formulas in a moment� Last but not least� we want B� andB� to be proper submatrices of B�

There are numerous ways to satisfy these requirements� In the mostgeneral case� both B� and B� intersect all four submatrices A�� A�� D� and� of B� and thus induce the following rather complicated�looking partitionof B� We explain the partition momentarily�

��

Y1

A2DD1

D12 D2

C2

C1B =

A1

Y2

Y1Y2

X1

X2

X1

X2

0

0

Partition of B displaying k�sum


In the notation of �� the submatrix B� of B� which is not explicitlyindicated� is indexed by X� �X� and Y��Y �� Furthermore� the submatrixB� is indexed by X� �X� and Y � � Y�� Hence� B� contains A�� intersectsA� in C�� and intersects D in �D� j D�� The submatrix B� contains A��intersects A� in C�� and intersects D in �D�D�� We assume that C� �resp�C�� is a proper submatrix of A� �resp� A�� This implies that both B� andB� are proper submatrices of B� Observe that D is the submatrix of Dcontained in both B� and B�� We assume that both submatrices �D� j D�and �D�D�� of D have GF��rank equal to k�� By Lemma �� thisimplies that D has GF��rank k � �� We display the matrices B� and B�

below�

��

Y1

DD1 C2

C1B1 =

A1

Y2

Y1

X1

X2

X10

0 A2D

D2

C2

C1

Y1 Y2

Y2

B2 =

X1

X2X2

0 0

Matrices B� and B� of k�sum

The decomposition of B into B� and B� corresponds to a decomposition ofM into two matroids� say M� andM�� that are represented by B� and B��We call that decomposition of M a k�sum decomposition and declare M�

and M� to be the components of the k�sum� The decomposition process isreadily reversed� All submatrices of B except for the submatrixD�� whichis indexed by X� �X� and Y� � Y �� are present in B� and B�� and thusare already known� For the computation of D�� from B� and B�� we rstdepict D and its submatrices�

�� DD1

D12 D2D =

X2X2

Y1

Y1

Partitioned version of D

Since GF��rank D � GF��rank D� there is a matrix F so that �D�� jD�� F � �D� j D�� Thus�

�� D� � F �D

and

�� D�� F �D�


We rst solve �� for F � The solution may not be unique� but anysolution is acceptable� Then we use F in �� to obtain D��

Suppose D is square and nonsingular� Then from �� we deduceF � D� � �D�� With that solution� we compute D�� according to ��as

�� D�� D� � �D�� D��

The reconstruction of B from B� and B� corresponds to a k�sum compo�

sition of M� and M� to M � Both the k�sum decomposition of M and thek�sum composition of M� and M� to M � we call a k�sum and denote it byM �M� �k M��

It is a simple exercise to prove that the �� and ��sums of Section ��are special instances of the above situation� In the ��sum case� D is a zeromatrix� Thus� we may select C�� C�� D�� D�� and D to be trivial or empty�whichever applies� Then by �� we have B� � A� and B� � A�� whichis the ��sum of Section �� In the ��sum situation� D is the � � � matrix� � �� C� and C� are trivial matrices� �D� j D� is row x of D� and �D�D�� iscolumn y of D� The formula �� for D is nothing but �� with thejust�dened D�� D�� and D�

The reader is well justied to wonder why we consider such complexk�sums� We argue in favor of our approach as follows� Suppose we intend toinvestigate a certain matroid property that is inherited under minor�taking�say graphicness� To be even more specic� let us assume that we want toobtain the minor�minimal matroids that are not graphic� Suppose theabove general conditions for k�sums admit a particular k�sum case wheregraphicness of the components implies graphicness of the k�sum� Then nominor�minimal nongraphic matroid can be such a k�sum� For if such a k�sumM exists� then its components� smaller as they are� are graphic� But bythe just assumed feature of the k�sum�M is graphic as well� a contradiction�Thus� we can restrict our search for the minimal nongraphic matroids toinstances that are not k�sums� a possibly very attractive insight� A secondsituation is as follows� We want to nd a construction for matroids havinga certain property� Indeed� we are willing to allow composition steps tobe part of the construction� In that case� we need composition rules thatpreserve the property of interest�

The k�sum compositions described here� complex as they may be� dopreserve a number of interesting matroid properties� Thus� for investi�gations into these properties� the k�sums are very useful� The evidencesupporting this claim will be presented in subsequent chapters�

So far we have expressed the k�sum decomposition or composition interms of representation matrices� It is interesting to consider k�sums interms of matroid minors as well� By the very derivation of B� and B��the components M� and M� of the k�sum are minors of M � It is easy to


see from �� that B� and B� share precisely the submatrix B given by�� below� Dene M to be the matroid represented by B� Clearly� Mis a minor of M � but more importantly� of M� and M� as well� Indeed�one may view the composition of M� and M� to be an identication of theminor M of M� with the minor M of M�� One might also say that M asminor of M forms the connection between the components M� and M� inM � In agreement with the latter terminology� we call M the connecting

matroid or connecting minor of the k�sum� By �� and the fact thatD and D have the same rank� the pair �X� � Y �� X� � Y �� is an exactk�separation of M � provided that jX� � Y �j� jX� � Y �j � k� The lattercondition is satised� for example� if both C� and C� are nonempty andnontrivial�

��

D C2

C1X1

X2

0=B

Y1 Y2

Submatrix B representing the connecting minor Mof the k�sum

Recall that we want important matroid properties to be preserved underk�sum composition� Evidently� M is the minor of M that decides whetheror not we achieve that goal� Thus� the selection of M requires great carewhen one desires k�sums suitable for the investigation of matroid properties�Note that a particular choice of M imposes constraints on the exact k�separations that are to be converted to a k�sum� For example� any k�separation �E�� E�� capable of producing a k�sum with a given M minormust satisfy for i � �� jEij � jXi�Y ij�� since otherwise the submatricesC� and C� of B are not proper submatrices of A� and A�� The selection ofM becomes even more complex when computational aspects are considered�For example� one might demand that the k�sums dened by M can belocated in polynomial time�

For this book� ��sums are of considerable importance� A good choicefor the matrix B representingM turns out to be

��1

D C2

C1 0=B =

D

0111

Matrix B of ��sum

where D is any �� GF��nonsingular matrix� If D is the �� identitymatrix� then by �� B represents up to indices M�W�� which is thegraphic matroid of the wheel with three spokes� If D contains exactly three


�s� the only other choice� then by one GF��pivot� say in C� � �� weobtain the former case� Thus� in all instances�M is anM�W�� minor ofM �With B of �� the matrix B of �� becomes the following matrix�

��

11

Y1

A2DD1

D12 D2

B =

A1

Y2

Y1Y2

X1

X2

X1

X2

0

01 1


The representation matrices B� and B� of the componentsM� andM� are

��

1 1

1

Y1

DD1

B1 =A1

Y2

Y1

X1

X2

X10

01

1

1

A2D

D2

Y1 Y2

Y2

B2 =

X1

X2X2

011

0


When does an exact ��separation �E�� E�� of a binary matroid M lead toa ��sum with this M� Appealing su�cient conditions turn out to be ��connectedness of M and the restriction that jE�j� jE�j � �� We state thisresult in the next lemma�

�� Lemma� Let M be a ��connected binary matroid on a set E�Then any ��separation �E�� E�� ofM with jE�j� jE�j � � produces a ��sum�and vice versa�

Proof� Let �E�� E�� be a ��separation of M with jE�j� E�j � �� SinceM is ��connected� the ��separation must be exact� Hence� by ��M has a binary representation matrix B given by �� below� withGF��rank D � �� In particular� any column submatrix of D containingfour or more columns must contain a zero column or two parallel columns�We claim that the row index subset X� of B of �� is nonempty� As�sume otherwise� By jX��Y�j � �� we have jY�j � �� But then D has a zerocolumn or two parallel columns� andM is not ��connected� a contradiction�Thus� X� �� and trivially Y� ��


��

A2

B =

A1

D

Y1 Y2

X1

X2

0

Matrix B with ��separation

Let A�� be a connected block of A�� There must be at least one such block�since otherwise M has coloops� To exhibit A�� we partition B of ��further as follows�

��

Y1Y11

A2

B =A11

D11

Y2

X1X11

X2

00

0

0 1

0 1

Matrix B of �� with partitioned submatrix A�

Note the column submatrix D�� of D corresponding to A�� We claim thatGF��rank D�� If this is not the case� then the pair �X�� Y�� E ��X�� Y�� constitutes a �� or ��separation of M � a contradiction�

Let us examine the submatrix �A��D�� of B of �� more closely�Consider the paths in the bipartite graph BG�A�� between all pairs ofnodes y and z in Y�� for which the columns y and z of D�� are GF��independent� Since A�� is connected and GF��rank D�� there is atleast one path� In a shortest path� all intermediate nodes in Y�� correspondto zero columns of D�� Suppose a shortest path has at least four arcs�With the path shortening technique of Chapter �� we reduce that path bypivots in A�� to one with exactly two arcs� Thus� we may assume that ashortest path of length � exists� say connecting nodes y and z in BG�A��Put di�erently� we may assume that A�� contains a row with two �s incolumns indexed by y� z � Y�� such that the columns of D�� indexed by yand z are GF��independent� The path shortening pivots� if any� can alsobe carried out in B of �� without a�ecting the entries of D and A��Thus� we may assume A� to contain a row with two �s in columns y and zsuch that the columns y and z of D are GF��independent�


By duality� we may suppose that A� contains a column having two �sin rows u and v such that the rows u and v of D are GF��independent�By Lemma �� GF��rank D � � implies that the rows u and v of Dand the columns y and z of D must intersect in a �� GF��nonsingularsubmatrix D of D� When we partition B of �� to exhibit the two �sof A�� the two �s of A�� and D� we get an instance of �� with theconnecting minor given by B of �� Thus� M is a ��sum�

The converse is obvious� since any ��sum given by �� producesthe ��separation �X� � Y��X� � Y�� with jX� � Y�j� jX� � Y�j � ��

Let us translate the ��sum operation of binary matroids to graphs�For simplicity� we conne ourselves to the case where a given graph G withedge set E is ��connected� In agreement with Lemma �� we assumethat G has a ��separation �E�� E�� where jE�j� jE�j � �� By that lemma�the graphic matroidM �M�G� has a ��sum decomposition induced by the��separation �E�� E�� The previously discussed results for the connectingminorM of that ��sum can be restated for the case at hand as follows� Thegraph G has a minor G on edge set E and with a ��separation �E�� E��such that E� E� and E� E�� Up to indices� G is the wheel W��

The deletions and contractions reducing G to G evidently involve theedges of E� �E� and E� �E�� Suppose in G we carry out these deletionsand contractions just for the edges of E� � E�� Declare G� to be theresulting graph� The process producing G� corresponds to the reductionof the ��sumM to the componentM�� Thus�M� �M�G�� By analogousreductions� this time conned to the edges of E� � E�� we obtain a graphG� for which M� � M�G�� Examine the representation matrices B� andB� of �� for M� and M�� Clearly� the set X� � Y � is a triangle inM�� and the set X� � Y � is a triad of M�� Correspondingly� E� must be atriangle in G�� and E� must be a cocircuit of G� of size �� Indeed� by thestructure of G and G�� that cocircuit of G� must be a ��star� The drawingbelow summarizes these conclusions for G� G�� and G��

�� H1 H2H1 H2


So far we have described the ��sum decomposition of G into componentgraphs G� and G�� The composition is even easier� We identify the threenodes of G� explicitly shown in �� with the three nodes of G� so thatthe triangle of G� and the ��star of G� form a copy of G� The resultinggraph is G plus that copy of G� We delete the edges of that copy of G andhave the desired G�


We use the term ��sum to describe the decomposition of G into G�

and G�� as well as the composition of the latter graphs to G� In agreementwith the matroid case� we denote this by G � G� �� G� and call G theconnecting graph or connecting minor of the ��sum of G� and G��

We saw earlier that certain separations can be converted to �� or ��sums� In the next section� we discuss how one may locate such separationsand thus �� and ��sums�

�� Finding �� and ��Sums

Suppose that for a given binary matroidM � we want to either nd a �� or ��sum decomposition� or conclude that there is no such decomposition�In this section� we solve this seemingly di�cult problem� Main tools arethe separation algorithm of Section �� some results about sequences ofnested minors of Section �� and the lemmas of Section �� that assure usof a �� or ��sum decomposition when certain separations are present�

We start with the simplest case� where the given binary matroid Mis not connected� By Lemma �� every representation matrix of Mis disconnected� Thus� we easily detect this situation using an arbitrarilyselected representation matrix B of M � From that B� we may deduce a��sum decomposition ofM as described in Section �� Thus� from now on�we may assume M to be connected� We also suppose that M has at leastfour elements�

To nd a �� or ��sum decomposition� if it exists� we rst rely on Lemma�� That result says that under the above assumptions� M has a ��separation or anM�W�� minor� Implicit in the proof of the lemma is a poly�nomial algorithm that decides which case applies� Suppose a ��separationis determined� Then Lemma �� tells us that this ��separation can beeasily converted to a ��sum decomposition of M � So we now assume thatwe detect an M�W�� minor� say N �

To M and N we apply Theorem �� which in turn cites The�orem �� Accordingly� there is a polynomial algorithm that nds a��separation of M � or that establishes a sequence of nested ��connectedminors M�� M�� Mt � M with the following features� M� is isomor�phic toN � For some s� � � s � t� the subsequenceM��M�� Ms consistsof wheel matroids and has gap �� and the subsequence Ms� Ms�� Mt

has gap �� We know already how to handle the ��separation case� Thus�we may assume that the prescribed sequence of nested ��connected minorsis found� The description of that sequence can be simplied by saying thatthe sequence starts with anM�W�� minor and has gap � or �� We use thatsequence to nd a ��sum decomposition for M or to prove that there is nosuch decomposition� The details are as follows�

� � Finding �� and ��Sums ��

First� we observe that by Lemma �� M has a ��sum decomposi�tion if and only if it has a ��separation �E�� E�� with jE�j� jE�j � �� Indeed�the decomposition can be e�ciently determined from such a ��separationusing the algorithm implicit in the proof of that lemma� Thus� our task isnding such a ��separation or proving that none exists�

FromSection �� we know that the intersection algorithm can solve thelatter problem in polynomial time� However� the order of the algorithm is solarge that the scheme is practically unusable unless the matroidM is quitesmall� In addition� it presently is not known how one might reduce the orderto an acceptable level� For this reason� we outline below another algorithmthat by suitable renements and appropriate implementation does becomevery e�cient� The method is based on the following observations� wherefor the moment we assume that M does have a ��separation �E�� E�� withthe desired property� For j � �� t� let Ej be the groundset ofMj � For

each j� and for i � �� dene Eji � Ej � Ei� Clearly� each pair �Ej

�� Ej��

satisfying jEj�j� jE

j�j � � is a ��separation of Mj � By its very derivation�

any such ��separation ofMj induces a ��separation of M with at least fourelements on each side� for example �E�� E��

For the moment� let j be the smallest index so that jEj�j� jE

j�j � ��

There is such an index since �Et�� E

t�� E�� E�� satises jEt

�j� jEt�j � ��

Note that Mj has necessarily at least eight elements� Since M� has onlysix elements� we conclude that j � �� Because of the minimality of j� wemust have jEj��

� j or jEj�� j � �� Since the gap of the sequence is at most

�� we deduce from the latter fact that jEj�j or jE

j�j � �� There are only

polynomially many pairs �Ej�� E

j�� for Mj where jE

j�j� jE

j�j � � and jEj

�j or

jEj�j � �� Indeed� there are only polynomially many pairs satisfying that

condition when we allow j to range over �� t�

We use these observations as follows� Note that we still assume Mto have a ��separation �E�� E�� where jE�j� jE�j � �� To nd such a ��separation� we rst locate for j � �� t all pairs �Ej

�� Ej�� satisfying

jEj�j� jE

j�j � � and jEj

�j or jEj�j � �� For each pair� we test whether it

is a ��separation of Mj � We discard the cases where the answer is �no��

For each remaining pair� say �Ej�� E

j�� of Mj � we check with the separation

algorithm of Section �� whether it induces a ��separation of M � For atleast one such pair� say �Ej

�� Ej�� of Mj � we must obtain an a�rmative

answer� The ��separation of M so found� say �E�

�� E�

�� satises for i � �� E�

i Eji and thus jE�

ij � jEji j � �� Hence� �E�

�� E�

�� is the sought�after��separation of M �

So far� we have assumed that M has a ��separation �E�� E�� wherejE�j� jE�j � �� If that is not the case� we can still carry out the abovepolynomial process� without success of course� But the lack of successproves that M has no ��separation �E�� E�� with jE�j� jE�j � ��

The next theorem summarizes the above discussion�


�� Theorem� There is a polynomial algorithm that accepts anybinary matroid M as input� and that outputs the conclusion that M isnot connected� or that M is connected but not ��connected� or that M is��connected� In the �rst case� the algorithm also supplies a ��sum decom�position� in the second case� a ��sum decomposition� and in the third case�a ��sum decomposition if it exists�

In the next section� we meet close relatives of ��sums� called �sumsand Y�sums�

�� Delta�Sum and Wye�Sum

There are two variations of the ��sum� We call them delta�sum and wye�sum� for short �sum and Y�sum� The relationships among �� and Y�sums are very elementary� That does not imply that they may be employedinterchangeably� Indeed� for some applications one denitely prefers onetype over another� Both �sum and Y�sum are derived from the ��sum�so for convenient reference we repeat to related matrices of �� and�� below�

��

11

Y1

A2DD1

D12 D2

B =

A1

Y2

Y1Y2

X1

X2

X1

X2

0

01 1


��

1 1

1

Y1

DD1

B1 =A1

Y2

Y1

X1

X2

X10

01

1

1

A2D

D2

Y1 Y2

Y2

B2 =

X1

X2X2

011

0


We proceed as follows� We rst dene � and Y�sums� Then we showthat the components of these sums are isomorphic to some minors of the

� Delta�Sum and Wye�Sum ��

sum� provided the sum is ��connected� Finally� we indicate why one wouldwant to use one type of these sums over another in certain applications�

We derive the �sum from the ��sum as follows� Let M be a ��connected binary matroid with a ��sum decomposition into M� and M��with representation matrices B� B�� and B� given by �� and ��Suppose inM� we perform a Y exchange that replaces the triad X� �Y �

by a triangle Z� as specied in �� The new matroid� say M�� isrepresented by

��X2

Y2Z2

A2dD

D2

11B2 =

Matrix B�� for M��

where the three columns of B�� indexed by Z� sum to � �in GF��We say that M has been decomposed in a �sum decomposition into

components M� andM�� The reverse process we call a �sum composition

of M� and M�� to M � We refer to both cases as a �sum and denote itby M � M� �� M�� The relevant triangle in M� or M�� i�e�� X� � Y �

or Z�� is the connecting triangle of M� or M�� It is not di�cult to check�say using the circuits of M � that the �� operator is commutative� Thus�M��M�� M��M�� We omit the details of the proof� since we willmake no use of this result�

The graph interpretation of the �sum is as follows� Recall that by�� the ��sum decomposition of a graph G into component graphs G�

and G� can be depicted as

�� H1 H2H1 H2


The Y exchange transformingM� to M�� becomes a replacement of theexplicitly shown ��star of G� by a triangle� Let G�� be the resulting graph�We may view the composition of G� and G�� to G to be an identicationof the triangle of G� with that of G�� followed by a deletion of the so�created triangle from the resulting graph� Borrowing from the terminologyof the matroid case� we call G a �sum of G� and G�� and denote this byG � G� �� G��

The Y�sum of binary matroids is dened in an analogous fashion� Asfor the �sum� we start with a ��sumM �M��M�� But now we convert


the triangle X� � Y � of M� by a Y exchange to a triad Z�� gettinga matroid M�Y� By B� of �� and the Y exchange of �� thefollowing matrix B�Y represents M�Y�

��1

DD1

A1

Y1

Z1

X1

1

e

B1Y =

Matrix B�Y for M�Y

Here� the three rows of B�Y indexed by Z� sum to � �in GF�� We callMa Y�sum with components M�Y and M�� and write M �M�Y �YM�� Therelevant triad in M�Y or M� is the connecting triad of M�Y or M�� TheY�sum operator can be veried to be commutative� as one would expect�

In the graph case of a Y�sum� we replace the explicitly shown triangleof G� of �� by a ��star� getting a graph G�Y� G is obtained fromG�Y and G� as follows� Let i� j� k �resp� p� q� r� be the three nodes ofattachment of the ��star in G�Y �resp� G�� Assume that i �resp� j� k�corresponds to p �resp� q� r�� Now connect the node i �resp� j� k� of G�Y

minus its ��star with the node p �resp� q� r� of G� minus its ��star� sayusing an edge e �resp� f � g�� The edges e� f � and g form a cutset in thenew graph� and contraction of that cutset produces the graph G� We callG a Y�sum of G�Y and G�� and denote this by G � G�Y �Y G��

The above Y�sum operation may seem cumbersome� In fact� the readermost likely has an alternate Y�sum composition in mind where the ��starof G�Y is identied with that of G�� and where the edges of the resulting��star are deleted� Both Y�sum processes produce the same outcome� butthey do di�er in the way in which the composition is carried out� We chosethe above� more complicated description because then the �sum and Y�sum composition steps are dual operations� The reader may want to tryto dualize the alternate Y�sum process to a new �sum process for graphs�This should turn out to be a di�cult task� Indeed� one can show that sucha dual composition process cannot exist for graphs� At any rate� we knowthat if M is the �sumM� ��M�� then M� is the Y�sum M�

� �Y M�

��The matrices B� and B� of �� are submatrices of B of ��

Thus� by denition�M� andM� are minors of M � One cannot expect M�Y

or M�� also to be minors of M due to the assigned index sets Z� and Z��However� one would hope that M�Y and M�� are isomorphic to minors ofM � The next result supports this notion�

�� Lemma� Let M be the ��connected matroid represented by the

� Delta�Sum and Wye�Sum ��

binary matrix B of �� Then M has minors isomorphic to M�Y andM��

Proof� By duality� it su�ces that we prove the claim for M�� For this�we may assume that the nonsingular submatrix D of B of �� is a �� identity matrix� since this can always be achieved by at most one GF��pivot in B on one of the �s indexed by X� and Y �� Given D as a � � �identity matrix� we rst perform in B two GF��pivots on the �s of D�These pivots convert B to the matrix B� below� We explain the structureof B� momentarily�

��

X2

(A2)'

(A1)'

B' =

Y2

Y2

X1

Y1

X1

abc

Y1~

X2~

Y1 = Y1 − Y1~

X2 = X2 −~

X2

0 1

0

Matrix B� derived from B of �� by two pivots

Note that the pivots in B are made within the submatrix B�� Thus� thesubmatrix of B� indexed by the rows of X� � Y � � �X� and the columns ofX� � Y� represents M�� Since �X� � Y��X� � Y�� is a ��separation of M �the submatrix D� of B� indexed by X� � Y � and X� � Y� has GF��rank�� Indeed� since X� � Y � is a triad of M�� the row subvectors a� b� c arenonzero and add �in GF�� to the zero vector� By Lemma �� pivotsin �A�� of B� are possible so that the new �A�� of the new B� has two �sin some column w and in rows u and v for which the rows u and v of D� areGF��independent� The submatrix of the new B� indexed by �X� � fu� vgand X� � Y� � fwg then is isomorphic to M�� by the rule �� for Yexchanges�

The simple relationships among ��sum� �sum� and Y�sum may de�ceive one into thinking that each such sum can be substituted for anotherone in all applications� A rst warning of the incorrectness of this conclu�sion comes from applications involving planar graphs� That is� a Y�sumof two planar graphs must be planar� while a �sum of two planar graphsneed not be planar� Planarity of the Y�sum is most easily seen when oneembeds each component on a sphere and combines the two drawings to onedrawing on another sphere� The �sum need not be planar� For example�the nonplanar graph K�� can be created from two copies of the planar


graph given below�

�� gf

e

Planar component of nonplanar �sum

The triangle fe� f� gg of one copy is identied with that of the other copy�Subsequently� the triangle created by the identication process is deleted�and K�� results� Other signicant applications where caution is in orderinvolve the max��ow min�cut matroids and the convex hull of the disjointunions of circuits of a binary matroid �the so�called cycle polytope�� On theother hand� when one examines matroid regularity� one may freely switchamong the three types of sums without penalty� Validity of these claims isproved in Chapters �� and ��

In the nal section� we link the material of this chapter to relatedresults and provide references�


Basic aspects of the composition or decomposition of matroids have beenexplored in a number of references� e�g� in Edmonds and Fulkerson ��Edmonds ��a�� Nash�Williams �� Bixby�� Brylawski �� Cunningham �� a�� b�� Cunningham and Edmonds �� Iri �� Na�kamura and Iri �� Seymour ��b�� Tomizawa and Fujishige ��Fujishige �� and Conforti and Laurent �� De�composing highly connected matroids and composing them again has notbeen treated extensively� Related to the approach taken here are the de�composition and composition of the graphs without K� minors in Wagner��a�� the modular constructions of Brylawski �� and the decompo�sition and composition of the regular matroids in Seymour ��b�� Theconcept of the connecting minor is taken from Truemper ��a�� where itis developed for general matroids using abstract matrices� instead of justfor binary matroids as done here� The material of Sections �� and �� isalso derived from general matroid results of that reference� The methodsof Section �� for nding �� and ��sums are from Truemper ��a��

Locating k�sums for general k � � is muchmore di�cult since one mustidentify a k�separation and an a priori specied connecting minor� We donot know how this can be e�ciently accomplished for binary matroids� let

� Extensions and References ��

alone for general matroids� Truemper ��a� treats a particular ��sumcase where the connecting minors have eight elements�

Most of the material of Section �� is based on Gr�otschel and Truem�per ��b�� That reference examines the �sum and Y�sum in much moredetail� The �sum of Section �� is popular in graph theory �e�g�� see Wag�ner ��a�� and is used in Seymour ��b� to e�ect the decomposition ofthe regular matroids� Lemma �� is proved by a quite di�erent methodin Seymour ��b��

We have omitted entirely some interesting decomposition results ofTruemper ��a��b�� These references contain a number of basic re�sults about k�sums and three decomposition classication theorems plusapplications� The theorems cover general matroids� binary matroids� andgraphs� respectively� Each of them says that any given matroid is decom�posable� or a number of elements can be removed in any order withoutloss of ��connectivity� or the matroid belongs to a small class of matroidswith few elements� The proofs are quite long and are the main reason thatwe have omitted these results� Signicant applications of the theorems arerather short proofs of the profound excluded minor theorems for planargraphs� graphic matroids� and regular matroids due to Kuratowski ��and Tutte �� The latter theorems are proved in this bookusing di�erent machinery in Chapters �� and ��

Chapter �

Matrix Total Unimodularity and

Matroid Regularity

�� Overview

At this point� we have assembled the basic matroid results and tools forthe remaining developments� We now begin the investigation into the twomain technical subjects� which are matrix total unimodularity and matroidregularity�

Total unimodularity and its many variants are very important for com�binatorial optimization� For a long time� matrix techniques evidently didnot permit any profound insight into total unimodularity� That fact mo�tivated the translation of the matrix property of total unimodularity intothe matroid property of regularity� The translation opened the way forthe application of powerful matroid techniques� After an e�ort spanningthree decades� matroid regularity and thus total unimodularity were atlast understood� The main results� in historical order� are due to Tutteand Seymour� The subsequent chapters contain these results as well asclosely related material�

We proceed as follows� In Section �� we de�ne total unimodularity�sketch important applications� and translate total unimodularity into ma�troid language� thus getting matroid regularity� Section �� contains the�rst major result for regularity� which is Tuttes characterization of theregular binary matroids� Recall that F� is the Fano matroid and F �

� itsdual� and that Um

n is the uniform matroid on n elements where any subsetwith at most m elements is independent� The characterization says that amatroid is regular if and only if it has no U�

�� F�� or F �

�minors� The origi�

nal proof of that characterization is long and complicated� Here we rely on

��

� � Basic Results and Applications ��

a proof of amazing brevity and clarity by Gerards� A minor modi�cationof the proof produces Reids characterization of the ternary matroids� i�e��the matroids representable over GF�� The characterization says that amatroid is ternary if and only if it has no U�

�� U�

�� F�� or F �

�minors� This

result is presented in Section �� The �nal section� �� contains extensionsand references�

The chapter requires knowledge of the material of Chapters � and �

�� Basic Results and Applications of

Total Unimodularity

In this section� we de�ne matrix total unimodularity and matroid regularityand establish elementary results about these two properties� We also pointout representative applications of total unimodularity�

A real matrix is totally unimodular if every square submatrix has � or�� as determinant� Thus� the entries of a totally unimodular matrix mustbe � or �� For example� the zero matrices and the identity matrices aretotally unimodular� Nontrivial examples are given by the next lemma�

�� Lemma� Let A be a real f��g matrix where every column

contains exactly one �� and one �� Then A is totally unimodular�

Proof� Let D be a square submatrix of A� We induct on the order of D�In the nontrivial case� D has order k � � and has no zero columns� If D hasa column with exactly one �� we use cofactor expansion and induction tocalculate the determinant as � or �� Otherwise� D has exactly one ��and one �� in each column� and thus the determinant is ��

De�ne F to be the support matrix of the matrix A of Lemma ��View F to be binary� Since A has exactly one �� and one �� in eachcolumn� the matrix F has exactly two �s in each column� Then accordingto the de�nitions of Section �� F is the node�edge incidence matrix ofsome graph G� Speci�cally� each row of F de�nes a node of G� and eachcolumn g of F � say with �s in rows i and j� de�nes an edge connectingnodes i and j of G� We now declare the matrix A to represent a certaindirected version of G� as follows� Assume that column g of A has a ��in row i and a �� in row j� Then we replace the undirected edge of Gconnecting the nodes i and j by a directed arc from i to j� Let H be theresulting directed graph� We call A the node�arc incidence matrix of H�By Lemma �� every node�arc incidence matrix is totally unimodular�

The next lemma summarizes elementary operations that maintain to�tal unimodularity�

�� Chapter � Total Unimodularity and Regularity

�� Lemma� Total unimodularity is maintained under the taking of

submatrices� transposition� pivots� and the adjoining of zero or unit vectors

or of parallel f��g vectors�

Proof� The pivot is the only nontrivial operation� Let a pivot convert atotally unimodularmatrix A to a matrix A�� Adjoin to A an identity matrixto get �I j A�� Now A is totally unimodular if and only if every basis matrixof �I j A� has �� as determinant� The latter property is maintained underthe elementary row operations in �I j A� that correspond to the pivot in A�and that� together with scaling by f��g factors and a column exchange�convert �I j A� to �I j A�� Thus� A� is totally unimodular�

According to the next lemma� counting may be used to check totalunimodularity for the f��g matrices whose bipartite graph BG�� is achordless cycle� i�e�� for the k � k real matrices� k � �� of the form

�� ±1±1 ±1±1 ±1

±1±1±1

.

. ..

Matrix whose bipartite graphis a chordless cycle

�� Lemma� The real matrix of �� is totally unimodular if and

only if its entries sum to ��mod ��

Proof� Let A be the matrix of �� Scaling of a row or column of A by�� does not a�ect total unimodularity and changes the sum of the entriesof A by a multiple of � Thus� for the proof of the lemma� we scale A tothe matrix A� given by

�� .-1-1 11 α

11-1

. ..

Particular matrix whose bipartite graphis a chordless cycle

By cofactor expansion and counting� we con�rm that detIRA� is � �resp� ��if and only if � � � �� which holds if and only if the entries of A�

sum to � �resp� �� The �only if� part of the proof is then immediate� The�if� part follows from Lemma ��

A matrix B is regular if it is binary and if its �s can be replaced by��s so that a real matrix results that is totally unimodular� By the above


discussion� every node�edge incidence matrix is regular� The signing of aregular matrix to achieve a real totally unimodular matrix is essentiallyunique� a fact proved next�

�� Lemma� Let A and A� be two totally unimodular matrices with

the same support matrix� Then A� may be obtained from A by a scaling

of the rows and columns by f��g factors�

Proof� We may assume A to be connected� Let T be a tree of BG�A�� andlet T � be the corresponding tree of BG�A�� Because of scaling� we maysuppose that A and A� agree on the entries corresponding to the edgesof T and T �� Suppose A and A� di�er� say on the �i� j� element� Thecorresponding edges e and e� of BG�A� and BG�A�� form cycles C and C �

with T and T �� respectively� Select T and e� and thus T � and e�� so thatthe cardinality of the cycles is minimum� Suppose C and C � have chords�By the minimality condition� the entries of A and A� corresponding to suchchords must agree� But then C and C � do not have minimum cardinality� acontradiction� Thus� C and C � are chordless cycles of BG�A� and BG�A��and the submatrices of A and A� corresponding to C and C � are of theform �� The sums of the entries of the two submatrices di�er by ��since they di�er on just one entry� But then one of the sums is ��mod ��and the other one is ��mod �� By Lemma �� one of A and A� is nottotally unimodular� a contradiction�

Lemma �� and its proof imply the following result�

�� Corollary� There is a polynomial algorithm that by signing

converts any regular matrixB to a real matrixA that is totally unimodular�

The signing can be carried out as follows� First some submatrix is signed�

Then� iteratively� the method signs an arbitrarily selected additional row

or column that is adjoined to the submatrix on hand�

Proof� Use Lemma �� and the arguments of its proof to establish thecorrect signs for the additional row and column�

A matroidM is regular if it has a regular representation matrix B� Theessentially unique totally unimodular matrix A deduced from the regularB may be used to represent M as follows�

�� Lemma� Let M be a matroid represented by a real totally

unimodular matrix A� De�ne A� to be a numerically identical copy of

A� but view A� to be a matrix over an arbitrary �eld F � Then M is

represented by A� over F �

Proof� Since all square submatrices of A have real determinant � or ��any such submatrix of A must be IR�nonsingular if and only if the corre�sponding submatrix of A� is F�nonsingular� Thus� A over IR and A� overF de�ne the same matroid�

Lemma �� leads to a short proof of the following theorem�


�� Theorem� The following statements are equivalent for a matroid

M �

�i� M is regular�

�ii� M has a real representation matrix that is totally unimodular�

�iii� M is representable over every �eld�

�iv� M is representable over GF�� and GF��v� M representable over GF�� and every representation matrix over

GF�� when viewed as real� is totally unimodular�

Proof� By de�nition� �i� �� ii�� and by Lemma �� ii� �� iii��Trivially� �iii� �� iv� and �v� �� ii�� We show �iv� �� v�� By �iv��some matrix C over GF�� and the binary support matrix B of C representM � Declare A to be a copy of C� but view A to be over the reals� We haveshown �v� once we have proved A to be totally unimodular� Suppose A hasa submatrix with real determinant di�erent from �� We may assumethat A itself is such a matrix� and that every proper submatrix of A istotally unimodular� If A is a � � � matrix� then by a trivial case analysiswe must have up to scaling by f��g factors in A and C�

��1 1 -1-1

11 1

1B = ;

1 1A = ;

1 1C =

A over IR� B over GF�� and C over GF��

Then det�B � � and det�C � �� We conclude that B and C representdi�erent matroids� a contradiction� Suppose the order of A is greater than�� We produce the � � � case by pivots in A� B� and C� as follows� Weperform an IR�pivot on any nonzero entry of A and delete the pivot rowand pivot column� In B and C� we carry out the corresponding operations�Let A�� B�� C � be so obtained from A� B� and C� It is easy to see that A�

is not totally unimodular� and that every proper submatrix of A� is totallyunimodular� Furthermore� B� is the binary support matrix of A�� and A�

and C � have their �s� ��s� and ��s in the same positions� By induction�the contradictory �� case applies�

The preceding results and arguments yield the following corollaries�

�� Corollary� For every regular matroid M � the following holds�

�a� Every binary representation matrix of M is regular�

�b� Every minor of M is regular�

�c� The dual of M is regular�

Proof� Lemma �� and Theorem �� imply �a�� b�� and �c��

�� Corollary� The graphic matroids as well as the cographic ones

are regular�


Proof� By Corollary �� we only need to consider the case of graph�icness� So let G be a graph producing a graphic matroid M � Add a newnode to G and connect it to all other nodes� Let G� be the resulting graph�The added edges constitute a spanning tree T � of G�� The representationmatrix of the graphic matroid M � of G� is nothing but the node�edge inci�dence matrix of G� with rows indexed by T �� By Lemma �� the lattermatrix is regular� Thus� M � is regular� By Corollary �� the minorM �nT �� which is M � is regular as well�

We know that the transpose of a graphic matrix need not be graphic�Examples are the representation matrices of M�K�� andM�K�� given by�� and �� By ��sum composition� we thus can create regularmatrices B that are not graphic and not cographic� For example� we maychoose as one block of B the matrix B� of �� and as the second blockof B the transpose of B�� There are less trivial ways of creating regularnongraphic and noncographic matrices� We present them in Chapter �� Atthis time� we introduce two regular nongraphic and noncographic matrices�called B�� and B�� that play a very special role in Chapters �� and ��The matrix B�� is

��

1

11

B10 =0

00

0

101

11

11

0

011

111

1 10 1

Matrix B�� of regular matroid R��

To prove regularity� we declare B�� to be over IR and check that all squaresubmatrices have real determinant equal to � or �� Brute�force checkingturns out to be quite tedious� But in Chapter �� we learn much aboutminimal f��g matrices that are not totally unimodular� The conditionspresented there almost immediately prove B�� to be totally unimodular�The regular matroid represented by B�� is called R�� We prove nongraph�icness and noncographicness of R�� as follows� Delete the last column fromB�� Up to indices� the matrix of �� results� Hence� R�� has anM�K�� minor� The matrix B�� is also its transpose� Thus� R�� alsohas an M�K�� minor� We conclude that R�� is not graphic and notcographic�

We have met the matrix B�� already in � � �� Below we includethat matrix in �� in partitioned form� for reasons explained shortly�Following that matrix� we display a signed version of B�� in �� Weclaim that the matrix of �� is totally unimodular� which implies thatB�� is regular� Analogously to the B�� case� one may verify the claim usingthe results of Chapter ��


�� B12 =

0

0

11

0

0101

110

1

1010

1

0101

1 11 11 00 1

1 0 01 0 0

Matrix B�� of regular matroid R��

��

0

0

11

0

0-10

-1

110

1

1010

1

0-10

-1

1 11 11 00 1

1 0 01 0 0

Totally unimodular version of B��

An easier method for verifying total unimodularity of the matrix of ��uses decomposition� as follows� By �� and �� the partition ofthe matrix B�� of �� induces a �sum decomposition with componentmatrices B� and B� given by �� below� A GF��pivot on the � in thelower right corner of B� converts that matrix� up to indices� to the matrixof �� The latter matrix represents up to indices M�K�� and sodoes B�� The matrix B� is the transpose of B�� Thus� the matroid of B�

is isomorphic to M�K��

�� B1 = ;

111

0

11

0

01

10

10

1

01

1 01 0 B2 =

110 110 1

0

0

1 11 11 00 1

1 0 0

Component matrices B� and B� of B��

These facts imply that R�� is not graphic and not cographic� They alsoestablish that R�� is a �sum of a copy of the regular M�K�� with one ofthe regularM�K�� In Chapter �� we see that any �sum of two regularmatroids is also regular� This fact proves R�� to be regular� The simplesigning algorithm implicit in the proof of Corollary �� con�rms thatthe signed version of B�� given by �� is indeed totally unimodular�


Total unimodularity is an important property for combinatorial opti�mization� Numerous problems of that area of mathematics can be expressedas an integer program of the form

��

min dt � xs� t� A � x � b

� � x � c

x integer

where A is a given integral matrix� b� c� and d are given integral columnvectors� and x is a column vector representing the solution� The dimensionsof the arrays are such that the indicated multiplications and inequalitiesmake sense� The abbreviation �s� t�� stands for �subject to�� In general�� is not easy� But if A is a totally unimodular matrix� then onecan e�ectively drop the integrality requirement from �� and solvethe resulting linear program� In this book� we do not dwell on details andimplications of this approach� which has been treated extensively elsewhere�In Section �� we provide appropriate references�

A special class of the problems �� involves as A the node�arcincidence matrices of directed graphs� or matrices derived from node�arcincidence matrices by pivots and deletion of rows and columns� Any prob�lem of �� in that class is called a network �ow problem� That class hasalso been treated extensively� and numerous special algorithms and famousinequalities exist� That these algorithms work and that the inequalitiesare valid can almost always be traced back to the total unimodularity ofnode�arc incidence matrices� Again� we must resist an even cursory treat�ment and must point to Section �� for references� However� in Section ��of the next chapter� we do present an e�cient algorithm for testing whetheror not a given f��gmatrix A is the matrix of some network �ow problem�

The importance of total unimodularity naturally leads us to ask anumber of questions� How can we recognize a totally unimodular matrix�Is there a simple construction of the entire class of totally unimodularmatrices� What are the characteristics of the matrices that are not totallyunimodular� but all of whose proper submatrices are totally unimodular�Are there other well�behaved problem classes of type �� where A isnot totally unimodular� but where A is related to some totally unimodularmatrix�

Theorem �� links matrix total unimodularity and matroid regu�larity� Thus� one may pose the following related matroid questions� Isthere a simple construction of the entire class of regular matroids� Whichare the nonregular binary matroids all of whose proper minors are regular�Which are the nongraphic binary matroids all of whose proper minors aregraphic� The latter two questions make sense since regularity as well asgraphicness are maintained under minor�taking� Are there other classes of


binary matroids that are not regular� but that are closely related to theregular matroids�

The above questions have complete answers� all of which are providedin the sequel� The answer to the question concerning the minimal nonreg�ular binary matroids is given in the next section�

�� Characterization of Regular Matroids

By Theorem �� every regular matroid is binary� We already havea characterization of the binary matroids� Theorem �� says that amatroid is binary if and only if it has no U�

�minors� Thus� a regular matroid

has no U�

�minors� By Corollary �� regularity is maintained under

minor�taking and dualizing� For a complete characterization of regularity�we thus must identify the minimal binary matroidsM that are not regular�In this section� we prove the famous theorem of Tutte according to whichthere is just one such M up to isomorphism and dualizing� That matroidis the Fano matroid F� represented by the matrix B� of �� below� Thename is due to the fact that the matroid is the Fano plane� which is theprojective geometry PG�� The seven elements of the matroid are thepoints of the geometry� We saw the Fano matroid earlier in Sections �and � � under �� and � � �� In the latter case� �con� and �del�labels were assigned to its elements� We have no such labels here�

��0

1

1 11B7 = 1 10

0 1 1

1

Matrix B� of Fano matroid F�

The matrix B� cannot be signed to become totally unimodular� as follows�The last three columns of B� give a totally unimodular submatrix� If B�

is regular� then by Corollary �� we can sign the �rst column of B� toachieve a totally unimodular matrix� But for all such ways� a �� or �matrix with real determinant equal to �� of �� is created� Thus� B� isnot regular� Tuttes theorem is as follows� The amazingly simple proof isdue to Gerards�

�� Theorem� A binary matroid is regular if and only if it has no

F� or F �

�minors�

Proof� For proof of the nontrivial �if� part� let M be a nonregular binarymatroid all of whose proper minors are regular� Evidently� any binaryrepresentation matrix B of M must be connected� BG�B� cannot be asingle cycle or a path� for the �rst case corresponds to a wheel matroid�

� � Characterization of Regular Matroids ��

and the second one to a minor of a wheel matroid� both matroids areregular� Thus� BG�B� is connected and has a node of degree � and hencehas a spanning tree with a degree node� Such a tree has at least three tipnodes� and hence has at least two tip nodes that correspond to two rowsor to two columns of B� Thus� deletion of two columns or of two rows�say indexed by p and q� reduces B to a connected matrix B� Because ofdualizing of M � we may assume the former case� Thus�

��g

p

h

q

B = B

Binary Matrix B of minimal nonregular matroid M

where the submatrices B� �g j B�� and �h j B� are connected� By Corollary�� we may sign B so that a real matrix A of the form

�� a

p

b

q

A = A

Matrix A derived from B of �� by signing

results where both �a j A� and �b j A� are totally unimodular� SinceM is notregular� A is not totally unimodular� By the construction� any submatrixof A proving the latter fact must intersect both columns p and q� Let D bea minimal such submatrix� If D is not a �� matrix� we may convert it toone by real pivots in A� The corresponding binary pivots in B must leadto a matrix that is the support of the one deduced from A� for otherwise�at least one of the matrices �a j A�� b j A� is not totally unimodular�Furthermore� by Lemma �� the pivots convert �a j A� and �b j A� tosome other totally unimodular matrices� and do not a�ect connectedness ofA� Thus� we may assume the �� D to be already present in A of �� That is� the columns a and b of A contain a �� matrix with determinantequal to �� or �� say in rows v and w�

Up to scaling� A can thus be further partitioned as shown in ��

below� The submatrix A has been subdivided into A� c� and e� The vectorsa and b have become a and b plus the explicitly shown ��s� The latterentries make up D�


��

v11

w1

-1

p q

A =

ec

a b A

Partition of matrix A displayingnon�totally unimodular �� submatrix

Examine the row subvectors c and e in rows v and w� Suppose both sub�vectors have �� entries in some column y � p� q� Then those two entriesplus the two ��s of column p or column q must form a non�totally unimod�ular �� matrix� a contradiction that both �a j A� and �b j A� are totallyunimodular� None of the subvectors c and e can be zero� since otherwise Ais not connected� By scaling� we thus can assume c and e to be as indicatedbelow�

��

1. .-1w 1 0 11v 1 1 1 0

p q

A =b

Y1 Y2

A

. .

a

Further partitioning of matrix A

Since A is connected� there is a path from some r Y � to some s Y � in

the bipartite graph of the submatrix A� By the path shortening techniqueof Chapter �� we may assume the path to have exactly two edges� Thus�we can re�ne A of �� to

��

cuv

-1w 11α

1 110

0

β

p q r s

A =

A

±1±1

e

a b

Final partitioning of matrix A

By scaling in row u� we may presume the entry in row u and column r to bea �� We now concentrate on the submatrix A� of A indexed by fu� v�wgand fp� q� r� sg� We show that submatrix below� The entries � and � of the

� � Characterization of Ternary Matroids ��

submatrix are yet to be determined�

��±1

w

u α βq

011 -1

11v

srp

0

1

1A' =

Submatrix A� of A displaying F� minor

If � � � � �� then by Lemma �� the � column submatrix ofA� indexed by fp� r� sg or the one indexed by fq� r� sg is not totally uni�modular� But then one of �a j A� and �b j A� is not totally unimodular� acontradiction� Suppose both � and � are nonzero� Due to column r andtotal unimodularity of �a j A� and �b j A�� we have � � � � �� But due tocolumn s and rows u and w� the entries � and � must have opposite sign�a contradiction� Thus� exactly one of � and � is zero� Then A� of ��has up to index sets the matrix B� as support� We conclude that M orM� is isomorphic to F��

During a �rst reading of the book� the next section may be skipped�There we characterize the ternary matroids�

�� Characterization of Ternary Matroids

The proof of Theorem �� of the preceding section can be extended toestablish the following characterization by R� Reid of the ternary matroids�i�e�� the matroids representable over GF�� Recall from Section � thatrepresentability over a given �eld is maintained under minor�taking anddualizing�

�� Theorem� A matroid is representable over GF�� if and only if

it has no F�� F�

�� U�

�� or U�

�minors�

Proof� We assume that the reader is quite familiar with the material onabstract representations of Section � � It is easy to check that the excludedminors are not representable over GF�� Indeed� it su�ces to verify thisfor F� and U�

�� since F �

�and U�

�are the duals of these matroids� Thus�

the �if� part holds� For proof of the converse� we argue as in the proofof Theorem �� except that B is an abstract matrix representing M �The graph BG�B� cannot be a cycle or path� since otherwiseM is a GF��representable wheel or whirl matroid or a minor of such a matroid� Thus� upto dualizing of M � B can be partitioned as in �� where B is connected�

We need the following auxiliary result� Two matrices over GF�� andwith the same support represent the same matroid if and only if either


one of the two matrices may be obtained from the other one by scalingof rows and columns by f��g factors� The proof is identical to that ofLemma �� except that the �nal sentence of the proof is replaced bythe observation that a matrix of �� has the GF��determinant equalto � if and only if its entries sum �in IR� to ��mod ��

Because of the auxiliary result and its proof� the signing of an abstractmatrix to achieve a representation matrix over GF�� can be done columnby column� provided of course the matroid is GF��representable� Thisobservation is the GF��analogue of Corollary �� Thus� for the caseat hand� we may deduce from B� which is partitioned as in �� a matrixA of �� over GF�� where �a j A� represents Mnq and where �b j A�represents Mnp� Since M is not representable over GF�� A does notrepresent M �

Due to abstract pivots in B of B and corresponding GF��pivots inA of A� we may assume that B and A satisfy the following two additionalconditions� First� A is the matrix of �� except that the explicitlyshown �� in row w and column q is a �� or �� We denote that entry by�� Second� the GF��determinant of the �� submatrix of rows v� w andcolumns p� q is not correct for M � That is� the GF��determinant of thatsubmatrix is zero �resp� nonzero�� i�e�� resp� � � �� if and only ifthe related �� submatrix of B has abstract determinant � �resp� �� Letus include the just�described matrix A for easy reference�

��

v11

w1γ

p q

A =

ec

a b A

Partition of matrix A displaying �� submatrixwith incorrect determinant

Due to the � entry� the ensuing arguments are a bit more subtle than thoseproving Theorem �� Assume that both row subvectors c and e in rowsv and w of A have ��s in a column y � p� q� Extract from A of �� thesubmatrix A� indexed by fv�wg and fp� q� yg� i�e��

�� 11

±1±1w γ

1vq yp

A' =

Submatrix A� of A displaying U�

� minor

That submatrix corresponds to a minor M � of M � According to the previ�ous discussion� p and q are parallel in M � if and only if � � �� Assume

� � Characterization of Ternary Matroids ��

� � �� By �� one of fp� yg� fq� yg� say fp� yg� contains two parallelelements of M �� and the other one is a base of M �� Since � � �� p and q

are parallel in M �� But �is parallel to� is an equivalence relation� so y andq must also be parallel in M �� a contradiction� Thus� � � �� Arguing bycontradiction as in the previous case� both fp� yg and fq� yg must be basesof M �� Then M � is isomorphic to U�

� � and we are done�We return to A of �� knowing now that it can be further parti�

tioned as in �� except that the explicitly shown �� in �� must

be replaced by �� By path�shortening GF��pivots in A and subsequentdeletion of rows and columns� we get the following analogue of the smallcase ��

��

w

u α β δ εq

011 γ

11v

srp

0 1A' =

Submatrix A� of A displaying U�

� � U�

� � or F� minor

Here �� while � and � are as�yet�undetermined f��g entries�We examine the possible cases for � and �� Let M � be the minor of Mcorresponding to A��

Case �� Suppose the �� submatrix A� of A� indexed by u� v

and p� q has incorrect GF��determinant for M �� Then the rows u� v andthe columns p� q� r are up to indices an instance of �� Thus�M � has aU�

� minor� Hence� we may assume the determinant of A� to be correct forM �� Similarly� the GF��determinant of the �� matrix indexed by u�w

and p� q may be assumed to be correct for M �� But then up to indices� thecolumns p� q constitute the transpose of the case �� thus giving an U�

�

minor�

Case �� We GF��pivot in A� on the � in row w and column s

to obtain case ��

Case � � � �� This is the last case� since � � �� issymmetric to it by scaling� We may assume � � � because of scaling ofrow u� If � � �� resp� � � �� we GF��pivot on the � in row v �resp�w� and column r �resp� s� of A� to get case �� Hence� assume � � � and� � �� If � � �� a GF��pivot on the � in row v and column r of A��followed by a GF��pivot on the � in row w and column s� produces case�� If � � �� then a simple analysis of the GF��determinants proves M �

to be isomorphic to F��The conclusions drawn via the preceding GF��pivot arguments are

valid for the following reasons� Let one of the above GF��pivots transformA� of �� to A�� Let B� be the abstract matrix that represents M � and


that corresponds to A�� We claim that the related abstract pivot in B� pro�duces a B�� that is linked to A�� as follows� First� the �� submatrix in thelower left corner has determinant � in B�� if and only if the correspondingsubmatrix of A�� has a nonzero GF��determinant� Thus� that submatrixof A�� has incorrect GF��determinant for M � as does that of A�� Second�let deletion of the �rst or second column reduce A�� and B�� to A�� and B��respectively� Then any square submatrix of B�� has determinant � if andonly if the corresponding square submatrix of A�� has GF��determinant�� Thus� that submatrix of A�� has a correct GF��determinant for M ��In particular� A�� and B�� have the same support� The preceding claimsfollow directly from the pivot rules for abstract determinants as describedin Section � �

Theorem �� implies Theorem �� by elementary arguments�By Theorem �� a binary matroid M is regular if and only if it isrepresentable over GF�� By Theorem �� a matroid is representableover GF�� if and only if it has no U�

� � U�

� � F�� F�

� minors� But U�

� and U�

�

are not binary� and thus Theorem �� follows�In the �nal section� we indicate extensions and cite references�


The prior work on total unimodularity can be roughly divided into twocategories� The �rst one contains papers investigating the structure oftotally unimodular matrices� of minimal f��g matrices that are not to�tally unimodular� and of closely related matrix classes� Relevant referencesare Cederbaum �� Ghouila�Houri �� Camion ��a�� b�� Heller �� Veinott and Dantzig �� Chandraseka�ran �� Commoner �� Gondran �� Padberg �� Brown �� Tamir �� Truemper �� b�� b�� b�� Kress and Tamir �� de Werra�� Cunningham ��a�� Chandrasekaran and Shirali �� Fonluptand Raco �� Seymour ��a�� Yannakakis �� Crama� Hammer�and Ibaraki �� Conforti and Rao �� and Hempel� Herrmann� H ol�zer� and Wetzel ��

The second category concerns applications of total unimodularity� Per�tinent references are Dantzig and Fulkerson �� Ho�man and Kruskal�� Ho�man and Kuhn �� Motzkin �� Heller and Tompkins�� Heller �� Ho�man �� Ford and Ful�kerson �� Heller and Ho�man �� Rebman �� Lawler ��Baum and Trotter �� Ho�man and Oppenheim �� Truemper andChandrasekaran �� Truemper and Soun �� Soun and Truemper�� Maurras� Truemper� and Akg ul �� Bixby ��a�� a��


Schrijver �� Tamir �� Recski �� and Ahuja� Magnanti� andOrlin �� A comprehensive treatment is given in Schrijver ��

Lemma �� is well known� while Lemmas �� and �� areimplicit in most references of the �rst category� Lemma �� is takenfrom Camion ��b�� It also follows from Brylawski and Lucas ��Lemma �� and Theorem �� though with a quite di�erent proof�are from Tutte ��

References given above for the second category contain numerous re�sults about problems of the form ��

The matroids R�� and R�� are the two central matroids in the proof ofthe regular matroid decomposition theorem in Seymour ��b�� The ma�troid R�� had appeared earlier in Ho�man �� See also Bixby ��Contrary to sometimes�voiced claims� the matroids R�� and R�� do arisefrom well�known combinatorial problems� We present two examples� The�rst one involves the real constraint matrices of two�commodity �ow prob�lems on directed graphs� In Soun and Truemper �� it is shown that anin�nite subclass of such problems has constraint matrices that give rise toregular matroids that are nongraphic and noncographic� Indeed� it can beshown that these matroids have R�� minors�

The second example is due to Chv!atal �� It involves the graph G

below�

��

Graph G

Construct the following real matrix B from G� Each column of B corre�sponds to a node of G� and each row to a clique �� maximal completesubgraph� of G� Each row i of B is then the incidence vector of the nodesin clique i� The matrix B is the clique�node incidence matrix of G� Theclique�node incidence matrices of graphs are very useful for the solution ofthe so�called independent vertex set problem� For the graph G at hand� theclique�node incidence matrix B can be proved to be totally unimodular�Indeed� it is not di�cult to verify that the regular matroid represented byB is nongraphic and noncographic and has an R�� minor�

Theorem �� is Tuttes famous characterization of the regular ma�troids �Tutte �� The beautiful proof is due to Gerards��a�� Variations of the theorem appear in Bixby �� and Truemper��b�� a�� Related is work on extremal matroids in Murty ��

Theorem �� is due to R� Reid� who never published his proof�Other proofs of that result appear in Bixby �� Seymour ��a��


Truemper ��b�� Kahn �� and Kahn and Seymour �� As re�marked in Section �� any proof of Theorem �� essentially constitutesa proof of Theorem �� as well� A characterization of the matroids rep�resentable over GF�� in terms of circuit signatures is given in Roudne�and Wagowski ��

Representability over GF�� and GF�q� is treated in Kung ��b��and Kung and Oxley �� Ternary matroids without M�K�� minors arecovered in Oxley ��c�� Partial results for representability over GF� �are given in Oxley �� a� and Kahn �� The di�cult charac�terization of representability over GF� � in terms of the excluded minors isprovided by Geelen� Gerards� and Kapoor �� Early examples of non�representable matroids are in MacLane �� Lazarson �� Ingleton�� and Vamos �� It has been conjectured �Rota ��that for any �nite �eld F � the number of matroids that are not repre�sentable over F � and that are minimal with respect to that property� is�nite�

General questions concerning representability are discussed in Vamos�� Kahn �� and Ziegler ��

The complexity of representability tests using various oracles is ex�amined in Robinson and Welsh �� Seymour ��c�� Seymour andWalton �� Jensen and Korte �� and Truemper ��a�� Virtu�ally every such test requires exponential time when the matroid is givenby an oracle that decides independence of sets� An exception is the test ofrepresentability over every �eld� A polynomial algorithm for that problemis given in Truemper ��a�� The algorithm relies on the regular matroiddecomposition theorem of Seymour ��b�� which is covered in Chapter��

Chapter 10

Graphic Matroids

10.1 Overview

At this point, we are ready to investigate the first complicated class ofbinary matroids treated in this book: the class of graphic matroids. Recallthe following definitions and results. Kn is the complete graph on n nodes,and Km,n is the complete bipartite graph with m nodes on one side andn nodes on the other side. For any graph G, the corresponding graphicmatroid is regular and is denoted by M(G). F7 denotes the nonregularFano matroid. Finally, the asterisk is used as dualizing operator.

In this chapter, we first identify certain minimal regular matroids thatare not graphic, or that are not graphic and not cographic. Specifically, inSection 10.2, we characterize the planar regular matroids, i.e., the matroidsproduced by planar graphs. In Section 10.3, we investigate the behavior ofnongraphic regular matroids with M(K3,3) minors. We build upon theseresults in Section 10.4. There we prove two profound theorems of matroidtheory: Tutte’s theorem that M(K5)∗ and M(K3,3)∗ are the minimal reg-ular matroids that are not graphic, and Seymour’s theorem that the twomatroids R10 and R12 defined in Section 9.2 are the minimal regular, 3-connected matroids that are not graphic and not cographic. The lattertheorem is one of the two central ingredients in the proof of Seymour’s pro-found decomposition theorem for regular matroids. We take up the lattertheorem and its proof in Chapter 11.

In Section 10.5, we introduce a simple but very useful decompositionscheme that will be used repeatedly in Chapters 11–13. Indeed, the scheme

205

206 Chapter 10. Graphic Matroids

is the second ingredient in the proof of the just-mentioned decompositiontheorem for regular matroids. In Section 10.5, we employ the scheme todeduce some graph decomposition theorems, among them Wagner’s famousdecomposition theorem for the graphs without K5 minors.

In Section 10.6, we present an efficient algorithm for deciding graph-icness of binary matroids and for deciding whether or not a real {0,±1}matrix A is the coefficient matrix of a network flow problem. Finally, inSection 10.7, we indicate extensions and provide references.

The chapter requires knowledge of Chapters 2, 3, and 5–8. We alsomake use of the easy part of Theorem (9.3.2), according to which the Fanomatroid is nonregular.

10.2 Characterization of Planar Matroids

In this section, we prove that a regular matroid is planar if and only if it hasno M(K3,3), M(K3,3)∗, M(K5), or M(K5)∗ minors. The result constitutesone of two preparatory steps toward proofs of the theorems by Tutte andSeymour cited in the introduction to this chapter. The latter results areestablished in Section 10.4.

Definition of Graph with T Nodes

For the arguments of this section and the next one, we need a convenientway to encode and manipulate 1-element binary additions of any graphicmatroid N . Let B be a binary representation matrix of N , say with rowindex set X and column index set Y . Thus, the matroid M = N+z isrepresented by the following matrix B.

(10.2.1)b

z

B = X

Y

B

Matrix B for matroid M = N+z

Let G be any connected graph for N , i.e., M(G) is N . The graph G neednot be unique. The row index set X of B is a tree of G. Suppose wepremultiply the matrix [I | B | b] with the node/edge incidence matrix ofthe tree X . By the results of Section 3.2, that multiplication turns thesubmatrix [I | B] into the node/edge incidence matrix, say F , of G. Thecolumn vector b becomes some vector, say d. Evidently, F and [F | d] havethe same GF(2)-rank. Accordingly, since every column of F has exactlytwo 1s or none, the vector d must have an even number of 1s. Since each

10.2. Characterization of Planar Matroids 207

row of F corresponds to a node of G, we can associate the 1s of d with anode subset T of G. Each node of T we call a T node. In drawings of G, wedenote each T node by a square box. By the construction, M is completelyrepresented by G and the set T .

Section 3.2 contains the following alternate way of representing M viaG. Each 1 of the vector b of (10.2.1) corresponds to an edge of the treeX . To single out these edges, we temporarily paint them red. In general,the red edges form a red subgraph X of G without cycles. The followinglemma links that subgraph to the set T .

(10.2.2) Lemma. A node of G is a T node if and only if the node hasan odd number of red edges incident.

Proof. In the matroid M , the element z forms a fundamental circuit Cwith X . Indeed, C−{z} is nothing but the red subgraph X . That subgraphindexes a column submatrix F of F . Thus, the columns of [F | d], whichare indexed by C, are GF(2)-mindependent, i.e., the columns are GF(2)-dependent, but any proper subset of the columns is GF(2)-independent.This is so if and only if the 1s of d are in the rows of F with an odd numberof 1s. Equivalently, a node of G is in T if and only if the node has an oddnumber of red edges incident.

Lemma (10.2.2) implies a convenient method for determining the Tnodes. Suppose we have B of (10.2.1) and a graph G for the matroid N ofB. For each 1 in the vector b, say in row x ∈ X of B, we temporarily paintthe edge x of G red. Then we define the nodes of G with odd number ofred edges incident to be the T nodes. Finally, we declare the red edges tobe unpainted again.

We claim that M is graphic if |T | = 2. Indeed, the red subgraph Xis then a red path, say from node u of G to node v. We thus may add anedge z connecting u and v to get a graph representing M .

Example Graphs with T Nodes

Below, we carry out the derivation of T for a few nongraphic matroidsthat are important for our purposes. We depict each instance using thefollowing scheme.

(10.2.3)

b

z

B = X

Y

B

Matrix B

−→

Graph Gwith redsubgraph X(indicated bybold edges)

−→

Graph Gwith T nodes(indicated bysquares enclo-sing nodes

Representation of M by matrix B, by graph Gwith red edges, and by graph G with T nodes


Here are the example matroids M .

(10.2.4)

a

b

e

d

c

fa

b

e

d

c

f111

011

101

d e f zabc

110

Fano Matroid F7 (defined by (9.3.1))

(10.2.5)

b

d

a

e

c

f

b

d

a

e

c

f

e1a

bcd

1 1 0110

1 1 1

0 1f z

Fano Dual F ∗7

(10.2.6)

a h

d

f

e

g

b

c

a h

d

f

e

g

b

c

1100 0 1 1 1

11 1 0

0 00 0

1 111

abcd

e f g h z

M(K3,3)∗ (defined by (3.2.46))

(10.2.7)

a

b

e

d

c

h

i

g f

a

b

e

d

c

h

i

g f

000111

011010

101001

g h i zabcdef

110100

M(K5)∗ (defined by (3.2.44))


(10.2.8)

ic e

h

d

ag

b

f

ic e

h

d

ag

b

f

1100 0 1 1 1

11111

1 1 0 11 0 0 10 0 1 1a

bcde

f g h i z

R10 (defined by (9.2.13))

(10.2.9)

f

ih

d

a

ec

b g

k j

i

a

f

h

d

ec

b g

k j

abcdef

00

0

11

0

0101

11z g h i j k

0

1

1010

1

0101

1 11 110 1

1 0 01 0 0

R12 (defined by (9.2.14); note thatcolumn z is the first column)

In each of the example cases, we have established the set T via aparticular tree X of G. Since T was originally defined via the matrix[F | d], any other tree of G would have produced the same set T . For thisreason, we may always select a particularly suitable tree X when discussingsome matroid operation and its impact on G and T . We have three suchoperations in mind: deletion of an element y �= z of M that is not a coloopof G, contraction of an element x �= z of M that is not a loop of G, andthe switching operation of Section 3.2. We take up these operations next.

Deletion, Contraction, and Switching in Graphswith T Nodes

Suppose we delete from M an element y �= z that is not a coloop of G.Thus, G has a tree X that is also a tree of G\y. Then G\y with the originalT node labels represents M\y.

Suppose in M we contract an element x �= z that is not a loop. Wemay assume that the tree X of G contains x. Then X − {x} is a tree ofG/x and a base of M/x. We claim that the red edges of X − {x} plus z


constitute the fundamental circuit that z forms with X −{x} in M/x. Fora proof, we delete row x ∈ X from the matrix B to obtain a representationmatrix for M/x. Inspection of the column z of the latter matrix verifiesthe claim. Thus, we may use the red edges of X − {x} to represent theelement z of M/x by a node subset T ′ of G/x analogously to the use of thered edges of X to define the node subset T of G. We deduce T ′ directlyfrom T by the following rules. Let i be any node of G different from thetwo endpoints u and v of x. Then i is in T ′ if and only if i is in T . Definew to be the node of G/x created from u and v of G when x is contracted,i.e., w = (u ∪ v)− {x}. Then w is in T ′ if and only if exactly one of u andv is in T . Validity of these rules follows directly from the just-mentionedfact about the red edges of X − {x}, and from a simple parity argumentinvolving the red edges of G incident at nodes u or v.

Recall the switching operation of Section 3.2. On hand must be a2-separation of G, say involving subgraphs G1 and G2. The graph G1 isremoved, turned over, and reattached to G2. The resulting graph G′ is2-isomorphic to G. Thus, G′ also represents M\z. We want to deduce theset T ′ for G′ from the set T of G. That is, G′ and T ′ are to represent Manalogously to G and T . Evidently, any tree X of G is one of G′, and thered edges of X may be used to deduce T ′ for G′. Thus, any node differentfrom the two nodes joining G1 and G2 is in T ′ if and only if it is in T .The rules for the latter two nodes are also quite simple. For the generalsituation, we leave their derivation to the reader. Instead, we just examinethe special case where the graph G has two series edges e and f with acommon endpoint w ∈ T with degree 2. Let u be the second endpoint ofedge e, and let v be that of edge f . Assume u �= v. The switching operationresequences e and f . Thus, u becomes u′ = (u−{e})∪ {f} and v becomesv′ = (v − {f}) ∪ {e}. For the derivation of T ′ of G′ from T of G, we maysuppose that the tree X of G includes both e and f . Since w ∈ T , exactlyone of the edges e and f is red. Correspondingly, the parity of the numberof red edges at u (resp. v) in G is different from the parity of the numberof red edges at u′ (resp. v′) in G′. Thus, u′ (resp. v′) is in T ′ if and onlyif u (resp. v) is not in T . An example case is depicted below. As before,nodes of T are indicated by squares.

(10.2.10)vu

we f u'wf e v '

switching

Graph G Graph G′

Effect of switching on T nodes

The set T ′ produced by any switching may have cardinality different fromthat of T . Thus, we are justified in assuming for convenience that |T | is


minimal under switchings. In that case, M is graphic if and only if |T | is0 or 2. The 0 case corresponds to z being a loop of M . It cannot occurwhen M is connected.

Characterization of Planar Matroids

We now have sufficient machinery to prove the main result of this section,which characterizes planarity of regular matroids in terms of excluded mi-nors.

(10.2.11) Theorem. A regular matroid M is planar if and only if Mhas no M(K3,3), M(K3,3)∗, M(K5), or M(K5)∗ minors.

Proof. The “only if” part holds since planarity is maintained under minor-taking and since M(K3,3), M(K3,3)∗, M(K5), and M(K5)∗ are not planar.For the proof of the nontrivial “if” part, let M be a regular matroid allof whose proper minors are planar. Thus, M is minimally nonplanar withrespect to the taking of minors. We must show that M is isomorphic toM(K3,3), M(K3,3)∗, M(K5), or M(K5)∗.

If M is graphic or cographic, then the desired conclusion is providedby Theorem (7.4.1), which characterizes nonplanar graphs by exclusion ofK3,3 and K5 minors. Thus, we may assume from now on that M is notgraphic and not cographic.

We claim that M is 3-connected. If that is not the case, then M hasa 1- or 2-separation. By Lemma (8.2.2) or (8.2.6), M is a 1- or 2-sum. Ineither case, the components of the sum are proper minors of M and thusplanar. But by Lemma (8.2.2) or (8.2.7), the latter conclusion implies Mto be planar as well, a contradiction.

By the census of Section 3.3, every 3-connected regular matroid on atmost eight elements is planar. Thus, M has at least nine elements.

We apply the binary matroid version of the wheel Theorem (7.3.3)to the 3-connected, nongraphic, and noncographic M on at least nine ele-ments. Accordingly, M must have an element z so that at least one of theminors M/z and M\z is 3-connected. If the M/z case applies, we replaceM by its dual. Since all assumptions made so far for M are invariant underdualizing, this change does not affect the proof. Thus, we may assume thatM\z is 3-connected. By the minimality of M , the 3-connected minor M\zis planar. Let G be the corresponding planar graph. We extend G to arepresentation of M by selecting an appropriate subset T of nodes of G forthe element z. Recall that the cardinality of T is necessarily even. SinceM is not graphic, |T | ≥ 4.

Suppose G is a wheel. It is easily verified that one can delete spokesfrom G and contract rim edges so that the wheel with four spokes and withfour T nodes results. Then we either have, up to indices, the M(K3,3)∗

case of (10.2.6) and are done, or we can delete one spoke and contract one


rim edge to obtain an instance of the F7 case of (10.2.4), which contradictsthe regularity of M .

We are left with the case where G is not a wheel. By the wheelTheorem (7.3.3), G has an edge e so that G/e or G\e is 3-connected.Assume the latter case. The deletion of the edge e from G leaves the numberof T nodes unchanged. Note that G\e plus these T nodes represents M\e.Since G\e is 3-connected and |T | ≥ 4, the minor M\e must be 3-connectedand nongraphic, a contradiction of the minimality of M . Thus, the caseof a 3-connected G/e must be at hand. We claim that the contraction ofthe edge e in G must reduce the number of T nodes to 2. If this is notthe case, then arguments analogous to those for M\e prove M/e to be3-connected and nongraphic, a contradiction. Since G/e has exactly twoT nodes, the graph G must have exactly four T nodes, two of which mustbe the endpoints of e, say u and v. Define i and j to be the other two Tnodes of G.

By the 3-connectedness of G and Menger’s theorem, there is a path Pfrom i to u and a second path Q from j to u such that these paths haveonly node u in common and do not involve node v. Imagine G drawn inthe plane. Then deletion of node u would create a new face. The boundarywould be a cycle, say C. Clearly, v has become a node of C. If in G the Tnode i does not lie on C, then we contract the edge of the path P incidentat i. After suitable repetition of this process, the T node has become anode of C. Then we declare that T node to be i again. Similarly, we makethe T node j a node of C. At that time, the edges incident at node uand those of the cycle C constitute a subdivision of a wheel with at leastthree spokes. The rim of that wheel subdivision contains the T nodes v, i,and j. The latter nodes induce a partition of the rim into three paths, sayP1 from i to v, P2 from j to v, and P3 from i to j. Simple case checkingconfirms that the arguments made earlier for the wheel case of G applyhere fully unless every edge incident at node u has its second endpoint injust one of the paths P1 or P2, say P1. Below, we show a typical instanceof that exceptional case, together with an additional path P4. That path isnothing but a portion of the previously defined path Q that in G connectednodes j and u. In the general case, the path P4 connects an interior nodeof the path P2 ∪ P3 with an interior node k of the path P1.

(10.2.12) u

i j

k

l

vP1

P2

P3P4

Wheel subdivision plus path P4

10.3. Regular Matroids with M(K3,3) Minors 213

The drawing does not show any edges of the previously defined path P .But there are still enough edges left from that path so that node u canbe reached from node i while avoiding all nodes of P4. This fact and theplanarity of G imply that an interior node l of the subpath of P1 from i tok must be connected with node u, as depicted in (10.2.12).

Evidently, we can contract enough edges of the paths P1–P4 and deletesome edges incident at node u so that node k becomes the center node ofa wheel graph with four spokes and with four T nodes on its rim. Thus,up to indices, we have an instance of the M(K3,3)∗ case of (10.2.6).

Related to Theorem (10.2.11) is the following pioneering combinatorialcharacterization of planar graphs due to Whitney. It constitutes the firstmatroid result about graph planarity.

(10.2.13) Corollary. A graph G is planar if and only if M(G)∗ isgraphic.

Proof. The “only if” part is clear. For proof of the “if” part, let M =M(G). Since M∗ = M(G)∗ is graphic, M∗ has no M(K5)∗ or M(K3,3)∗

minors. Thus, M has no M(K5) or M(K3,3) minors. Since M is graphic,M has no M(K5)∗ or M(K3,3)∗ minors. Thus, by Theorem (10.2.11), Mand G are planar.

One may derive Corollary (10.2.13) directly from Theorem (7.4.1),without use of Theorem (10.2.11). Indeed, Theorem (7.4.1) is the graphversion of Corollary (10.2.13). Whitney’s contribution is the deductionof this result from Kuratowski’s original planarity characterization, whichinvolved subdivisions of K5 and K3,3.

In the preceding chapters, we took great care when we dualized planargraphs. Each time, we embedded a given planar graph in the plane, thendualized that plane graph. Corollary (10.2.13) frees us from this adherenceto planar embeddings. We now may take the following viewpoint. Givena planar graph G, let H be any graph for M(G)∗. Thus, the matroidsM(G) and M(H) are duals of each other. We declare H to be a dualgraph of G. For our purposes, any H satisfying M(H)∗ = M(G) willdo. By this definition, any two such graphs are 2-isomorphic. Thus, byTheorem (3.2.36), H is unique if it, or equivalently G, is 3-connected, thecase typically of interest to us. In that situation, we are justified to call Hthe dual of G. Suppose H is 2-connected. By Theorem (3.2.36), any othergraph for M(G)∗ is related to H by switchings. We leave it to the readerto explore this issue further. Whitney first recognized these relationshipsamong embeddings, 2-isomorphism, and switchings.

We move on to the next section, where we investigate regular matroidswith M(K3,3) minors.


10.3 Regular Matroids with M(K3,3) Minors

The title of this section may seem strange. Why would one be interestedin M(K3,3) minors of regular matroids? Early in this section, we give apartial answer in the form of two lemmas. More satisfactory as answer arethe arguments of the next section, which prove M(K3,3) to play a centralrole in the analysis of regular matroids.

Following the two lemmas and some other preparatory material, weintroduce the main theorem of this section. It says that any 3-connected,regular, nongraphic and noncographic matroid with an M(K3,3) minor hasa minor isomorphic to one of the nongraphic and noncographic matroidsR10 and R12. This profound result is due to Seymour. In the next sectionwe combine it with Theorem (10.2.11) to obtain two important character-izations of nongraphic regular matroids by Seymour and Tutte.

To start, we recall the splitter definition of Section 7.2. Let M be aclass of binary matroids that is closed under isomorphism and under thetaking of minors. Let N be a 3-connected matroid of M on at least sixelements. Then N is a splitter of M if every connected matroid M ∈ Mwith a proper N minor is 2-separable. Intuitively and informally speaking,the presence of an N minor forces M to split. By Theorem (7.2.11), thegraph K5 is a splitter of the graphs without K3,3 minors. That result hasthe following matroid extension.

(10.3.1) Lemma. M(K5) is a splitter of the regular matroids withoutM(K3,3) minors.

Proof. By Theorem (7.2.1), we only need to show that every 3-connectedregular 1-element extension of M(K5) has an M(K3,3) minor. This isaccomplished by a straightforward case analysis. To assist the reader, wesketch one way of checking.

By (3.2.38), the matrix

(10.3.2) 1

0

00

0

10

11

11

0

01

1 00 1

1 1 00 0 1B =

Matrix B for M(K5)

represents M(K5). Suppose we adjoin a column vector b that representsan added element z. If b has at most two 1s, then z is a coloop or a parallelelement. If b has three or four 1s, then [B | b] contains a submatrix repre-senting the Fano matroid F7, which is nonregular. Thus, no 3-connectedregular 1-element addition is possible.

Now assume we adjoin a row vector c that represents a 1-elementregular and 3-connected expansion by an element e. We may view B as


the node/edge incidence matrix of a graph H that is isomorphic to K4. Itis convenient to encode each 1 of the row vector c by a red edge of H. SinceM(K5)&e is 3-connected, H must have at least two red edges. We analyzethe possible configurations of red edges.

Suppose that H contains exactly two red edges, and that these edgesshare an endpoint. Then M(K5)&e is easily confirmed to be graphic, withan M(K3,3) minor. Next, suppose H contains exactly four red edges thatform a cycle C. Let y and z be the edges of H that are not in C. Then(M(K5)&e)\{y, z} is isomorphic to M(K3,3).

For the remaining configurations of red edges, one proves the presenceof an F ∗

7 minor. Specifically, if the red edges form a triangle, then thecolumns of [B/c] corresponding to that triangle establish the presence ofan F ∗

7 minor. The other cases are slightly more difficult to prove. As anexample case, let us examine the 1-element expansion M of M(K5) by theelement e given by

(10.3.3)1a

bcde

f g h i j k

1

1 1

00 0

0 0 000

1 11 1 1

0 011 0 0 0 10 0 1 1 0

Matrix of 1-element expansion M of M(K5)

It turns out that we can prove nonregularity without using the last columnk. So let us consider that column deleted. One readily verifies that theremaining matrix is represented by

(10.3.4)a

c

d

g

bf e h

i

Graph plus T nodes for M\k

where the T nodes correspond to the element j. From the graph of (10.3.4),we delete the edges a and c, and contract the edge b. The resulting graphwith adjusted T set is given by (10.3.5) below. By (10.2.5), that graphrepresents an F ∗

7 minor. Thus, the matroid given by the matrix of (10.3.3)


is nonregular.

(10.3.5)

d

e

i

f

h

g

Graph plus T nodes for M/{b}\{a, c, k}

The remaining open cases are handled the same way.

For later reference, we include another lemma about M(K3,3). Weomit the elementary case analysis via graphs plus T sets.

(10.3.6) Lemma. Every 3-connected binary 1-element expansion ofM(K3,3) is nonregular.

The proof of the next result about M(K3,3) requires a bit of prepara-tion. Recall from Section 3.3 the following definition. Let M be a binarymatroid and z be an element of M . Then M c©z is the matroid obtainedfrom M/z by deletion of all elements but one from each parallel class. Theminor M d©z is derived from M\z by contraction of all elements but onein each series class. For convenient reference, we restate Lemma (3.3.31),which links M c©z and M d©z to 3-connectedness.

(10.3.7) Lemma. Let M be a 3-connected binary matroid on a set E.Take z to be any element of E. Then M c©z or M d©z is 3-connected.

Define a line of a graph to be a path of maximal length where allinternal nodes have degree 2. A corner node of a graph is a node of degreeat least 3. Let G be a subdivision of a 3-connected graph with at leastfour corner nodes. Evidently, each line of G is a series class and viceversa. We emphasize the latter fact, since generally a series class of agraph need not be a line. This is due to the definition of Section 2.2, wheretwo edges are declared to be in series if they form a cocycle. A priori,the same subtle point needs to be considered when one specializes Lemma(10.3.7) to graphs. In that case, the notation G c©z and G d©z is interpretedanalogously to that of M c©z and M d©z.

We now show that the just-mentioned complications concerning seriesedges do not arise when we apply Lemma (10.3.7) to graphs. Let G be3-connected, and let z be an edge of G, say with endpoints u and v. Recallthat u and v are edge subsets. By Lemma (10.3.7), one of G c©z, G d©z is3-connected. Let G c©z be that graph. The contraction of z may introduceparallel edges only at the new vertex (u ∪ v) − {z}. Thus, G c©z is readilydetermined. Now let G d©z be 3-connected. We claim that G\z contains


series classes with more than one edge only if u or v has degree 3, andthat such series classes correspond to paths with two edges. Indeed, if uhas degree 3, then u − {z} is one such class. Similarly, v may producea second series class. No other series class is possible, since 2-connectedseries expansions of a 3-connected graph can only produce a subdivision ofthat graph. Thus, G d©z is readily determined.

The above conclusion is valid only if G d©z is 3-connected. Indeed, ifG d©z is not 3-connected, then G\z may have a series class that is not aline. Fortunately, we never deal with the latter case since we make use ofG d©z only when that graph is 3-connected.

Here is a second preparatory lemma about G c©z and G d©z.

(10.3.8) Lemma. Let H be a subdivision of a 3-connected graph. As-sume H has at least four corner nodes. Let G be any graph derived fromH by the addition of nonloop edges. No such added edge is to connect twonodes of a line of H. Also, G is not allowed to have parallel or series edges.Then (a)–(c) below hold.

(a) G is 3-connected.

(b) G d©z is 3-connected for every arc of G that is not in H.

(c) G c©z is 3-connected for every arc of H both of whose endpoints havedegree 2 in H.

Proof. Clearly, G of part (a) is 2-connected. Suppose G has a 2-separation.We know that H is a subdivision of a 3-connected graph. Thus, any 2-separation of H has on one side a subset of one line of H. Assume that the2-separation of G induces one in H. Then G has series edges, or G has anedge that is not in H and that connects two nodes of one line of H. Bothcases contradict the assumptions. If the 2-separation of G does not induceone in H, then G must have parallel edges, again a contradiction. Thus, Gis 3-connected.

Under the given assumptions, G d©z and G c©z of parts (b) and (c)satisfy the construction rules imposed on G. By (a), these minors are3-connected.

The next result concerns graphs with K3,3 minors.

(10.3.9) Theorem. Let G be a 3-connected graph with a K3,3 minor.

(a) If G contains a triangle formed by edges e, f , and g, then G has oneof the graphs of (10.3.10) below as a minor. The bold edges denote e,f , and g.

(b) If G has a node u of degree 3, then G has as subgraph a subdivisionof K3,3 that has u as corner node.


(10.3.10)

Two extensions of K3,3 containing a triangle

Proof. First we show part (a). Due to minor-taking, we may assume thatevery proper minor of G is not 3-connected, or does not have a K3,3 minor,or does not contain the triangle {e, f, g}. We say that G is minimal todenote this fact. Let u, v, and w be the nodes of the triangle. Denote byH any subgraph of G that is a subdivision of K3,3.

Suppose u is not a node of some H, say of H1. Thus, u has an arcz �= e, f, g incident that is not in H1. By Lemma (10.3.7), one of G c©z,G d©z is 3-connected. Assume G c©z to be 3-connected. In G/z we candelete parallel edges so that the triangle {e, f, g} is retained. Now supposeG d©z is 3-connected. If u has degree 3, then in G\z two edges of e, f , gare in series, and G d©z has two edges of e, f , g in parallel, a contradiction.Thus, u has degree of at least 4, and G d©z contains the triangle {e, f, g}.Clearly, both G c©z and G d©z have K3,3 minors. But these facts contradictthe minimality of G.

We conclude that u, v, and w are nodes of every H. If all three nodesoccur on one line of some H, then there exists another H that avoids oneof the three nodes. Therefore, at most two of the nodes occur on any oneline of any H. Accordingly, we can always delete and contract edges in Gsuch that e, f , g are retained and such that their endpoints become cornernodes of some K3,3 minor. That process produces one of the graphs of(10.3.10).

For part (b), we once more define H to be any subgraph of G that is asubdivision of K3,3. Suppose the given degree 3 node u of G is not a cornernode of some H. Then by a ΔY exchange (see Section 4.3), the 3-star ucan be replaced by a triangle {e, f, g}. It is easily seen that upon deletionof edges parallel to e, f , or g, we have a 3-connected graph with a K3,3

minor. Apply part (a) to the latter graph. Thus, that graph has as minorone of the graphs of (10.3.10). Now replace {e, f, g} by the 3-star u again.The resulting graph is a minor of G and is readily verified to have a K3,3

minor G with u as a corner node. In straightforward fashion, we extend Gto a subgraph of G that is a subdivision of K3,3 and that has u as a cornernode.

We are ready to state and prove the main result of this section.


(10.3.11) Theorem. Let M be a 3-connected regular matroid with anM(K3,3) minor. Assume that M is not graphic and not cographic, but thateach proper minor of M is graphic or cographic. Then M is isomorphic toR10 or R12.

Proof. Let M be a smallest regular matroid that satisfies the assumptionsof the theorem but not its conclusion. By Lemma (10.3.6), M does nothave a 3-connected 1-element expansion of any M(K3,3) minor. Take N tobe any M(K3,3) minor of M . Apply Theorem (7.3.4) to M and N . Thattheorem establishes the existence of a certain nested sequence of minors.In particular, the result implies that M has an element z for which theminor M\z is a series extension of a 3-connected matroid with an M(K3,3)minor. The proof of Theorem (10.3.11) consists of a thorough analysis ofM\z and of the role of the element z. We begin with two simple claimsabout M\z.

First, we claim that each series class of M\z has at most two elements.Suppose otherwise. Thus, the connected M\z has a representation matrixwith three parallel rows. Adjoin a column z to that matrix to get a repre-sentation matrix for M . Regardless of the entries of column z, the matrixfor M has two parallel rows. Thus, M is not 3-connected, a contradiction.

Second, we claim that M\z is graphic. This is so since each properminor of M is graphic or cographic, and since M\z has an M(K3,3) minor,which is not cographic.

By the first claim, a graph G for M\z is obtained from a 3-connectedgraph by subdividing each edge at most once. Hence, each line of G has oneor two edges. We represent M by G plus a node subset T that handles theextra element z. Recall that |T | is even. Choose G so that |T | is minimal.Since M is not graphic, we have |T | ≥ 4.

The strategy in the remainder of the proof is as follows. We attemptto reduce G and T to G′ and T ′, where G′ has fewer edges than G or|T ′| < |T |. The graph G′ is to have an M(K3,3) minor, and |T ′| ≥ 4 is tohold. Indeed, G′ and T ′ must represent a nongraphic matroid. At times,we simply say that we reduce G to denote this process. Evidently, any suchreduction contradicts the minimality of M or T , and thus is not possible.The reduction attempts reveal enough structural information about M toprove that matroid to be isomorphic to R10 or R12, contrary to the initialassumption. We present the details.

Claim 1. G is 3-connected.

Proof. Suppose G is not 3-connected. We know that G is a subdivisionof a 3-connected graph. Indeed, each line of G with at least two edges hasexactly two edges. Let G have m such lines. If a midpoint of one of theselines is not in T , then in M the elements corresponding to the two edges ofthat line are in series. Thus, M is not 3-connected, a contradiction. Hence,the midpoints of the m lines are in T .


If m ≥ 4, we contract in one of the m lines an edge. By switchings,the set of T nodes may now be reduced, say to T ′. But the midpoints ofthe remaining m − 1 lines must remain in T ′, so |T ′| ≥ 4. Thus, G hasbeen reduced. Similarly, one may handle the cases m = 1 and 2, and alsom = 3 when the three lines do not form a cycle.

Consider the remaining case, where m = 3 and where the three linesform a cycle. If any node other than those of the three lines is in T , again wecan reduce G. Thus, all such nodes are not in T . Temporarily contract oneedge of each line, getting a graph G. The three lines have become a triangle.According to Theorem (10.3.9), the graph G has a minor isomorphic to oneof the two graphs of (10.3.10). The bold lines of those graphs are those ofthe triangle. We can further reduce each one of the two graphs to

(10.3.12)

Minor of the two graphs of (10.3.10)

where again the bold lines indicate the triangle. The graph of (10.3.12) isstill a minor of G if we replace each triangle edge by the corresponding linewith appropriate designation of T nodes. By the minimality of |T |, thissubstitution must result in

(10.3.13)e f

Minor of G with three lines

which is not graphic. Contract the edges labeled e and f in (10.3.13). Theresulting G′ and T ′ has |T ′| = 2 and represents a matroid with an M(K3,3)minor. Thus, the matroid of (10.3.13) is not cographic. But that matroidis a proper minor of M since the graph of (10.3.12) is a proper minor of thetwo graphs of (10.3.10). Thus, we have a contradiction of the minimalityof M . Q. E. D. Claim 1

Denote by H any subgraph of G that is a subdivision of K3,3. Re-call that G c©e is obtained from G by contraction of edge e and deletionof parallel edges. Similarly, G d©e is produced by deletion of edge e andcontraction of each line with two edges to just one edge. Indeed, in G\e, atmost two lines of length 2 may exist. In each such line, we may choose theedge to be contracted as is convenient. This aspect is important when we


want to preserve T nodes. As was done before, we use squares to denotenodes in T . We assign a question mark to a node if that node may or maynot be in T . We say that a node is cubic if it is a 3-star.

Claim 2. Every node v of G that is not part of some H is cubic, andthat node and two of its neighbors are in T . Furthermore, |T | = 4 in thatcase.

Proof. Let e be an edge incident at v, and u be its other endpoint. IfG d©e is 3-connected, one of the situations below must prevail; otherwise areduction is possible, as is readily checked.

(10.3.14)

e

v u

? e

v u

?

u

e

v

Case (i) Case (ii) Case (iii)u cubic v cubic Both u, v cubic|T | = 4 |T | = 4 |T | = 6

If G c©e is 3-connected, then u, v ∈ T and |T | = 4, since otherwise G c©ewith adjusted T set is a smaller case. We call the last situation case (iv).Thus, we have a total of four possible cases for the edge e.

Suppose v �∈ T . Apply the above arguments to every edge e incidentat v. In each instance, case (i) of (10.3.14) must apply, since cases (ii),(iii), and (iv) demand v to be in T . Thus, |T | = 4, and all neighbors of vand their neighbors (except v) must be in T . In addition, the neighbors ofv must be cubic. A simple case analysis shows that these conditions implyG to be 2-separable. Thus, v ∈ T .

Suppose v is not cubic. Again by arguments for each edge e incidentat v, we have |T | = 4, and v and all neighbors of v must be in T . Byassumption, there are at least four neighbors, so this is not possible.

So far we know v to be cubic and to be in T . If |T | > 4, then case (iii)of (10.3.14) must hold for every edge e incident at v, and |T | = 6. Oncemore, a case analysis shows that G is not 3-connected. Thus, |T | = 4.

Suppose v and its three neighbors are in T . Then no other node of Gis in T . Take any node w different from these four nodes. There exist threeinternally node-disjoint paths in G from v to w. Then clearly G and Tcan be reduced to a graph isomorphic to that of (10.2.5). The latter graphrepresents the nonregular matroid F ∗

7 , a contradiction. Q. E. D. Claim 2

Claim 3. There exists an H such that

(i) G has no node beyond those of H, and


(ii) no edge of G that is not an edge of H connects two nodes of a line ofH.

Proof. Suppose some H violates (ii). We then can find another one vio-lating (i). Thus, G has a node u not in some H. By Claim 2, u is cubic,u and precisely two of its neighbors are in T , and |T | = 4. By Theorem(10.3.9), there is another H with u as corner node. Pick a minimal such H,say H1. Clearly, H1 satisfies (ii) of Claim 3. We now prove that (i) holdsas well. If not, then G has, again by Claim 2, a cubic node v not in H1

such that v and precisely two of its neighbors are in T . Thus, G has thefollowing subgraph, where dashed lines represent internally node-disjointpaths.

(10.3.15)

v

u

w = u

Graph H plus node v

By Menger’s Theorem, there is a path from w to a non-T node of the dashedgraph. Thus, we can reduce G to produce one of the following graphs.

(10.3.16)

uuuv

e

f

v

v

Graph G1 Graph G2 Graph G3

From G1, delete the edge e and contract the edge f . This produces thegraph

(10.3.17) u

v

Graph for R12


which by (10.2.9) represents R12. It is easily checked that both G2 and G3

can be reduced to the graph

(10.3.18)

Graph for F7

which by (10.2.4) represents the nonregular F7. Q. E. D. Claim 3

Claim 4. |T | ≤ 6. Take any H that satisfies (i) and (ii) of Claim 3.Suppose G has an edge e that is not in H. Then at least one endpoint vof e is cubic, and v and the neighbors linked to v by edges of H are in T .

Proof. Suppose an edge e exists that is not in H. By part (b) of Lemma(10.3.8), G d©e is 3-connected. If |T | ≥ 8, or if the remaining conclusionsof Claim 4 do not hold, then G d©e provides a smaller case. If G has noedges beyond those of H, then G is isomorphic to K3,3, and |T | ≤ 6 holdstrivially. Q. E. D. Claim 4

Claim 5. There is an H satisfying (i) and (ii) of Claim 3 such that Hcontains all edges of G but at most one.

Proof. Let H be the graph of Claim 3. Simple case checking confirms thatH must contain all edges of G except for possibly two edges. Assume thereare two such edges. By Claim 4, one of these arcs has a cubic endpoint,say u, such that u and the neighbors linked to u by arcs of H are in T .

Now select a minimal H, say H2, that has u as corner node. By theproof of Claim 3, H2 may be assumed to satisfy (i) and (ii) of that claim.Suppose again there are two edges in G, say e and f , that are not in H2.Let v and w be the cubic endpoints of e and f that have the propertiesdescribed in Claim 4. Clearly |T | = 6. If v and w are on one line of H2,we can contract an intermediate arc of that line, and by part (c) of Lemma(10.3.8) have a smaller case. Otherwise, one endpoint of e or f , say of e,is not in T , and G d©e produces a smaller case. Thus, H2 is the desiredgraph. Q. E. D. Claim 5

Let H be the graph of Claim 5. If every edge of G is in H, then G andH are isomorphic to K3,3. If |T | = 6, we have the graph of (10.2.8), and Mis isomorphic to R10. If |T | = 4, then G can be reduced to the nonregularinstance of (10.3.18).

Finally, assume just one edge of G, say e, is not in H. By Claim 4, atleast one endpoint of e, say u, is cubic, and u and the neighbors linked to


u by arcs of H are in T . If the second endpoint v of e is cubic or |T | ≥ 6,then we can reduce G to a smaller case. If v ∈ T , then G can be reducedto the nonregular case of (10.3.18). Thus, G must be

(10.3.19)

uv = r, s or t

r

s t

Graph H plus one arc

If v = r or s, then the nonregular case of (10.3.18) can be produced. Ifv = t, we have by (10.2.9) an instance of R12.

We are prepared for the next section, where we prove two profoundexcluded minor theorems of Seymour and Tutte.

10.4 Characterization of Graphic Matroids

We prove profound characterizations of two classes of regular matroids interms of excluded minors. The first characterization is due to Tutte. Itsays that a regular matroid is graphic if and only if it has no M(K3,3)∗ orM(K5)∗ minors. The second characterization is due to Seymour. Accord-ing to that result, a 3-connected regular matroid is graphic or cographic ifand only if it has no R10 or R12 minors. The latter matroids are definedby (10.2.8) and (10.2.9). In this section, we prove these characterizationsand deduce some related material. Given the results of the preceding twosections, it is advantageous for us to start with the second characterization.

(10.4.1) Theorem. A 3-connected regular matroid is graphic or co-graphic if and only if it has no R10 or R12 minors.

Proof. The graphs, T sets, and representation matrices of (10.2.8) and(10.2.9) for R10 and R12 prove these matroids to be nongraphic and isomor-phic to their respective duals. Thus, R10 and R12 are also not cographic.These observations establish the easy “if” part.

For proof of the converse, let M be a 3-connected regular matroidthat is not graphic and not cographic. Thus, M is not planar, and by The-orem (10.2.11) has a minor isomorphic to M(K5), M(K3,3), M(K5)∗, orM(K3,3)∗. By Lemma (10.3.1), M(K5) is a splitter for the regular matroidswithout M(K3,3) minors. By these results, M has a minor isomorphic toM(K3,3) or M(K3,3)∗, or M is isomorphic to M(K5) or M(K5)∗. The

10.4. Characterization of Graphic Matroids 225

latter case is a contradiction. Thus, M or M∗ has an M(K3,3) minor. ByTheorem (10.3.11), M or M∗ has an R10 or R12 minor. Since R10 and R12

are isomorphic to their duals, M itself has an R10 or R12 minor.

We turn to the characterization of regular matroids that are graphic.

(10.4.2) Theorem. A regular matroid is graphic (resp. cographic) if andonly if it has no M(K5)∗ or M(K3,3)∗ minors (resp. M(K5) or M(K3,3)minors).

Proof. Lemma (3.2.48) implies the easy “only if” part. For proof of theconverse, let M be a nongraphic regular matroid all of whose proper minorsare graphic. If M is 1- or 2-separable, then arguments analogous to those ofthe proof of Theorem (10.2.11) establish M to be graphic, a contradiction.Thus, M is 3-connected.

Suppose M is not cographic. By Theorem (10.4.1), M has an R10

or R12 minor. The drawings of (10.2.8) and (10.2.9) clearly establish thatboth R10 and R12 have proper M(K3,3) minors. Now R10 and R12 areisomorphic to their duals. Thus, M has an M(K3,3)∗ minor.

Now consider M to be cographic, i.e., consider M∗ to be graphic. IfM∗ is planar, then M is planar, and hence graphic, a contradiction. IfM∗ is nonplanar, then by Theorem (7.4.1), M∗ has an M(K5) or M(K3,3)minor. Thus, M has an M(K5)∗ or M(K3,3)∗ minor, as desired.

The parenthetic claim of the theorem follows by duality.

We complete this section with two corollaries. The first one effectivelyrestates Corollary (10.2.13).

(10.4.3) Corollary. A matroid is planar if and only if it is graphic andcographic.

Proof. Apply Corollary (10.2.13), or compare the excluded minors of The-orems (10.2.11) and (10.4.2) to obtain the conclusion.

For the second corollary, we need the following auxiliary result.

(10.4.4) Lemma. Every 1-element reduction of R10 or R12 produces amatroid with an M(K3,3) or M(K3,3)∗ minor.

Proof. For R10, the proof is easy. One first shows that R10 is highlysymmetric as follows. Consider a binary matrix D with five rows. Eachcolumn of D has exactly three 1s, with each possible case occurring. Thus,D has

(53

)= 10 columns. Index the columns of D by a 10-element set

E. We claim that E and the subsets of E indexing GF(2)-independentcolumns of D define a matroid isomorphic to R10. For a proof, we performrow operations in D until a 5 × 10 matrix [I | B] results. Simple checkingconfirms that the matrix B is either up to indices the matrix of (10.2.8)


for R10, or is the matrix of (10.4.5) below. A pivot on the 1 in the (1, 2)position of the latter matrix converts it to the former one.

(10.4.5)

11 1 1 0 0

0111 1 1

11001

100 0

1 0 0 1

Alternate matrix for R10

By (10.2.8), the minor R10\z of R10 is isomorphic to M(K3,3). By the just-proved symmetry and duality, every 1-element deletion (resp. contraction)in R10 produces a matroid isomorphic to M(K3,3) (resp. M(K3,3)∗).

The proof for R12 is about as easy. That matroid is also isomorphic toits dual. Thus, we can confine ourselves to the contraction case. With theaid of (10.2.9), one can prove presence of an M(K3,3) or M(K3,3)∗ minorafter each 1-element contraction.

The proof of Lemma (10.4.4) contains the following result about R10.

(10.4.6) Lemma. Up to indices, the matrices of (10.2.8) and (10.4.5)are the only binary representation matrices for R10.

Related to Lemma (10.4.4) is the following theorem about 1-elementregular extensions of planar matroids. The proof of the theorem relies onTheorem (10.4.1), Lemma (10.4.4), and two results of Chapter 11: Theo-rem (11.3.2), which establishes R10 to be a splitter of the class of regularmatroids, and Theorem (11.3.14), which is the regular matroid decompo-sition theorem.

(10.4.7) Theorem. Every regular 1-element extension of a 3-connectedplanar matroid is graphic or cographic.

Proof. Let N be a 3-connected planar matroid and M be a regular exten-sion of N by an element z. By duality, we may assume that N = M/z.

If M is not 3-connected, then by the 3-connectivity of N , the elementz is a loop, coloop, or series element of M , and planarity of N impliesplanarity of M . Hence, assume that M is 3-connected.

If M is not graphic and not cographic, then by Theorem (10.4.1), Mhas an R10 or R12 minor. If M itself is that minor, then by Lemma (10.4.4),N = M/z must have an M(K3,3) or M(K3,3)∗ minor and cannot be planar.Hence, assume that M has a proper R10 or R12 minor.

Theorem (11.3.2) establishes R10 to be a splitter of the class of regularmatroids. Hence, the 3-connected M cannot have a proper R10 minor.

In the remaining case, M is 3-connected and has a proper R12 minor.By Theorem (11.3.14), M has an R12 minor whose 3-sum decomposition

10.5. Decomposition Theorems for Graphs 227

as displayed by B12 of (11.3.11) and shown here for ready reference,

(10.4.8)B12 =

X1

X2

Y2Y1

0

0

11

0

0101

110

1

1010

1

0101

1 11 11 00 1

1 0 01 0 0

Matrix B12 for R12

induces a 3-separation, indeed a 3-sum decomposition, of M . Thus, arepresentation matrix B of (8.3.10) of the 3-sum M exists that displaysB12 of (10.4.8) such that each index set Xi or Yi of B of (8.3.10) containsthe corresponding set Xi or Yi of B12 of (10.4.8).

We use the matrix B to analyze the 1-element contractions of M thatresult in a 3-connected minor. Straightforward arguments show that eachsuch minor has an M(K3,3) or M(K3,3)∗ minor. That conclusion contra-dicts the fact that N = M/z is 3-connected and planar.

The assumption of 3-connectivity of N in Theorem (10.4.7) is essential.For a proof, adjoin to the matrix B12 of (10.4.8) the column [0, 0, 0, 1, 0, 1]t

and assign z as index to that column. Define M to be the matroid repre-sented by the resulting matrix. It is easily checked that M is 3-connected,regular, not graphic, and not cographic, and that N = M/z is planar andconnected, but not 3-connected. Hence, the assumption of 3-connectivityof N in Theorem (10.4.7) cannot be reduced to connectivity.

In the next section, we introduce a major switch of topic. We examinea simple yet powerful idea that produces interesting graph decompositiontheorems. In Chapters 11–13, we rely on the matroid generalization of thisidea to prove profound matroid decomposition theorems.

10.5 Decomposition Theorems for Graphs

In subsequent chapters, we make repeated use of a recursive construction ofmatroid decomposition theorems. In this section, we use a graph exampleto motivate and explain that construction. In the process, we deduce byrather elementary checking a famous decomposition theorem of Wagnerabout the class of graphs without K5 minors.

The basic idea of the construction is as follows. Given is a class G ofconnected graphs. The class is closed under isomorphism and under thetaking of minors. On hand are also two subclasses L and H of G. The two


subclasses are so selected that each graph G ∈ G is in L or has, for someH ∈ H, an H minor.

One iteration of the construction is as follows. We select a graph Hof H and use one of the decomposition theorems of Chapter 6 or 7 toestablish a result of the following type: If a graph G ∈ G has an H minor,then G has a certain decomposition caused by H, or G is a member of acertain collection of graphs, say {L1, L2, . . . , Lm}, or G has a minor that isa member of a certain second collection of graphs, say {H1, H2, . . . , Hn}.We now derive from L the set L′ = L ∪ {L1, L2, . . . , Lm}, and from Hthe set H′ = (H − H) ∪ {H1, H2, . . . , Hn}. By the derivation of L′ andH′, we have established the following theorem: Each graph G ∈ G can bedecomposed, or belongs to L′, or has an H ′ minor for some H ′ ∈ H′.

At this point, we have completed one iteration of the construction.If H′ is empty, we stop; most likely the cited theorem is an interestingdecomposition result for the graphs of G. If H′ is nonempty, we have twochoices: We may stop, or we may carry out another iteration by declaringL′ to be L and H′ to be H. Evidently, the recursive construction is nothingbut a concatenation of decomposition results, each of which is deduced fromsuitably selected theorems of Chapters 6 and 7.

An example will help to clarify the construction. We want a decom-position theorem for the graphs without K5 minors. That class of graphsturns out to be important for a number of combinatorial problems. Detailsare included in Section 10.7. We ignore the applications for the time being,and concentrate on the construction of a decomposition theorem for thatclass. In agreement with the preceding outline, we define G to be the setof connected graphs without K5 minors. The subclass L is the collectionof connected planar graphs, and H is the set {K3,3}. By Theorem (7.4.1)and the exclusion of K5 minors from the graphs of G, each graph G ∈ G isplanar or has a K3,3 minor. Thus, L and H do satisfy the condition statedearlier, i.e., each graph G ∈ G is in L or has an H minor for the singlegraph H = K3,3 of H.

We are ready for the first iteration of the construction. We will relyon the splitter theorem for graphs of Section 7.2, listed there as Corollary(7.2.10). We repeat that result below.

(10.5.1) Theorem. Let G be a class of connected graphs that is closedunder isomorphism and under the taking of minors. Let H be a 3-connectedgraph of G with at least six edges.

(a) If H is not a wheel, then H is a splitter of G if and only if G doesnot contain any graph derived from H by one of the following twoextension steps:

(1) Connect two nonadjacent nodes of N by a new edge.

(2) Partition a vertex of degree at least 4 into two vertices, each ofdegree at least 2, and connect these two vertices by a new edge.


(b) If H is a wheel, then H is a splitter of G if and only if G does notcontain any of the extensions of H described under (a) and does notcontain the next larger wheel.

The application of the splitter theorem involves several graphs, whichwe define next. The graphs are K3,n, n ≥ 3, as well as certain variantsof K3,n, called K1

3,n, K23,n, and K3

3,n. The graph K33,n is given by (10.5.2)

below. The other graphs are obtained from K33,n by deletion of some edges.

In the notation of (10.5.2), K23,n is the graph K3

3,n\c, and K13,n is the graph

K33,n\{b, c}. At times, we refer to K3,n as K0

3,n. We collect the graphs justdefined in a set K = {Ki

3,n | 0 ≤ i ≤ 3, n ≥ 3}.

(10.5.2)

1

2

n

a

b

c

Graph K33,n

We need one additional graph V defined by

(10.5.3)

v1

v2

v4

v3

v5

v6

v7

Graph V

For the moment, the reader should ignore the dashed line in the drawingof V . It indicates a 3-separation that we will utilize later. Note that V hasno K5 minors. Indeed, V has eleven edges and seven vertices, and if V hasa K5 minor, then such a minor must be produced by a single contractionor deletion. But any such reduction results in a graph with at least sixvertices.

The splitter Theorem (10.5.1) involves 3-connected 1-edge extensionsof graphs. The next lemma supplies information about such extensions forthe graphs of K.


(10.5.4) Lemma. For any graph G of K, any 3-connected 1-edge exten-sion of G is isomorphic to another graph of K, or has a K5 minor, or hasa V minor.

Proof. The proof involves a simple checking of cases for the graphs Ki3,n,

0 ≤ i ≤ 3, with n fixed, plus induction. As examples, we cover the casesof K3,3 and K1

3,3. Since every vertex of K3,3 has degree 3, any 3-connected1-edge extension must be an addition. Indeed, just one addition case exists,the graph K1

3,3. We depict that graph below.

(10.5.5)

v1

v2

v3

v4

v5

v6

Graph K13,3

The 3-connected 1-edge extensions of K13,3 are as follows. We start with the

expansion cases. By symmetry, all cases are isomorphic to the followingone, where we split the vertex v1. The resulting graph is isomorphic to V ,as is evident from the next drawing.

(10.5.6)

v11

v12

v2

v3

v4

v5

v6

=

v4

v5

v12

v2

v3

v11

v6

3-connected 1-edge expansion of K13,3

We turn to the addition cases. All such instances are isomorphic to twographs, one of which is K2

3,3, while the second one becomes K5 uponcontraction of one edge. We leave the verification of this claim to thereader.

With these preparations, we carry out the first iteration of the con-struction process. We must select the single graph K3,3 of H as H. Fromthe splitter Theorem (10.5.1) and Lemma (10.5.4), we deduce the followingdecomposition result.


(10.5.7) Theorem. Every connected graph without K5 minors is 1-or 2-separable, or is planar, or is isomorphic to one of the graphs Ki

3,n,0 ≤ i ≤ 3, n ≥ 3, or has a V minor.

Proof. Let G be any graph without K5 minors. In the nontrivial case, Gis nonplanar. By Theorem (7.4.1), we know that G has a K3,3 minor, andthus a Ki

3,n minor with an edge set of maximum cardinality. If G itself isthat minor, we are done. Otherwise, we apply the splitter Theorem (10.5.1)to the class of graphs consisting of all minors of G and their isomorphicversions. The graph Ki

3,n plays the role of the splitter. Thus, G is 2-separable, or is 3-connected and has a 3-connected 1-edge extension of aKi

3,n minor. In the first case, we are done. In the second case, we know byLemma (10.5.4) and the assumed maximality of the edge set of Ki

3,n thatG has a V minor.

We have reached the end of the first iteration. In the notation of thegeneral construction process, the current set L′ contains the planar graphs,and all graphs Ki

3,n plus their isomorphic versions. The set H′ containsjust one graph, V .

We begin the second iteration. The current H, i.e., the set H′ ofiteration 1, contains just V . Thus, that graph is selected as H. We intendto invoke the induced separation result for graphs of Section 6.3, listedthere as Corollary (6.3.26). We include that result below.

(10.5.8) Theorem. Let G be a class of connected graphs that is closedunder isomorphism and under the taking of minors. Let a 3-connectedgraph H ∈ G have a 3-separation (F1, F2) with |F1|, |F2| ≥ 4. Assume thatH/F2 has no loops and H\F2 has no coloops. Furthermore, assume thatfor every 3-connected 1-edge extension of H in G, say by edge z, the pair(F1, F2∪{z}) is a 3-separation of that extension. Then for any 3-connectedgraph G ∈ G with an H minor, the following holds. Any 3-separation of anysuch minor that corresponds to (F1, F2) of H under one of the isomorphismsinduces a 3-separation of G.

We need a 3-separation (F1, F2) of V of (10.5.3) for the application ofTheorem (10.5.8). Thus, we define F1 to be the set of edges of V incidentat node v1 or v2, and declare F2 to be the set of the remaining edges. The3-separation (F1, F2) of V is informally indicated in (10.5.3) by the dashedline.

We will encounter one additional graph G8 given by (10.5.9) below.For the moment, the unusual indexing of the node labels of G8 should beignored. It will make sense shortly. The graph G8 has twelve edges, andevery vertex has degree 3. We claim that G8 does not have K5 minors;otherwise, the contraction of two suitably selected edges could produce agraph where at least five vertices have degree of at least 4. But that is not


possible.

(10.5.9)

v1v5

v7

v6

v3

v2v42

v41

Graph G8

The decomposition result for the second iteration is as follows.

(10.5.10) Theorem. Let G be a 3-connected graph without K5 minors,but with a V minor. Then the 3-separation of that minor defined from(F1, F2) of V induces a 3-separation of G, or G has a G8 minor.

Proof. We apply Theorem (10.5.8) with the class of connected graphswithout K5 minors as G, and with the graph V of (10.5.3) as H. We readilyverify that V /F2 has no loops and that V \F2 has no coloops. Thus, byTheorem (10.5.8), the claimed induced 3-separation exists, or V can beextended by one edge z to a 3-connected graph for which (F1, F2 ∪ {z}) isnot a 3-separation. We consider all such 3-connected extensions of V byan edge z.

We start with the expansion case. Since V &z is to be 3-connected, wemust split the single degree 4 vertex v4 of V and insert z. There are threeways to do this. The corresponding graphs V1, V2, V3 are given below.

(10.5.11)

v1

v2

v3

z

v41

v42

v5

v6

v7

z

v1

v2

v3

v41

v42

v5

v6

v7

z

v1

v2

v3

v41

v42

v5

v6

v7

Graph V1 Graph V2 Graph V3

Evidently, (F1, F2 ∪ {z}) is not a 3-separation for V1 or V2, but this is sofor V3. Thus, V1 and V2 are the graphs of interest to us. The graphs V1

and V2 are isomorphic. Indeed, one isomorphism from the vertices of V1

to those of V2 is an identity except that it takes v3, v5, v6, v7 of V1 to v5,v3, v7, v6 of V2, respectively. Furthermore, a comparison of V1 with G8 of(10.5.9) proves these two graphs to be identical.


We turn to the addition case. Since V +z is to be 3-connected, theadded edge z cannot be parallel to another edge. Also, (F1, F2∪{z}) is notto be a 3-separation of V +z. Thus, one endpoint of z must be v1 or v2.The second endpoint must be v1, v2, v6, or v7. Suppose v1 is one endpointand v2 the second one. Contract in V +z the edges (v3, v6) and (v5, v7).A K5 minor results, which is a contradiction. Suppose v1 is one endpointand v6 is the second one. Contract in V +z the edges (v2, v3) and (v5, v7).Again a K5 minor results. The remaining cases are isomorphic to the latterone.

We conclude that in each case G has a G8 minor as desired.

We combine Theorems (10.5.7) and (10.5.10) to the following result.

(10.5.12) Theorem. Every connected graph without K5 minors is 1- or2-separable, or has a 3-separation with at least four edges on each side, oris planar, or is isomorphic to a graph Ki

3,n, 0 ≤ i ≤ 3, n ≥ 3, or has a G8

minor.

Proof. By Theorem (10.5.7), we may assume G to have a V minor. ThenTheorem (10.5.10) establishes presence of the 3-separation or of a G8 mi-nor.

We have reached the end of the second iteration. In terms of thegeneral description of the construction, the current set L′ contains theplanar graphs, and all graphs Ki

3,n plus their isomorphic versions. The setH′ contains just one graph, G8.

The third iteration involves an application of the splitter Theorem(10.5.1). The result so produced is as follows.

(10.5.13) Theorem. G8 is a splitter for the graphs without K5 minors.

Proof. We only need to show that every 3-connected 1-edge extension ofG8 has K5 minors. Since every vertex of G8 has degree 3, a 3-connected 1-edge expansion is not possible. Up to isomorphism, just two addition casesare possible. Both cases have K5 minors. We leave the easy verification ofthis claim to the reader.

We combine Theorems (10.5.12) and (10.5.13) with the results for 1-,2-, and 3-sums of Chapter 8 to obtain Wagner’s famous decompositiontheorem for the graphs without K5 minors.

(10.5.14) Theorem. Every connected graph without K5 minors is a 1-,2-, or 3-sum, or is planar, or is isomorphic to K3,3 or G8.

Proof. Assume G to be a connected graph without K5 minors. Theo-rems (10.5.12) and (10.5.13) imply that G is 1- or 2-separable, or has a3-separation with at least four edges on each side, or is planar, or is iso-morphic to G8 or to a graph Ki

3,n, 0 ≤ i ≤ 3, n ≥ 3. If G is isomorphic to agraph Ki

3,n different from K3,3, then by (10.5.2), G has a 3-separation with


at least four edges on each side. We apply some results of Chapter 8. If Gis 1- or 2-separable, then according to Section 8.2, G is a 1- or 2-sum. IfG has a 3-separation with at least four edges on each side, then by Lemma(8.3.12) and the discussion following that lemma, G is a 3-sum. Thus, wemay conclude that G is a 1-, 2-, or 3-sum, or is planar, or is isomorphic toK3,3 or G8, as claimed in the theorem.

Note that at the end of the third iteration, the set H′ is empty. Thus,the construction process stops.

Recall the Δ-sum decomposition of Section 8.5. In a connected graphG, such a decomposition is carried out as follows. Given is a 3-separation ofG with at least four edges on each side. Let H1 and H2 be the correspondingsubgraphs of G. Thus, both H1 and H2 are connected subgraphs of G.Identification of three connecting nodes of H1 with three connecting nodesof H2 produces G. For i = 1, 2, we enlarge Hi by attaching a triangle tothe three connecting nodes. Let Gi be the resulting graph. Then G is aΔ-sum of G1 and G2, denoted by G = G1 ⊕Δ G2. The components G1 andG2 of a 3-connected Δ-sum G1 ⊕Δ G2 are 3-connected, except possibly foredges parallel to the edges of the connecting triangle in G1 or G2.

Also recall the 2-sum decomposition of Section 8.2. The graphs H1 andH2 have two connecting nodes each. For i = 1, 2, we enlarge Hi by joiningthe connecting nodes by an edge. Let Gi be the resulting graph. Then Gis a 2-sum of G1 and G2, denoted by G = G1 ⊕2 G2. The components G1

and G2 are 2-connected if G is 2-connected.By inverting the above operations, we obtain the Δ-sum and 2-sum

compositions. At times, one may desire to construct a graph recursivelyby these operations. Initially, one combines two graphs G1 and G2 in a 2-or Δ-sum. Then one composes the resulting graph in a 2- or Δ-sum witha graph G3. Continuing in this fashion, one recursively enlarges the graphon hand by G4, G5, etc. We call the Gi, i ≥ 1, the building blocks of thisprocess.

We may use Theorem (10.5.14) to establish such a construction processfor the graphs without K5 minors. The details are specified in the nexttheorem.

(10.5.15) Theorem. Any 2-connected graph without K5 minors is pla-nar, or isomorphic to K3,3 or G8, or may be constructed recursively by2-sums and Δ-sums. The building blocks of that construction are as fol-lows.

2-sums: planar graphs, and graphs isomorphic to K3,3 or G8.Δ-sums: planar graphs.

The proof of Theorem (10.5.15) utilizes the following two lemmas.

(10.5.16) Lemma. Let G be a 3-connected nonplanar graph without K5

minors and not isomorphic to K3,3 or G8. Assume G to have a triangle C.


Then G has a 3-separation (E1, E2) where |E1|, |E2| ≥ 4 and where one ofE1, E2 contains C.

Proof. We use induction. If G has ten edges, the smallest case, then directchecking proves the lemma. Otherwise, G is by Theorem (10.5.14) a 3-sum,and thus has a 3-separation (E1, E2) where |E1|, |E2| ≥ 4. If one of E1, E2

contains C, we are done. Thus, we may assume that just one edge of C,say c, is in E2. If |E2| ≥ 5, we shift the edge c from E2 to E1 and have thecase where one side of a 3-separation contains C. Thus, we assume that|E2| = 4. It is easy to see that G must be of the form

(10.5.17)

ecfg

E1 E2

3-separation (E1, E2) of G with |E2| = 4 and c ∈ E2

If G has an edge that forms a triangle with the explicitly shown edges e andg, or with f and g, then we exchange c and such an edge between E1 andE2, and again have the desired 3-separation. Otherwise, the minor G/ghas no parallel edges. Note that {e, f, c} is a triangle of G/g. Indeed, G/gis isomorphic to one of the components of the Δ-sum induced by (E1, E2),and thus is 3-connected. We consider two cases, depending on whetherG/g is planar.

If G/g is planar, draw it in the plane. If the triangle {e, f, c} lies onone face, then it is easily seen that G itself is planar, a contradiction. Thus,{e, f, c} partitions the plane into two regions, both of which contain at leastone vertex of G/g. Then we readily confirm that G/g, and hence G, has a3-separation of the form claimed in the lemma.

For the second case, we assume G/g to be nonplanar. Since G/g hasa triangle while K3,3 and G8 do not, G/g cannot be isomorphic to eitherone of the latter graphs. We apply induction and see that G/g as well asG have the desired 3-separation as well.

(10.5.18) Lemma. Let G be a 3-connected nonplanar graph withoutK5 minors and not isomorphic to K3,3 or G8. Assume G to have either adesignated triangle C or a designated edge e. Then G is a Δ-sum G1⊕ΔG2,where G1 contains C or e, whichever applies, and where G2 is planar.

Proof. We prove the case for the triangle C and leave the easier situationwith the edge e to the reader. We use induction. The smallest case, whichhas ten edges, is handled by direct checking. For larger G, we apply Lemma(10.5.16). Thus, G is a Δ-sum G1⊕Δ G2 where C is part of the component


G1. If the second component G2 is planar, we are done. Otherwise, wemay assume G2 to be 3-connected. We define C′ to be the triangle ofG2 involved in the Δ-sum. Due to the presence of the triangle C′, thegraph G2 cannot be isomorphic to K3,3 or G8. By induction, G2 has aΔ-sum decomposition G21 ⊕Δ G22 where C′ is in G21, and where G22 isplanar. Evidently, G1 ⊕Δ G21 and G22 are the components of a Δ-sumdecomposition of G of the desired form.

Proof of Theorem (10.5.15). Let G be any 2-connected graph withoutK5 minors and not isomorphic to K3,3 or G8. If G is 3-connected, the resultfollows from Lemma (10.5.18). Otherwise, G is a 2-sum. Choose the 2-sumdecomposition, say G1 ⊕2 G2, so that G2 has a minimal number of edges.Evidently, any 2-separation of G2 contradicts the minimality assumption,so G2 is 3-connected. If G2 is planar or isomorphic to K3,3 or G8, we aredone. Otherwise, let e be the edge of G2 that is identified with an edgeof G1 in the 2-sum composition creating G. By Lemma (10.5.18), G2 is aΔ-sum G21⊕ΔG22, where G21 contains e and where G22 is planar. Clearly,G is a Δ-sum where one component is G1 ⊕2 G21, and where the secondcomponent is the planar G22, as demanded in the theorem.

The recursive construction scheme described at the beginning of thissection produces a number of additional decomposition theorems. We in-clude two example theorems that may be obtained that way. The firsttheorem refers to the graph G9 of (10.5.19) below, and to K5\y, which isK5 minus an arbitrarily selected edge y.

(10.5.19)

Graph G9

(10.5.20) Theorem. Every 3-connected graph G with at least six edgesand without K5\y minors is, for some k ≥ 3, isomorphic to the wheel Wk,or is isomorphic to G9 or K3,3.

Proof. One first shows that each 3-connected 1-edge extension of anywheel graph with at least four spokes has a K3,3 or G9 minor. Then onesuitable application of the splitter Theorem (10.5.1) proves the result.

The next theorem is much more complicated. We omit the proof, sinceit involves rather tedious calculations. The theorem refers to a number ofgraphs, which are listed subsequently under (10.5.22).

(10.5.21) Theorem. Every connected graph without G12 minors is a 1-,2-, or 3-sum, or is planar, or is isomorphic to K5, K3,3, G8, G13, G1

14, G214,

G115 G2

15, G315, or G4

15.


Here are the graphs mentioned in Theorem (10.5.21).

(10.5.22)

G12

G151

G141

G153 G15

4

G142G13

G152

Graphs of Theorem (10.5.21)

The graph G415 is the well-known Petersen graph.

From Theorems (10.5.20) and (10.5.21), one may deduce the following2- and Δ-sum construction results.

(10.5.23) Theorem. Any 2-connected graph on at least two edges andwithout K5\y minors may be constructed recursively by 2-sums. Eachbuilding block is a cycle with two or three edges, or is isomorphic to Wk,k ≥ 3, G9, or K3,3.

Proof. Let G be a 2-connected graph on at least two edges and withoutK5\y minors. If G is not isomorphic to one of the listed graphs, then Ghas by Theorem (10.5.20) a 2-sum decomposition G1 ⊕2 G2. Choose thedecomposition so that G2 has a minimum number of edges. Then G2 mustbe 3-connected, and by Theorem (10.5.20) must be one of the prescribedbuilding blocks.

(10.5.24) Theorem. Any 2-connected graph without G12 minors isplanar, or isomorphic to K5, K3,3, G8, G13, G1

14, G214, G1

15, G215, G3

15,or G4

15, or may be constructed recursively by 2-sums and Δ-sums. Thebuilding blocks are as follows.

2-sums: planar graphs, and graphs isomorphic to K5, K3,3, G8, G13,G1

14, G214, G1

15, G215, G3

15, or G415.

Δ-sums: planar graphs and graphs isomorphic to K5.

Proof. We use appropriately modified Lemmas (10.5.16) and (10.5.18),and the proof of Theorem (10.5.15). Below, we indicate the necessary


adjustments. First, the graphs K3,3 and G8 must throughout be replacedby K5, K3,3, G8, G13, . . . , G4

15. We note that no graph of that new listhas a triangle except for K5. Second, the proof of the modified Lemma(10.5.16) must be adjusted as follows. In the case of a nonplanar graphG/g, that graph cannot be isomorphic to K5, K3,3, G8, G13, . . . , G4

15,except for K5. For the exceptional case, one directly shows G to have a 3-separation of the type demanded by the modified Lemma (10.5.16). Third,in the modified Lemma (10.5.18) and its proof, one now permits the graphG2 to be isomorphic to K5. The analogous change applies to the graphG22 of the proof of Theorem (10.5.15).

The decomposition tools provided in this section plus those cited in thereferences should enable the reader to construct additional decompositiontheorems as they are needed. We sketch representative applications for thepreceding decomposition theorems in Section 10.7.

Once more we switch topics, and turn to the problem of decidinggraphicness of a binary matroid.

10.6 Testing Graphicness of Binary

Matroids

Chapters 3, 5, 7, and 8 implicitly contain a quite efficient algorithm fortesting graphicness of binary matroids, the topic of this section. Thus, thissection is mainly a synthesis of material gleaned from those chapters. Wealso cover the related problem of deciding whether or not a real {0,±1}matrix is the coefficient matrix of a network flow problem.

We start with the graphicness test. Let B be a binary representationmatrix of the matroid M to be tested. Small instances are easily decided,so assume that M has at least six elements. If B is not connected, then itclearly suffices that we test each connected component of B for graphicness.Indeed, for some m ≥ 2, let G1, G2, . . . , Gm be the graphs correspondingto the connected components of B. From each Gi, we select some vertexvi. Then we combine G1, G2, . . . , Gm to a connected graph G for M andB by identifying the vertices v1, v2, . . . , vm to one vertex.

Suppose that B is connected, and that we know of a 2-separation of B.By Lemma (8.2.6), M is a 2-sum of two matroids M1 and M2. Let B1 andB2 be the respective submatrices of B representing the latter matroids.By Lemma (8.2.7), M is graphic if and only if B1 and B2 are graphic.Thus, we may analyze B1 and B2 instead of B. Indeed, let G1 and G2

be graphs for B1 and B2, respectively. Then the 2-sum composition of G1

and G2 displayed in (8.2.8) produces a graph G for B. In general, G is not

10.6. Testing Graphicness of Binary Matroids 239

unique. But by Theorem (3.2.36), we know that any other graph for B canbe obtained from G by a sequence of switchings.

The case remains where B is 3-connected. By Corollary (5.2.15),M has an M(W3) minor. Furthermore, by Theorem (7.3.4), M has a3-connected 1-element expansion of an M(W3) minor, or has a sequence ofnested 3-connected minors M0, M1, . . . , Mt = M , where M0 is an M(W3)minor, and where each Mi+1 is obtained from Mi by some series expansions,possibly none, followed by a 1-element addition.

By the census of Section 3.3, every 3-connected 1-element expansion ofan M(W3) minor must be an F ∗

7 minor, which is not regular, and hence notgraphic. Thus, evidence of such a minor proves M to be nongraphic. Onthe other hand, graphicness of each matroid of the sequence M0, M1, . . . ,Mt can be efficiently decided as follows. Clearly, the M(W3) minor M0 isgraphic, and a graph for it is readily found. Suppose for given 0 ≤ i < t,we know Mi to be graphic. By Theorem (3.2.36), the 3-connectedness ofMi implies that just one graph exists for M , say Gi. We assume that wehave that 3-connected graph on hand. Now Mi+1 is obtained from Mi bysome series expansions and a 1-element addition. The expansions stepscan also be done in Gi, so they preserve graphicness. By Lemma (3.2.49),there is a polynomial, indeed very simple, subroutine for deciding whetherthe addition step preserves graphicness as well. In the affirmative case, thesubroutine also produces the graph for Mi+1.

So far, we have assumed that we can locate 2-separations and M(W3)minors, as well as a 3-connected 1-element expansion of a given M(W3)minor or the sequence M0, M1, . . . , Mt. But these tasks can be efficientlyaccomplished by the path shortening technique of Chapter 5 and by the effi-cient method sketched in the proof of Theorem (7.3.6). These two methodsplus the simple graphicness testing subroutine of Lemma (3.2.49) constitutean efficient way of deciding whether a given binary matroid is graphic.

We turn to the second testing problem covered in this section. We areto decide whether or not a real {0,±1} matrix is the coefficient matrix ofa network flow problem. Recall from Section 9.2 that a node/arc incidencematrix of a directed graph is a real {0,±1} matrix where each column hasonly 0s or exactly one +1 and one −1. Furthermore, recall that a realmatrix is defined to be the coefficient matrix of a network flow problem ifit is the node/arc incidence matrix A of a directed graph, or is derived fromsuch a matrix by pivots and deletion of rows and columns. The supportmatrix B of A is the node/edge incidence matrix of the undirected versionof that graph. View B to be over GF(2).

Lemmas (9.2.1), (9.2.2), (9.2.6), (9.2.8), and Corollary (9.2.7) permitthe following conclusions about A and B. The matrix A is totally unimod-ular, and B is regular. Any matrix A deduced from A by real pivots anddeletion of rows and columns is totally unimodular. The support matrixB of A can be deduced from B by the corresponding GF(2)-pivots. Thus,


B is graphic. Finally, an elementary signing process exists by which onemay determine from B the signs of the entries of A, up to a scaling of somerows and columns by −1.

Because of these results, we may test a given real {0,±1} matrix A forthe network flow property as follows. View the support matrix B of A tobe over GF(2). Test B for graphicness with the efficient method describedearlier in this section. If B is not graphic, then A cannot have the networkflow property, and we stop. Otherwise, sign B to convert it to a totallyunimodular matrix A′. By a simple variation of that signing process, whicheffectively is the proof procedure of Lemma (9.2.6), determine whether ornot A′ can be converted to A by scaling. Then A is a network flow problemif and only if the answer is affirmative. This procedure can be improvedby use of the undirected graph on hand once B has been determined to begraphic. We leave the details to the reader.

A variation of the preceding problem is as follows. We are given areal matrix A. We are to settle whether by row scaling, or by row andcolumn scaling, that matrix can be converted to a {0,±1} matrix with thenetwork flow property. The question can be reduced to the above situationas follows. If BG(A) is not connected, we apply the test given below to thesubmatrices of A that correspond to the connected components of BG(A).Hence, assume A to be connected. First, we determine whether by scalingunder the assumed restrictions (i.e., scaling of rows only, or scaling of rowsand columns), the given A can be converted to a {0,±1} matrix. This isreadily accomplished by a scaling of the entries of A corresponding to anarbitrarily selected tree of BG(A). We leave it to the reader to fill in thesimple details. If a {0,±1} matrix cannot be produced, then the networkflow property cannot be attained by scaling under the assumed restrictions.If a {0,±1} matrix is obtained, we test that matrix for the network flowproperty. The answer for the original question is affirmative if and only ifthe scaled matrix has the network flow property. The conclusion is validsince the scaled matrix is unique up to scaling of some rows and columnsby −1, as is readily confirmed via the proof of Lemma (9.2.6).

We summarize the above discussion in the following theorem.

(10.6.1) Theorem. There are polynomial algorithms for each one of theproblems (a), (b), and (c) below.

(a) Given is any binary matrix B. Let M be the binary matroid rep-resented by B. It must be decided whether M is graphic. In theaffirmative case, an undirected graph G must be produced so thatM(G) = M .

(b) Given is any {0,±1} real matrix A. It must be decided whether A hasthe network flow property. In the affirmative case, a directed graph Gmust be produced so that the node/arc incidence matrix of G can byrow operations be transformed to the matrix [I | A].

10.7. Applications, Extensions, and References 241

(c) Given is any real matrix A. It must be decided whether A can by rowscaling, or by row and column scaling, be transformed to a {0,±1}matrix. In the affirmative case, the {0,±1} matrix must be produced,and for that matrix, problem (b) must be solved.

In the final section, we sketch applications and extensions and citerelevant references.

10.7 Applications, Extensions, and

References

We link the material of this chapter to prior work, and point out applica-tions and extensions.

The representation of a binary 1-element addition of a graphic matroidin Section 10.2 by a graph plus a node subset T is taken from Seymour(1980b). It is one of the many innovative concepts and ideas of that ref-erence. The planarity characterization of Theorem (10.2.11) is properlyimplied by Tutte’s characterization of the graphic matroids (Tutte (1958),(1959), (1965)). The same applies to Corollary (10.2.13), which originallywas proved in Whitney (1932), (1933b).

Lemma (10.3.1) is taken from Seymour (1980b). Theorem (10.3.9)has been generalized to nongraphic 3-connected matroids with triangles inAsano, Nishizeki, and Seymour (1984). Seymour (1980b) contains the dif-ficult Theorem (10.3.11). The proof given here is a considerably shortenedversion of the one of that reference. The key difference lies in the repeatedapplication of Theorem (10.3.9).

The profound characterizations of Section 10.4 in historical order aredue to Tutte and Seymour. Theorem (10.4.2) is due to Tutte (1958), (1959),(1965), and Theorem (10.4.1) is due to Seymour (1980b). The latter the-orem is one of two main ingredients in the proof of the decompositiontheorem of regular matroids, which is covered in the next chapter. Rea-sonably short proofs of Theorem (10.4.2) are given in Seymour (1980a),Wagner (1985a), and Gerards (1995).

The material of Section 10.5 is based on Truemper (1988). That ref-erence contains details about induced graph decompositions. The famousTheorem (10.5.14) of Wagner was originally proved by quite different ar-guments (Wagner (1937a), (1970)). Short proofs are given in Halin (1964),(1967), (1981), Ore (1967), and Young (1971). When it was first proved,Theorem (10.5.14) established the equivalence of Hadwiger’s conjectureabout graph coloring and the four-color conjecture for planar graphs (nowa theorem). The details are as follows. A graph G is colorable with n colorsif the vertices can be colored with n colors so that any two vertices with


same color are not connected by an edge. The chromatic number of a graphis the least number of colors that permit a coloring of the graph. Hadwiger’sconjecture (Hadwiger (1943)) says that any graph with chromatic numbern has a Kn minor.

The conjecture is readily seen to be correct for n = 1, 2, and 3. Forn = 4, it was proved in Dirac (1952). Indeed, by Theorem (4.2.6), a 2-connected graph is a series-parallel graph if and only if it has no K4 minor.It is an elementary exercise to show that any graph whose 2-connectedcomponents are series-parallel graphs is colorable with three colors. Thus,a graph with chromatic number equal to 4 must have a K4 minor.

The case n = 5 is more complicated. The first step toward a proofwas accomplished in Wagner (1937a) with Theorem (10.5.14). That resultallows one to prove that the graphs without K5 minors are colorable withfour colors if this is so for all planar graphs, as follows. Suppose the graph,say G, is 3-connected. By Theorem (10.5.14), G is planar, or is isomorphicto K3,3 or G8, or has a 3-separation with at least four edges on each side.The planar case is handled by the assumption. The graphs K3,3 and G8

are colorable with three colors. In the 3-separation case, G is a Δ-sum,say G1 ⊕Δ G2. By Lemma (8.5.6) or by direct checking, G1 and G2 areminors of G. By induction, they have a coloring with at most four colors.In G1, the colors of the nodes of the connecting triangle must be distinct.The same holds for G2. By a suitable renaming of the colors of G2, thegraph G can thus be colored with four colors. The case of a 2-separable or1-separable graph G is even simpler.

The second step in the proof of the case n = 5 involves showing that allplanar graphs can be colored with four colors. The conjecture of that resultwas open for about one hundred years. It finally was proved in Appel andHaken (1977), and Appel, Haken, and Koch (1977). An exposition of theproof is included in Saaty and Kainen (1977). Thus, Hadwiger’s conjectureis correct for n ≤ 5. For n ≥ 6, the conjecture is still open.

Theorem (10.5.15) is part of Wagner (1937a). Theorem (10.5.20) isproved in Wagner (1960). Related decomposition results are described inHalin (1981). Theorem (10.5.21) is taken from Truemper (1988). Thesetheorems are useful for the solution of combinatorial problems via decom-position. An interesting instance is the max cut problem. Given is a graphG with nonnegative edge weights. One must find a disjoint union C ofcocycles such that the sum of the weights of the edges of C is maximum.This problem is solved in Barahona (1983) for graphs without K5 minorsusing Theorem (10.5.15). The same approach applies to graphs withoutG12 minors when Theorem (10.5.24) is substituted for Theorem (10.5.15).

There are numerous other ways in which graphs may be decomposed.The number of results is so large that we cannot even sketch the manyideas, theorems, and applications. Thus, we cite some representative refer-ences, but omit details. For example, the ear decomposition reduces a given

10.7. Applications, Extensions, and References 243

2-connected graph to a cycle by removing one path at a time while main-taining 2-connectedness. This decomposition is simple, but with its aidsignificant results have been proved (e.g., in Lovasz and Plummer (1975),Kelmans (1987), Lovasz (1983), and Frank (1993a)).

For some classes of graphs or for particular graph problems, specialdecompositions have been developed. Examples are decompositions forperfect graphs (e.g., in Burlet and Uhry (1982), Bixby (1984b), Burletand Fonlupt (1984), Whitesides (1984), Chvatal (1985), (1987), Chvataland Hoang (1985), Hoang (1985), Cornuejols and Cunningham (1985),Hsu (1986), (1987a), (1987b), (1988), and Chvatal, Lenhart, and Sbihi(1990)) and circle graphs (e.g., in Gabor, Supowit, and Hsu (1989)), andfor optimization problems (e.g., in Uhry (1979), Boulala and Uhry (1979),Cunningham (1982c), Edmonds, Lovasz, and Pulleyblank (1982), Ratliffand Rosenthal (1983), Cornuejols, Naddef, and Pulleyblank (1983), (1985),Bern, Lawler, and Wong (1987), Lovasz (1987), Mahjoub (1988), Coullardand Pulleyblank (1989), Fonlupt and Naddef (1992), and Barahona andMahjoub (1994a)–(1994d)). Diverse graph decomposition ideas are con-tained in Lovasz and Plummer (1986). An axiomatic treatment of graphdecompositions, indeed of decompositions of general combinatorial struc-tures, is given in Cunningham and Edmonds (1980).

The 3-connected components of a graph can be found in linear time byan algorithm of Hopcroft and Tarjan (1973). A decomposition of minimally3-connected graphs is given in Coullard, Gardner, and Wagner (1993).

The first polynomial test for graphicness of binary matroids was givenby Tutte (1960). The graphicness test of Section 10.6 is a simplified versionof a scheme of Truemper (1990). Other relevant references have alreadybeen cited in Section 3.6. The problem of deciding presence of the networkflow property was treated completely and for the first time in Iri (1968).That material appears also in Bixby and Cunningham (1980). An efficientmethod for deciding graphicness of a matroid not known a priori to bebinary was first proposed in Seymour (1981c). The matroid is assumed tobe specified by a black box for deciding the independence of subsets of thegroundset. Related material is included in Bixby (1982a), and Truemper(1982a). The easier case where all circuits are explicitly given is coveredin Inukai and Weinberg (1979). The recognition problem of generalizednetworks, which constitute an extension of directed graphs, is treated inChandru, Coullard, and Wagner (1985).

Last but by no means least, we should mention a truly astoundingproof by Robertson and Seymour of the following daring conjecture due toWagner: If a given graph property is maintained under minor-taking, thenthe number of nonisomorphic minor-minimal graphs that do not have theproperty is finite. The proof of that result involves a powerful graph decom-position concept called tree decomposition that we cannot treat here. Thelength of the proof is extraordinary. Together with numerous applications,


it is being published in a long series of papers (Robertson and Seymour(1983), (1984), (1985), (1986a)–(1986c), (1988), (1990a)–(1990c), (1991),(1994), (1995a)–(1995c), (1996), (1998a)–(1998h)). Related to this workon graphs are the matroid results of Kahn and Kung (1982), and Kung(1986a), (1986b), (1987), (1988), (1990a), (1990b). A survey of applica-tions is made in Fellows (1989). There is some evidence supporting thefollowing conjecture for the matroids that are representable over a givenfinite field: For any property that is maintained under minor-taking, thenumber of nonisomorphic minor-minimal matroids not having the propertyis finite. Truemper (1986) contains an encouraging related result about in-duced decompositions.

Chapter ��

Regular Matroids

�� Overview

Building upon the material on graphic matroids of Chapter �� we analyzein this chapter the class of regular matroids� Already� we know that classquite well� or so at least it seems� By Theorem �� a binary matroidis regular if and only if it has no F� or F �

� minors� By Corollary ��every graphic matroid is regular� There are also regular matroids that arenot graphic and not cographic� If such a matroid is ��connected� then byTheorem �� it has an R�� or R�� minor� These results are interesting indeed� the �rst and third one are profound� But they do not tell us howto construct regular matroids� or how to test binary matroids e�cientlyfor regularity� That gap in our knowledge is �lled by the extraordinarydecomposition theorem of regular matroids due to Seymour� The entirechapter is devoted to that theorem and to some of its rami�cations andapplications�

We proceed as follows� In Section �� we prove that �� and ��sumcompositions produce regular matroids when the components are regular�As a lemma for the ��sum case� we also establish that the �Y exchangede�ned in Section �� maintains regularity�

Section �� contains the main result of this chapter� the regular ma�troid decomposition theorem� It essentially says that every regular matroidcan be produced by �� and ��sums where the building blocks are graphicmatroids� cographic matroids� and copies of R�� Furthermore� only regularmatroids can be generated by this process�

��

�� Chapter �� Regular Matroids

In Section �� we develop e�cient tests for regularity of binary ma�troids and for total unimodularity of real matrices� We obtain the regularitytest by combining the method for �nding �� and ��sums of Section ��with the graphicness test of Section �� That scheme is then rather easilyextended to an e�cient test for total unimodularity�

The uses and implications of the regular matroid decomposition the�orem are far�ranging� In Section �� we describe representative applica�tions� In the �nal section� �� we list extensions and references�

The chapter requires knowledge of Chapters ��

�� and ��Sum Compositions

Preserve Regularity

In this section� we show that any �� or ��sum with regular componentsis regular� This result is the comparatively easy part of the regular matroiddecomposition theorem� We also prove that the �Y matroids of Section ��are regular�

The reader may wonder why we con�ne ourselves to �� and ��sums�and do not treat general k�sum compositions� It turns out that the �� sum cases have simple proofs and actually are the only k�sums needed forthe regular matroid decomposition theorem� On the other hand� for k�sumswith k � �� the situation becomes much more complicated� In Section ��we sketch what is known about that case�

Recall from Section �� that a real matrix is totally unimodular if all ofits determinants are �� Furthermore� a binary matrix is regular if it canbe signed to become a totally unimodular real matrix� By Lemma ��and Corollary �� the signing is unique up to scaling by f��g factors�Furthermore� the signing can be accomplished by signing one arbitrarilyselected row or column at a time�

We need some elementary facts about the real f��g matrices thatare not totally unimodular� but each of whose proper submatrices has thatproperty� We call such a matrix a minimal violation matrix of total uni�

modularity� for short minimal violation matrix� First� a minimal violationmatrix is obviously square� and its real determinant is di�erent from ��This fact implies that a � minimal violation matrix contains four ��s�Next� let a minimal violation matrix have order k � �� Suppose we per�form a real pivot in that matrix� then delete the pivot row and column� Asimple cofactor argument proves that the resulting matrix is also a minimalviolation matrix�

We are prepared for the proofs of this section� The �rst lemma dealswith the case of �� and �sums�

�� and ��Sum Compositions Preserve Regularity ��

�� Lemma� Any �� or �sum of two regular matroids is also regu�lar�

Proof� Let M� and M� be the given regular components of a �� or �sumM � We rely on the matrices of �� and �� repeated belowfor convenient reference�

�� X1B =

Y1

A1

0X2

Y2

A2

0

Matrix B for ��sum M

��

Y1y

A2

B =

A1

Y2

x

X1

X2

0

0

1all1s

A2B2 =

y Y2x

X210

Y1y

B1 = A1

x

X1

0 1 1

1

Matrices B� B�� and B� for �sum Mwith components M� and M�

In the ��sum case� M� and M� are represented by A� and A� of ��Since M� and M� are regular� we can convert A� and A� by signing tototally unimodular real matrices� say �A� and �A�� Derive a matrix �B fromB of �� by replacing A� and A� by �A� and �A�� Evidently� �B is atotally unimodular signed version of B� Thus� M is regular�

The �sum case is slightly more complicated� De�ne D to be thesubmatrix of B of �� indexed by X� and Y�� We sign B� and B� of�� so that totally unimodular matrices �B� and �B� result� The signingconverts the submatrices A� and A� of B� and B� to� say� �A� and �A�� Nextwe compute a signed version �D of D by the formula


�� D � �column y of �B� � �row x of �B�

Then �A�� A�� and �D de�ne a signed version �B of the matrix B forM �By the construction� the submatrices � �A�� D� and � �D j �A�� of �B are totallyunimodular�

We complete the proof by showing that the entire matrix �B is to�tally unimodular� Suppose it is not� Then �B contains a minimal violationmatrix V that intersects �A�� A�� and �D� Thus� V also intersects the �submatrix of �B indexed by X� and Y�� and accordingly must have orderof at least �� We perform a real pivot in �B on a �� that is in both �A�

and V � The resulting real matrix �B� contains a smaller minimal viola�

tion matrix� The pivot changes �A� and �D� say to �A��

and �D�� but leaves�A� unchanged� Since � �A�� D� is totally unimodular� and since� by Lemma

�� pivots do not destroy total unimodularity� the matrix � �A��

� �D�� istotally unimodular� If we perform the corresponding GF��pivot in B� wethus get an unsigned version of �B�� Furthermore� each column of �D� is ascaled version of a column of �D� Thus� � �D� j �A�� is totally unimodular� Asuitable repetition of the preceding reduction process eventually producesthe contradictory case of a minimal violation matrix contained in a totallyunimodular matrix�

We turn to the ��sum case� By �� and �� we may assumeM � M�� and M� to be represented by the matrices B� B�� and B�� respec�tively� as follows�

��

11

Y1

A2DD1

D12 D2

B =

A1

Y2

Y1Y2

X1

X2

X1

X2

0

01 1

1 1

1

Y1

DD1

B1 =A1

Y2

Y1

X1

X2

X10

01

1

1

A2D

D2

Y1 Y2

Y2

B2 =

X1

X2X2

011

0

Matrices B� B�� and B� for ��sum Mwith components M� and M�

�� and ��Sum Compositions Preserve Regularity ��

We may convert the component M� by a �Y exchange of Section ��to a matroid M�� that is represented by the following matrix B�� takenfrom ��

��X2

Y2Z2

A2dD

D2

11B2 =

Matrix B�� for M��

We �rst introduce a lemma that links regularity of M� to regularity ofM��

�� Lemma� M� of �� is regular if and only ifM�� of ��has that property�

Proof� For the �only if� part� let �B� be a totally unimodular version ofB� so that the two columns of �B� indexed by Y � do not contain any ��This is possible� since we may begin the signing process with these two

columns� Denote by �D and �D� the submatrices of �B� that correspond tothe submatrices D and D� of B� Declare �d to be the real di�erence of thetwo f�� g columns of � �D� �D�� Thus� �d is a f��g vector that� togetherwith the submatrices of �B�� de�nes a signed version �B�� of B

�� We aredone once we prove �B�� to be totally unimodular� If this is not the case�then �B�� contains a minimal violation matrix V � By the construction� the

latter matrix must intersect �d� The two columns of � �D� �D�� and the vector�d are IR�dependent� so V intersects � �D� �D�� in at most one column� But in�B�� we can produce a scaled version of V as submatrix by a real pivot onone of the explicitly shown �s in the �rst row� as may be readily checked�The latter fact contradicts the total unimodularity of �B��

We prove the �if� part using duality� First we observe thatM�

� may bederived from M�

��by replacing a triad by a triangle� We just established

that such a change maintains regularity� Thus� M�

�is regular if M�

��is

regular�

Lemma �� implies the following result�

�� Corollary� �Y exchanges maintain regularity�

We are ready for the ��sum case�

�� Lemma� Any ��sum of two regular matroids is regular�

Proof� We start with B� B�� and B� of �� for the assumed ��sumMwith regular componentsM� andM�� By Lemma �� the matrix B��

of �� is regular� Let d be the column vector displayed in B�� Then


up to indices� �d j D j A�� is the regular matrix B�� plus possibly parallelcolumns� Thus� �d j D j A�� may be signed to become a totally unimodularmatrix � �d j �D j �A�� By the regularity of B� and duality� �A��D� is alsoregular� and thus can be signed� starting with the submatrixD� to a totallyunimodular matrix � �A�� D�� Clearly� the matrices �A�� A�� and �D de�ne asigned version �B of B�

We are done once we show �B to be totally unimodular� We accomplishthis by essentially the same arguments as for the �sum case� Thus� if �Bis not totally unimodular� then it has a minimal violation matrix V thatintersects �A�� A�� and �D� The order of V must be at least �� In �B� wepivot on a �� that is in both �A� and V � The pivot changes � �A�� D� to

another totally unimodular matrix� say � �A��

� �D�� It is easy to verify thatthe columns of �D� are nothing but scaled versions of the columns of � �d j �D��Thus� the matrix � �D� j �A�� is up to scaling and parallel columns a submatrixof the previously de�ned � �d j �D j �A�� and hence is totally unimodular� Bya suitable repetition of the above reduction process� we eventually get thecontradiction that a minimal violation matrix is contained in a totallyunimodular matrix�

For future reference� we combine Lemmas �� and �� to thefollowing theorem�

�� Theorem� Any �� or ��sum of two regular matroids isregular�

We return to a class of binary matroids introduced in Section ��the class of �Y matroids� Each of these matroids is constructed from thematroid represented by B � � � � by repeated SP �� series�parallel and �Yexchanges� We do not repeat the details of these operations here� Thus�the reader may want to review Section �� before proceeding� There thefollowing result is claimed but not proved� We supply the proof next�

�� Theorem� The �Y matroids are regular�

Proof� We use induction on the number of SP extensions and �Y ex�changes used in the construction of a binary �Y matroid M � The binarymatroid represented by B � � � � is regular� By Lemma �� any SP ex�tension of M can be viewed as a �sum composition where M is one of thecomponents� and where the second component is a regular matroid withthree elements� By Theorem �� a �sum is regular if its compo�nents are regular� Thus� SP extensions maintain regularity� By Corollary�� the same conclusion holds for �Y exchanges�

Section �� contains two variations of the ��sum� called ��sum andY�sum� The latter sum is the dual of the ��sum� A ��sum decompositionis obtained from a ��sum M� ��M� by replacing M� by M�� For the ��


sum composition� we invert this process� Theorem �� and Corollary�� thus imply the following corollary for ��sums and Y�sums�

�� Corollary� Any ��sum or Y�sum M of two regular matroidsis regular�

In Section �� it is mentioned that the class of �Y matroids includes��connected matroids that are nongraphic and noncographic� By Lemma�� every connected minor of a �Y matroid is also a �Y matroid� By�� the regular matroid R�� has no triangles or triads� and thus is nota �Y matroid� Indeed� R�� is ��connected� We conclude that no binarymatroid with an R�� minor is a �Y matroid� The situation is di�erent forR�� In �� the latter matroid is represented by a graph plus a nodesubset T � With the aid of that representation� one easily proves R�� to bea �Y matroid� a fact already claimed in Section �� Speci�cally� a triangleof that graph is a triangle of R�� and a ��star that is not a T node is atriad� Because of these facts� a �Y sequence reducing R�� to the matroidrepresented by B � � � � can be computed by graph operations� First� onereduces R�� while retaining the T set until a graphic or cographic matroidis attained� Second� one switches representations by selecting a suitablegraph and �nds the remaining reductions� We leave the details to thereader�

Theorems �� and �� enable us to construct many regularmatroids from the matroids we already know to be regular� which are thegraphic matroids� their duals� and R�� and R�� In the next section� we seethat we can construct all regular matroids that way� In fact� we require onlya subset of the above initial matroids and a subset of the above constructionsteps to produce all regular matroids�

�� Regular Matroid Decomposition

Theorem

In this section� we prove Seymour�s profound decomposition theorem forregular matroids� It essentially says that every regular matroid can be ob�tained by �� and ��sums where the building blocks are graphic matroids�cographic matroids� and matroids isomorphic to R�� The theorem thusprovides an elegant and useful construction for the entire class of regularmatroids�

For the proof of the decomposition theorem� we rely on two ingredi�ents� The �rst one we already know� It is Theorem �� That resultsays that any ��connected regular nongraphic and noncographic matroid


has an R�� or R�� minor� The second ingredient consists of decompo�sition results of Sections �� and �� speci�cally Corollary �� andthe splitter Theorem �� We could cast the results of this section interms of the recursive construction of decomposition theorems introducedfor graphs in Section �� We will not do so� But we should mention that�nding decomposition theorems such as the one of this section is facilitatedby that recursive construction scheme� We discuss this aspect further inChapter ��

First� we analyze the in�uence of R�� and R�� minors� So assume R��

to be a minor of a regular matroid� For the analysis� we need the splitterTheorem �� which we repeat next for convenient reference�

�� Theorem �Splitter Theorem� Let M be a class of binarymatroids that is closed under isomorphism and under the taking of minors�Let N be a ��connected matroid of M on at least six elements�

�a If N is not a wheel� then N is splitter of M if and only if M does notcontain a ��connected ��element extension of N �

�b If N is a wheel� then N is a splitter of M if and only if M does notcontain a ��connected ��element extension of N and does not containthe next larger wheel�

The matroid N is called a splitter of the classM of matroids� Here isthe result for R��

�� Theorem� R�� is a splitter of the classM of regular matroids�

Proof� By Theorem �� we only need to show that every ��connected��element extension of R�� is nonregular� Since R�� is isomorphic to itsdual� it su�ces that we consider ��element additions� The case checking isconveniently accomplished when we represent R�� by a graph plus a T setas in �� i�e�� by

��

Graph plus T set representing R��

Up to isomorphism� just three distinct ��connected ��element additions arepossible�

In the �rst case� we join two nonadjacent nodes of the graph of ��


by an edge shown in bold below�

��

e

��element addition of R�� case �

We contract edge e� The two T endpoints of e become a non�T node�Evidently� the resulting graph contains a subdivision of

��

Graph plus T set representing F�

which by �� represents the Fano matroid F�� Thus� that extension ofR�� is nonregular�

The remaining two cases involve ��element additions that are non�graphic even when one deletes from R�� the element represented by the setT � For both cases� we depict the additional element by a subset T � of thenode set with jT �j � �� The two possible ways are easily reduced to aninstance of �� and thus are also nonregular�

Now we assume R�� to be a minor of a regular matroid� This timewe rely on Corollary �� We need a bit of preparation before wecan restate that result� Let N be a binary matroid with a k�separation�X� � Y��X� � Y� given by

��

Y1 Y2

X1BN =

A1

DX2 A2

0

Matrix BN for N with k�separation

Consider the following three ways of extending N as depicted by the rep�resentation matrices below�


�a The ��element expansions represented by

��x

Y2Y1

X1 A1

DX2 A2

e f

0

Matrix of ��element expansion of N

In row x� e is not spanned by the rows of D� and f is nonzero�

�b The ��element additions given by

��

Y1

X2

Y2

X1 A1

D A2

0

y

g

h

Matrix of ��element addition of N

In column y� g is nonzero� and h is not spanned by the columns of D�

�c The �element extensions with representation matrix

��

Y1

x

Y2

X1 A1

DX2 A2

e f

0

y

g

h

α

Matrix of �element extension of N

Either �c�� or �c� below holds for e� f � g� h� and ��

�c�� e is not spanned by the rows of D f � � g � � h �� eis not parallel to a row of A�� If column z Y� of A� is nonzero�then e is not a unit vector with � in column z�

�c� g �� h is spanned by the columns of D e is spanned by the rowsof D f � � implies e �� e j �� is not spanned by the rows of�D j h�� If D� the matrix obtained from D by deletion of a columnz Y�� has the same GF��rank as D� then �g�h� is not parallelto column z of �A��D�� If the rows of D do not span a row z X�

of A�� then �g�h� is not a unit vector with � in row z�

We restate Corollary ��


�� Theorem� Let M be a class of binary matroids that is closedunder isomorphism and under the taking of minors� Suppose that N givenby BN of �� is in M� but that the �� and �element extensions of Ngiven by �� and the accompanying conditions arenot in M� Assume that a matroid M M has an N minor� Then anyk�separation of any such minor that corresponds to �X� � Y��X� � Y� ofN under one of the isomorphisms induces a k�separation of M �

We intend to use Theorem �� with R�� as N and with the classof regular matroids asM� For the detailed arguments� we need the binaryrepresentation matrix B�� of �� for R�� We include that matrixbelow� According to that matrix� the pair �X� � Y��X� � Y� constitutes a��separation of R��

��B12 =

X1

X2

Y2Y1

0

0

11

0

0101

110

1

1010

1

0101

1 11 11 00 1

1 0 01 0 0

Matrix B�� for R��

Theorem �� leads to the following decomposition result for theregular matroids with R�� minor�

�� Theorem� Let M be a regular matroid with an R�� minor�Then any ��separation of that minor corresponding to the ��separation�X� � Y��X� � Y� of R�� see �� under one of the isomorphismsinduces a ��separation of M �

Proof� We verify the su�cient conditions of Theorem �� As apreparatory step� we calculate all ��connected regular ��element additionsof R�� By the symmetry of B�� of �� and thus by duality� thisresult e�ectively gives us all ��connected ��element expansions as well� Theaddition cases are collected as columns in the following matrix C�

��

X2 0

0 0 00

0

11

0

1010

0

0101

0

1111

0

C =

X1

Y1

Matrix C of ��connected regular additions to R��


Veri�cation of the claim about C involves simple but somewhat tedious casechecking� We omit the details� but should mention that the representationof R�� by a graph plus T set as in �� greatly simpli�es the task� Theadded element can be represented by a subset T � of the node set�

We verify the conditions of Theorem �� The cases depicted in�� rule out �� and �� They also narrow down the caseanalysis for �c�� and �c� of �� as follows�

According to �c�� e is not spanned by the rows of D� f � �� g � ��h �� and e is not parallel to a row of A�� If column z Y� of A�

is nonzero� then e is not a unit vector with � in column z� We apply theseconditions to B�� By �� and symmetry of B�� we see that e mustbe the vector �� Furthermore� h must be a unit vector� or must beparallel to a column of A� or D� or must be the subvector of any column ofC of �� indexed byX�� All such choices lead to nonregular matroids�as desired�

Suppose �c� applies� We ignore the conditions on e� f � and �� Theremaining conditions are as follows� The vector g is nonzero� and h isspanned by the columns ofD� IfD� the matrix obtained fromD by deletionof a column z Y�� has the same GF��rank asD� then �g�h� is not parallelto column z of �A��D�� If the rows of D do not span a row z X� of A

��then �g�h� is not a unit vector with � in row z� We apply these conditionsto B�� Thus� we determine that the matrix of �� minus row x� whichis B�� of �� plus the column vector �g�h�� represents a ��connected��element addition of R�� By �� any such addition with g �� isnonregular�

Thus� all conditions of Theorem �� are satis�ed� The conclusionof Theorem �� then proves the result�

At long last� we have completed all preparations for the regular ma�troid decomposition theorem�

�� Theorem �Regular Matroid Decomposition Theorem� Everyregular matroidM can be decomposed into graphic and cographic matroidsand matroids isomorphic to R�� by repeated �� and ��sum decomposi�tions�

Speci�cally� if M is ��connected and not graphic and not cographic�thenM is isomorphic to R�� or has an R�� minor� In the latter case� any ��separation of that minor corresponding to the ��separation �X��Y��X��Y�of R�� see �� under one of the isomorphisms� induces a ��separationof M �

Conversely� every binary matroid produced from graphic matroids�cographic matroids� and matroids isomorphic to R�� by repeated �� and ��sum compositions is regular�

Proof� Let a regular matroid M be given� Assume M to be nongraphicand noncographic� If M is ��separable� it is a ��sum� If M is �separable�


then it is by Lemma �� a �sum� Hence� assumeM to be ��connected�By Theorem �� M has an R�� or R�� minor� In the �rst case� Mmust by Theorem �� be isomorphic to R�� In the second case� Mhas by Theorem �� an induced ��separation� as claimed� By Lemma�� M is a ��sum� Finally� Theorem �� establishes the conversepart�

In Theorem �� one may want to rely on ��sums or Y�sumsinstead of ��sums� The next corollary supports that substitution�

�� Corollary� The claims of Theorem �� remain validwhen instead of ��sums one speci�es ��sums or Y�sums�

Proof� By Section �� any ��sum decomposition may be converted to a��sum or Y�sum decomposition by one �Y exchange involving one of thecomponents� By Lemma �� such an exchange maintains regularity�Indeed� if M is ��connected� then by Lemma �� the components ofany ��sum or Y�sum decomposition are isomorphic to minors of M � Atany rate� the above arguments prove that we may substitute ��sums orY�sums for ��sums in Theorem ��

Recall that the graph decomposition theorems �� and ��imply a construction of certain �connected graphs via � and ��sums�Each time� one of the components is a member of a class of building blocks�and the second component is a graph obtained by prior construction steps�We establish the analogous result for the connected regular matroids usingTheorem �� We treat ��sums as well as Y�sums and ��sums� Theterminology is adapted from that of Section ��

�� Theorem� Any connected regular matroid is graphic� co�graphic� or isomorphic to R�� or may be constructed recursively by �sums and ��sums �or Y�sums� or ��sums� using as building blocks graphicmatroids� cographic matroids� or matroids isomorphic to R��

The proof of Theorem �� is similar to that of Theorem ��We con�ne ourselves to compositions involving �sums and ��sums� TheY�sum case is handled by duality� and the ��sum case is proved by a simpleadaptation of the proof for ��sums� We begin with two lemmas�

�� Lemma� Let M be a ��connected� regular� nongraphic� andnoncographic matroid that is not isomorphic to R�� Assume M to have atriangle C� Then M has a ��separation �E�� E� where jE�j� jE�j � � andwhere one of E�� E� contains C�

Proof� By Theorem �� M has a ��separation �E�� E� induced bythe ��separation of an R�� minor� The latter ��separation corresponds tothe one of �� for R�� and thus has six elements on each side� Hence�jE�j� jE�j � �� If C is contained in E� or E�� we are done� Otherwise� we


shift one element of C from one side of the ��separation �E�� E� to theother one� to get a ��separation �E�

�� E�

� where C is contained in E�

�or

E�

�� It is easily checked that the ��separation of R�� depicted in ��

becomes an exact ��separation when any one element is shifted from oneside to the other� Thus� the element of C that we have shifted from E� orE� to the other set cannot be an element of the R�� minor inducing the��separation �E�� E�� We conclude that jE�

�j� jE�

�j � �� as desired�

�� Lemma� Let M be a ��connected� regular� nongraphic� andnoncographic matroid that is not isomorphic to R�� Assume M to haveeither a designated triangle C or a designated element y� Then M is a��sumM��M� whereM� contains C or y� whichever applies� and whereM� is graphic or cographic�

Proof� We prove the case for the triangle C and leave the easier situationwith the element y to the reader� We use induction� The smallest caseinvolves anM isomorphic toR�� By Lemma �� the matroidR�� is a��sum where one component contains C� and where the second componenthas nine elements and thus is graphic or cographic� Thus� we are done�For larger M � we again apply Lemma �� Thus� M is a ��sumM��M� where C is part of the componentM�� and where both M� andM� have at least nine elements� If the second component M� is graphic orcographic� we are done� Otherwise� we may assume M� to be ��connected�We de�ne C � to be the triangle of M� involved in the ��sum� Becauseof the presence of the triangle C �� or by the splitter result for R�� M�

cannot be isomorphic to the ��connected R�� By induction� M� has a ��sum decompositionM��M�� whereM�� contains C �� and whereM�� isgraphic or cographic� Via a representation matrix for M displaying the ��separations involved in the ��sumsM��M� andM��M�� we readilyverify that M is a ��sum with M� ��M�� and M�� as components� That��sum has the desired properties�

Proof of Theorem �� Let M be any connected� regular� non�graphic� and noncographic matroid that is not isomorphic to R�� If G is��connected� the result follows from Lemma �� Otherwise� G is a�sum� Choose the �sum decomposition� say M� �� M�� so that M� hasa minimal number of elements� Evidently� any �separation of M� contra�dicts the minimality assumption� so M� is ��connected� If M� is graphic�cographic� or isomorphic to R�� we are done� Otherwise� let y be the el�ement of M� that together with an element of M� de�nes the �sum� ByLemma �� M� is a ��sum M�� M�� where M�� contains theelement y� and where M�� is graphic or cographic� Via a representationmatrix for M � we con�rm that M is a ��sum where one component isM� �� M�� and where the second component is the graphic or cographicM�� as demanded in the theorem�

�� Testing Matroid Regularity and Matrix Total Unimodularity ��

An easy generalization of Theorems �� and �� is possibleby the following splitter result�

�� Lemma� F� �resp� F �

�� is a splitter of the binary matroids

without F �

��resp� F�� minors�

Proof� According to the census of small ��connected binary matroids inSection �� there are just two ��connected nonregular matroids on eight ele�ments� with representation matrices given by �� and �� Clearly�both matroids have F� and F �

� minors� Indeed� the matroids are selfdual�Thus� every ��connected ��element extension of F� �resp� F �

� has bothF� and F �

�minors� The result then follows from the splitter Theorem

��

We leave it to the reader to rewrite Theorems �� and ��so that they become results for the matroids without F� minors� or for thematroids without F �

� minors�One may concatenate matroid decomposition theorems of this section

and graph decomposition theorems of Section �� Later in this chapter�in Section �� we meet one such case� At any rate� such concatenation iseasy� and the reader may want to try his�her hand at producing potentiallyuseful theorems�

In the next section� we use Theorem �� to assemble e�cientalgorithms to decide regularity of binary matroids and total unimodularityof real matrices�

�� Testing Matroid Regularity and

Matrix Total Unimodularity

Prior to the introduction of the regular matroid decomposition Theorem�� no e�cient algorithm was known for testing a binary matroidfor regularity� or for deciding total unimodularity of real matrices� Weknow from Chapter � that these two tests are intimately linked� In fact�the results of Section �� imply that an e�cient test for one of the twoproblems can be easily converted to one for the other problem�

In this section� we construct the desired tests� relying� of course� on theregular matroid decomposition Theorem �� In addition� we invokethe algorithm of Section �� for �nding �� and ��sums� as well as thegraphicness test of Section ��

We begin with the regularity test for binary matroids� Let M be thebinary matroid for which regularity is to be decided� We �rst apply thepolynomial algorithm of Theorem �� to determine whether or not Mis a �� or ��sum� In the a�rmative case� we decompose M into two


components� Then we apply the algorithm to each component� etc�� untilwe eventually have a collection of binary matroids� say M�� M�� Mn�none of which is a �� or ��sum� It is not di�cult to prove that n� thenumber of such matroids� is bounded by a function that is linear in thenumber of elements of M �

The regular matroid decomposition Theorem �� says that M isregular if and only if each one of the matroidsM�� M�� Mn is graphic�or cographic� or isomorphic to R�� We settle graphicness� and if necessary�cographicness� of each one of the matroids with the polynomial algorithm ofTheorem �� Deciding whether or not a matroid is isomorphic to R��

is trivial� If one of the matroids is found to be nongraphic� noncographic�and not isomorphic to R�� then M is not regular� Otherwise�M has beenproved to be regular�

The preceding algorithm is clearly polynomial� It can be made very ef��cient by an appropriate implementation� The algorithm is readily adaptedto test total unimodularity of real matrices as follows� Given is a real matrixA� If A is not a f��g matrix� we declare A to be not totally unimodular�So assume A to be a f��g matrix� Let B be the support matrix of A�We view B to be binary� With the algorithm just described� we test thematroid M represented by B for regularity� If M is not regular� we knowA to be not totally unimodular� So assumeM � and hence B� to be regular�With the signing algorithm of Corollary �� we deduce from B a totallyunimodular matrix A�� By Lemma �� A is totally unimodular if andonly if A� can be obtained from A by scaling of some rows and columns by�� Implicit in the proof of the lemma is an algorithm that �nds the ap�propriate scaling factors� or determines that A� cannot be scaled to becomeA� Accordingly� we declare A to be totally unimodular or not�

We just have encountered one very important application of Theorem�� We introduce several others in the next section�

�� Applications of Regular Matroid

Decomposition Theorem

The uses of the regular matroid decomposition Theorem �� rangefrom the obvious to the unexpected� In this section� we cover representativeinstances�

Construction of Totally Unimodular Matrices

We begin with the most obvious case� the construction of the real totallyunimodular matrices� These matrices represent precisely the regular ma�troids� To �nd a construction for them� we only need to translate the

�� Applications of Regular Matroid Decomposition Theorem ��

��sum version of the construction Theorem �� into matrix language�First� we identify the matrix building blocks� We know that �� and�� provide the two possible GF��representation matrices for R��With the scheme of Corollary �� we sign these two matrices to obtainthe following totally unimodular representation matrices�

�� B10.1 = B10.2 =;

1 0 0 1 11 1 0 0 10 1 1 0 1

00 11 11 1 1 1 1

1 1 0 0 11 1 1 0 00 1 1 1 0

00 11 11 0 0 1 1

--

--

-

The two totally unimodular matrices representing R��

By Lemma �� B�� and B�� are� up to scaling of rows and columnswith f��g factors� the unique totally unimodular representation matricesfor R��

By the proof of Corollary �� all graphic matroids are minors ofthe graphic matroids that are represented by the binary node�edge matri�ces� The latter matroids are also represented by the real totally unimodularnode�arc incidence matrices� where each nonzero column has exactly one � and one �� The minor�taking translates to IR�pivots in node�arc inci�dence matrices and to the deletion of rows and columns� Call any matrixso produced a network matrix� The e�ect of the IR�pivots can be readilyestablished by graph operations� We leave it to the reader to work out thedetails�

At this point� we have the matrix building blocks that correspond tothe matroid building blocks of Theorem �� We now must under�stand the �sum and ��sum decomposition�composition� The task is madecomplicated by the fact that we must interpret these operations in GF��representation matrices without the use of GF��pivots� Only that waycan we deduce decomposition rules for totally unimodular matrices thatdo not involve IR�pivots� So let us assume B to be an arbitrary GF��representation matrix of a regular � or ��sumM with componentsM� andM�� The underlying � or ��separation is �X� � Y��X� � Y�� We assumeM � and hence B� to be connected�

We start with the �sum case� The �separation manifests itself in Bas depicted below� up to a switching of the roles of X� � Y� and X� � Y��The submatrix D of B has GF��rank ��

�� X1

Y1

A1

DX2

Y2

A2

0B =

Matrix B with �separation


Correspondingly� the component matroids M� and M� are� by �� and�� represented by B� and B� below once appropriate indices are as�signed to the row vector a of B� and the column vector b of B�� Thesetwo vectors are nonzero vectors of D� Since GF��rank D � �� the twovectors are unique up to indices� Indeed� because of permutations of rowsand columns in D� we may assume D � b � a�

�� X1

Y1

A1

aB1 = ; B2 =

Y2

A2bX2

Matrices B� and B� of �sum

We sign B� B�� and B� to get totally unimodular matrices� To simplifythe notation� we now assume the matrices B� B�� B� to be already suchsigned totally unimodular versions� We have the following �sum matrixcomposition rule�

�� Matrix �Sum Rule� Given are B� and B� of �� Thenwe derive B of �� from B� and B� by letting D � b � a �in IR��

The ��sum situation is more complicated� but yields to the same ap�proach� At the outset� we assume all matrices to be binary� The underlying��separation manifests itself in one of two ways� up to a switching of therole of X� �Y� and X� �Y�� Below� we indicate by B and �B the two cases�where GF��rank D � and GF��rank R � GF��rank S � ��

�� X1B = ;

Y1

A1

DX2

Y2

A2

0 X1

Y1

A1

RX2

Y2

A2

S B =~

Matrices B and �B with ��separation

We claim that� correspondingly� the component matroids M� and M� arerepresented below by B� and B�� or by �B� and �B�� once the vectors a� b� c�d� e� f � g� h of these matrices are appropriately de�ned and missing indicesare added� Speci�cally� the vectors a and b of B� �resp� c and d of B�are two arbitrarily selected GF��independent row �resp� column vectorsof D� Let D be the � submatrix of D created by the intersectionof the row vectors a and b of D with the column vectors c and d of D�By Lemma �� D is nonsingular� Because of �� and rowand column permutations� we may assume D � �c j d� � �D�� a�b� �inGF�� We de�ne the vector e of �B� �resp� g of �B� to be a nonzero


vector of R� and f of �B� �resp� h of �B� to be a nonzero vector of S� SinceGF��rank R � GF��rank S � �� the vectors e� g� f � h are unique up toindices� Furthermore� because of row and column permutations in R andS� we may assume R � g � e and S � f � h�

�� b

0

a 11

A1X1

Y1

B1 = ; B2 =c d

11

Y2

A2

0

X2

f;

~B1 =

~B2 =

X2g g

01Y2

A2

hf

e 0 1

A1X1

Y1

Matrices B�� B� as well as �B�� B� of ��sum

The case of B� B�� and B� is our customary way of displaying ��sums�The second one� with �B� �B�� and �B�� we have not presented before� Weshow it to be correct by GF��pivots that transform it to the �rst case�Speci�cally� suppose the submatrix S of �B has a � in the lower left�handcorner� Correspondingly� the last element of the leftmost vector f in �B�

and the �rst element of h in �B� are �s� Perform GF��pivots on these �sin the respective matrices� Each such GF��pivot exchanges a row indexagainst a column index� After the pivots� one readily con�rms that the��separation �X� � Y��X� � Y� of the new �B corresponds to the �rst case�and that the new �B� and new �B� are the component matrices for that case�

Consider the case of B� B�� and B� of �� and �� We maysign B to obtain a totally unimodular matrix� In fact� by �� and�� the signing may be selected so that in the corresponding signingof B� and B� the explicitly shown �s are not negated� The same signingconvention may be followed for �B� �B�� and �B�� As a matter of convenience�let us now assume that B� B�� B�� B� �B�� and �B� are the signed totallyunimodular matrices�

The above discussion validates the following matrix ��sum construc�tion�

�� Matrix ��Sum Rule� Given are B� and B�� or �B� and �B��of �� In the �rst case� we calculate D � �c j d� � �D�� a�b� �in IR�to determine B of �� In the second case� we compute R � g � e andS � f � h �in IR� to �nd �B of ��

The next result summarizes the above discussion�

�� Lemma� The matrix � and ��sum composition rules ��and �� correspond precisely to the regular matroid � and ��sum com�positions�


The desired construction theorem for totally unimodular matrices cannow be stated and proved�

�� Theorem� Any connected totally unimodular matrix is up torow and column indices and scaling by f��g factors a network matrix� or isthe transpose of such a matrix� or is the matrix B�� or B�� of �� ormay be constructed recursively by matrix �sums and ��sums� The rules aregiven by �� and �� The building blocks are network matrices�their transposes� and the matrices B�� and B�� of ��

Proof� The conclusion follows directly from Theorem �� and Lem�ma ��

Construction of f�� g Totally Unimodular Matrices

Closely related to the construction of totally unimodular matrices is that off�� g totally unimodular matrices� So suppose B is a regular matrix thatrequires no signing to achieve total unimodularity� The matrix �sum ruleobviously can be adopted without modi�cation� The two ��sum cases withB and �B of �� are more troublesome� Even though B or �B requiresno signing� the matrices B�� B�� B�� or �B� of �� may not be totallyunimodular without signing of some entries� By Corollary �� it is easyto see that such signing can be con�ned to the explicitly shown �s of thematrices of �� Suppose such signing is needed for B� of �� Wemay assume that the � in the lower right corner of B� must become a ��From now on� let B� denote that real totally unimodular matrix� with thejust�de�ned �� We modify the last column of B� and add one additionalrow and column to get the following real matrix !B��

��

b0

0 0

a 101

11

0

A1X1

Y1

B1 =^

Matrix !B� derived from B�

We claim that the !B� is totally unimodular� For a proof� we perform anIR�pivot on the � of !B� in the lower right hand corner� We obtain B� assubmatrix� Indeed� we see by the pivot that !B� represents a ��sum� Onecomponent is the matroid of B�� The second component is isomorphic toM�W�� the graphic matroid of the wheel with four spokes� Similarly� if


needed� we modify B� to

�� 1

c0 d

10

01 1

Y2

A2

00

X2

B2 =^

Matrix !B� derived from B�

Consider the ��sum case involving �B� and �B�� Assume the explicitly shown� in one or both of these matrices requires signing� The modi�cations are

as follows� where !�B� �resp� !�B� corresponds to �B� �resp� �B��

��

0

1

g g

00

01 1

Y2

A2

h0

X2

~B2 =^~

B1 =^ f f 0

e0

00

01

11

A1X1

Y1

Matrix !�B� and !�B� derived from �B� and �B�

We leave it to the reader to verify by IR�pivots that the modi�cations are

appropriate� Indeed� !�B� �resp� !�B� of �� represents a ��sum onecomponent is represented by �B� �resp� �B� of �� and the secondcomponent is isomorphic to M�W��

The reader likely anticipates that the analogue of Theorem ��holds for f�� g totally unimodular matrices� where this time we permit for��sums the cases of �� besides the ones of �� Thatguess is correct� But the proof requires some care� For instance� we mustshow that the matrices of �� are always smaller than thematrix of �� produced by them� We must also establish that any oneof the matrices of �� is graphic or cographic if the relatedmatrix of �� is graphic or cographic� The latter task may seem verydi�cult� Indeed� in general it cannot be accomplished� as we �nd out inthe proof of the next theorem�

�� Theorem� Any connected f�� g totally unimodular matrixis up to row and column indices a f�� g network matrix� or the transposeof such a matrix� or the matrix B�� of �� or may be constructedrecursively by matrix �sums and ��sums� The �sum rule is given by�� The ��sums are speci�ed by the ��sum rule �� except that we

also allow !B�� !B�� !�B�� !�B� of �� as component matrices�The building blocks are f�� g network matrices� their transposes� and thematrix B�� of ��


Proof� We invoke Theorem �� and the above derivation of !B�� !B��!�B�� !�B�� The case of B�� is not possible since that matrix cannot be

scaled to become a f�� gmatrix� Thus� we are done� unless !B�� !B�� !�B�� or!�B� is needed� and is of the same size as B or is not graphic or cographic�Because of pivots� symmetry� and duality� we may select any one of theabove matrices� say !B�� to eliminate these concerns�

First� we prove that !B� is smaller than B� By the proof of Theorem�� the ��separation �X� �Y��X� �Y� of B inducing the ��sum thatin turn produces !B� plus B� or !B� satis�es jX� � Y�j� jX� � Y�j � �� But!B� has length jX� � Y�j � and thus is smaller than B�

The second part is more complicated� We want to show that B�

graphic or cographic implies !B� graphic or cographic� This goal turnsout to be attainable in all cases but one� That exception we handle by aswitch to a di�erent ��separation of B�

Suppose B� is graphic� As stated above� !B� represents a ��sum� Onecomponent is the matroid of B�� The second component is isomorphic toM�W�� We claim that ��sum to be graphic� Indeed� any graph G for B�

can be converted to one for !B� as follows� By �� the edges of G notindexed by X� � Y� form a triangle C� say with vertices u� v� w� Then onan appropriately selected edge of C� say with endpoints u and v� we placea midpoint and connect that new node with w by a new edge� One easilyveri�es that this construction does produce a graph for !B��

We examine the second possibility� where B� is cographic and notgraphic� Thus� B� is nonplanar� Let H be any graph for the transpose ofB�� By �� the edges of H corresponding to the unlabeled rows andcolumns of B� form a cocircuit C� of H of cardinality ��

Assume C� to be a ��star of H� Analogously to the earlier case� onecon�rms !B� to be cographic� as desired�

Assume C� to be not a ��star of H� Thus� removal of the three edgesof C� transforms H to two connected nonempty graphs H� and H�� Ac�cordingly� the minor HnC� is not �connected� That minor corresponds tothe transpose of the submatrix A� of B�� Thus� the transpose of A�� andhence A�� are not connected� Indeed� A� is a ��sum of two matrices A��

and A�� where A�� corresponds to H�� and A�� to H�� We derive from Ha graph �H� �resp� �H� by contracting the edges of H� �resp� H�� In �H�

and �H�� the set C� is a ��star� If both �H� and �H� are planar� then oneeasily veri�es H to be planar as well� a contradiction of the fact that B�

is cographic and not graphic� Thus� we may assume �H� to be nonplanar�and hence to have at least nine edges� We now rede�ne the ��separationof B by shifting the submatrix A�� of A� to A�� For the new ��separation�the new B� corresponds to �H�� and thus is still cographic and not graphic�But C� is now a ��star of �H�� so the earlier case applies�

So far� we have seen two applications that involve a rather obvious


translation of Theorem �� The next application makes a more so�phisticated use of that theorem�

Characterization of Cycle Polytope

Given is a connected binary matroidM with groundsetE� To each elemente E� a real weight we is assigned� We want to �nd a disjoint union Cof circuits of M so that

Pe�C we is maximized� This problem occurs in a

number of settings� We describe one� and reference others in Section ��Let G be a connected graph with edge set E and real weights we�

e E� We want to partition the vertex set V of G into sets V� and V�such that the sum of the weights of the set C� of edges connecting V� andV� is maximized� The set C� is readily seen to be a disjoint union of thecocircuits of the graphic matroidM�G of G� Conversely� any such disjointunion of cocircuits of M�G is the set C� for some partition of V � Thus�the graph problem� which usually is called the max cut problem� becomesthe earlier matroid problem with M �M�G� �

The matroid problem can be easy or di�cult� depending onM and onthe sign of the weights� In general� the problem is known to be NP�hard�We sketch a polyhedral approach for a special subclass� Let xC be thecharacteristic vector for a disjoint union C of circuits of M � Thus� xC isindexed by E� and the entry in position e E of xC is equal to � if e C�and equal to � otherwise� We view the xC vectors to be in IRE� De�neP �M to be the convex hull of the xC vectors� Usually� P �M is called thecycle polytope of M � By results of polyhedral combinatorics� we can solvethe given problem e�ciently if we can determine whether or not a vectorx IRE is in P �M� and if for x � P �M we can provide a hyperplanethat separates x from P �M� As a �rst step� we thus strive to �nd linearinequalities that are satis�ed by all points of P �M� We say that suchinequalities are valid for P �M� Notationally� for any subset E E� letx�E �

Pe�E

xe�Obviously� the inequalities

�� xe � � e E

are valid for P �M� We derive additional valid inequalities as follows�By Lemma �� each circuit of M intersects each cocircuit in an evennumber of elements� Let C be a disjoint union of circuits� C� be a cocircuit�and F be a subset of C� of odd cardinality� We claim that the followinginequality is valid for P �M�

�� x�F � x�C� n F � jF j � � F C� � cocircuit jF j odd�

Since P �M is the convex hull of the xC vectors� we may establish validityof �� by proving it for x � xC � C being an arbitrary disjoint union


of circuits of M � If F is not contained in C� then x�F � jF j � �� and�� clearly holds� So assume F C� We know that F C�� SincejF j is odd and jC�C�j is even� C has at least one element in C� nF � Thus�once more �� holds� We conclude that �� is valid�

De�ne Q�M to be the subset of IRE de�ned by the inequalities of�� and �� When is P �M � Q�M" We sketch the answer tothis question and its proof� First� one shows that P �M �� Q�M if M hasa minor isomorphic to F �

�� M�K�� or R�� Second� one establishes that

P �M � Q�M ifM is graphic or isomorphic to F��M�K�� orM�G��where G is given by �� Third� one proves that P �M � Q�M if Mis a �sum or Y�sum of two matroidsM� andM� for which P �M� � Q�M�and P �M� � Q�M�� Fourth and last� one concatenates the dual versionof Theorem �� and Theorem �� to the following result�

�� Theorem� Let M be a connected binary matroid having noF �

� or M�K�� minors� Then M is graphic or isomorphic to F�� M�G��M�K�� orR�� or may be constructed recursively by �sums andY�sums�The building blocks are graphic matroids and matroids isomorphic to F��M�G�� M�K�� and R��

The four ingredients clearly imply the following conclusion�

�� Theorem� Let M be a connected binary matroid� ThenP �M � Q�M if and only if M has no F �

� � M�K�� or R�� minors�

References for additional material about P �M andQ�M are includedin Section ��

We describe some additional applications of the regular matroid de�composition theorem without proofs� The �rst one concerns the numberof nonzeros in the rows of totally unimodular matrices having fewer rowsthan columns�

Number of Nonzeros in Totally Unimodular

Matrices

Let A be a totally unimodular matrix� say of size m � n with m � n� Fori � �� m� let pi be the number of nonzeros in row i of A� De�nep� � minpi� Trivially� p� is bounded from above by n� the number ofcolumns of A� That upper bound is attained by the m � n matrix Acontaining only �s� The example matrix has parallel columns� Thus� onemay conjecture that a tighter upper bound on p� may exist in the absenceof parallel columns� The next theorem con�rms this conjecture by provingm� the number of rows of A� to be an upper bound on p� when A has noparallel columns�


�� Theorem� Let A be a totally unimodular matrix of sizem�nwith m � n� For i � �� m� let pi be the number of nonzeros in row iof A� and de�ne p� � minpi� If A has no parallel columns� then p� �m�

Triples in Circuits

The next theorem addresses the following question� Given are three ele�ments e� f � g of a binary matroidM � When does some circuit ofM includethem all"

It is not di�cult to reduce the problem to the case where the matroidis su�ciently connected� For that situation the answer is as follows�

�� Theorem� Let e� f � g be distinct elements of a ��connectedbinary matroid M � Assume that M does not have a ��separation with atleast four elements on each side� Then there is no circuit ofM containing e�f � and g if and only if fe� f� gg is a cocircuit ofM or the following conditionholds� M is graphic� and in the corresponding graph� the edges e� f � andg are edges with a common endpoint�

Odd Cycles

Given is an undirected graph G� What is the structure of G if every twocycles of G� each of odd length� have a node in common" Here� too� itis not di�cult to reduce the problem to the situation where the graph issu�ciently connected� The following theorem provides the answer for thatcase�

�� Theorem� Let G be a �connected graph on at least sixvertices and without parallel edges� Assume that G does not have a �separation with at least four edges on each side� Then every two cycles ofG� each with an odd number of edges� have a common vertex if and onlyif G observes �i or �ii below�

�i Deletion of some vertex or deletion of the edges of a triangle from Gresults in a bipartite graph�

�ii G can be drawn in the projective plane so that every region is boundedby an even number of edges�

Another application concerns the construction of the connected� undi�rected� and signed graphs without so�called odd�K� minors� Relevant def�initions and details of the construction are included in Chapter ��

In the last section� we point out some extensions and identify relevantreferences�



The main reference for the entire chapter is Seymour ��b� which con�tains Seymour�s decomposition theorem for regular matroids�

Lemmas �� and �� constitute the easy part of the regularmatroid decomposition theorem� They are due to Brylawski �� Fork � �� k�sums with regular components are not necessarily regular� Anexample for k � � is given in Truemper ��b� Intuitively� one is temptedto argue that regularity of k�sums must be assured when the connectingmatroid is su�ciently large and structurally rich� Indeed� this notion can bemade precise by su�cient conditions that� for any k � �� assure regularityof k�sums with regular components� Because of space limitations� we omita detailed treatment�

All decomposition results of Section �� either are taken directly fromSeymour ��b� or are implied by that reference� We have described themusing �� and Y�sums� Seymour ��b relies on ��sums�

A very e�cient version of the regularity�total unimodularity test ofSection �� is presented in Truemper �� Indeed� the algorithm de�scribed there has the currently best worst�case bound of all known schemes�Other polynomial algorithms� using di�erent ideas� are given in Cunning�ham and Edmonds �� Bixby� Cunningham� and Rajan �� andRajan �� We should mention that Seymour ��b already containsimplicitly a polynomial� though not very e�cient� testing algorithm� Apolynomial test for deciding regularity of matroids a priori not known tobe binary is given in Truemper ��a�

Theorems �� and �� which are matrix versions of Theo�rem �� are given without proof in Seymour ��a� and Nemhauserand Wolsey �� Seymour ��a uses the hypergraph terminology of

Berge �� The polytope question P �M� Q�M answered by Theorem

�� is just one example of numerous questions concerning �ows� cir�cuits� and cutsets in matroids� Seymour ��a contains a wide�ranginginvestigation of these issues� One of them concerns the sum of circuits

property �rst de�ned in Seymour ��b� It is shown in Seymour ��athat this property holds for a matroid M if and only if M is binary andhas no F �

� � M�K�� or R�� minors� That result� Theorems �� and�� and an amazing symmetry of P �M allow one to completely re�

solve the P �M� Q�M question by Theorem �� which is due to

Barahona and Gr#otschel �� The proof sketched here is from Gr#otscheland Truemper ��b� which contains additional results about P �M andtreats computational aspects� Earlier results for special matroid classes andapplications are described in Orlova and Dorfman �� Edmonds andJohnson �� Hadlock �� Barahona �� Barahona and Mahjoub�� and Barahona� Gr#otschel� J#unger� and Reinelt �� Additional


results for the cycle polytope are given in Gr#otschel and Truemper ��a�Theorem �� is proved in Bixby and Cunningham �� Theorem�� is from Seymour ��a� Theorem �� is due to Lov$asz andis included in Gerards� Lov$asz� Schrijver� Seymour� and Truemper ��References for the odd�K� result are given in Chapter �� Additional ap�plications of the regular matroid decomposition theorem are described inSeymour ��d� ��f� Bland and Edmonds �� have used the de�composition to reduce linear programs with totally unimodular constraintmatrix to a sequence of maximum �ow and shortest route problems�

Chapter ��

Almost Regular Matroids

�� Overview

So far in this book� we have always used matrices to understand matroids�We have selected a base of the matroid to be investigated� Then we haveconstructed for that base a representation matrix over some �eld or even anabstract matrix� Finally� we have analyzed the matrix structure to deducematroid results� In the third step� we have employed pivots� in particularin the path shortening technique� to modify the matrix in agreement withsome change of the matroid base� We have also deleted or added rows andcolumns to represent matroid reductions or extensions�

Occasionally� we have reversed the just�described roles of matroids andmatrices� An example is the test of total unimodularity in Section �� via atest of matroid regularity� A second example is the analysis of the structureof totally unimodular matrices in Section �� via the structure of regularmatroids� But generally� it seems to be dicult to answer matrix questionswith matroid techniques� In particular� the taking of matroid minors� amost useful matroid operation� has a cumbersome translation into matrixlanguage unless one permits pivots� But the pivot operation almost alwayschanges the matrix structure rather drastically� Thus� pivots often impedethe understanding of matrix structure�

The preceding arguments seem to lead to the inescapable conclusionthat matrix properties generally cannot be conveniently analyzed with ma�troids� In particular� one is inclined to accept that conclusion in the follow�ing setting� The matroid property in question� say P� is inherited under

��


submatrix�taking� One wants to understand the minimal matrices that donot have P� We call any such matrix a minimal violation matrix of P�

In this chapter� we show that the above reasoning� awless as it mayseem� is not valid for the investigation of the minimal violation matricesof certain properties� Indeed� we describe a general matroid techniquefor a class of such problems where P observes certain conditions� Wedemonstrate the technique for the case whereP is the property of regularity�The method deduces a matroid formulation that involves a class of binarymatroids called almost regular� An analysis of that class produces the�Y matroid construction stated earlier in Theorem �� Additionalanalysis leads to a number of matrix constructions of surprising simplicity�Two of the constructions generate the minimal violation matrices for thefollowing two properties� regularity and total unimodularity�

At the outset� the sections of this chapter may appear to be like thepieces of a puzzle� diverse and seemingly unrelated� Only toward the end ofthe chapter do the relationships and functions of the puzzle pieces emerge�We sketch now the content of each section�

In Section �� we determine for undirected graphs G the minimalsubgraphs that rule out success for a certain signing of the edges of G� Thesigning process is supposed to convert G to a graph G� with edge labels�� and �� The labels are to be such that in every chordless circuitof G�� the edge labels sum �in IR� to prescribed values speci�ed in a givenvector �� If such signing is possible� then we call the resulting graph G�

��balanced� The signing condition is then linked to the set N of the binaryminimal violation matrices of regularity� That way� we determine completecharacterizations for two proper subsets of N � We also �nd an admittedlyweak characterization of the remaining matrices of N �

In Section �� we shift our attention from graphs and the set N toseveral matrix classes� In particular� we de�ne the class U of real comple�ment totally unimodular matrices� the classes A and B of almost represen�tative matrices over GF�� and GF�� respectively� and the class V of realminimal violation matrices of total unimodularity� The de�nitions of theseclasses will give the impression that they are quite unrelated� But in a way�these classes as well as N are di�erent manifestations of one and the samephenomenon� the absence of matroid regularity plus certain minimalityconditions�

Another puzzle piece is introduced in Section �� There we describethe previously mentioned technique for the investigation of the minimalviolation matrices of certain matrix properties P� We specialize the generalmethod to the case where P is the property of regularity� As a result�we de�ne the class of almost regular matroids� We state and sketch aproof of the construction of the almost regular matroids via �Y extensionsequences� The latter result should be familiar since a summary is includedin Section ��

�� Chapter �� Almost Regular Matroids

The odd and unrelated puzzle pieces of Sections �� are mergedto a rather beautiful picture in Section �� The almost regular matroidsare seen to be the matroid manifestation of the matrices of the classes ofA� B� N � U � and V� In particular� the construction of the almost regularmatroids via �Y extension sequences produces an elementary construc�tion of the class U of complement totally unimodular matrices� With U inhand� we very easily construct the remaining classes A� B� N � and V� Wethus have a construction for the binary minimal violation matrices of reg�ularity� and for the real minimal violation matrices of total unimodularity�In the �nal section� �� we sketch applications and extensions� and citereferences�

The chapter assumes knowledge of Chapters �� and ��

�� Characterization of Alpha�Balanced

Graphs

We are given an undirected graph G� For each chordless circuit C of G�we are given an integer number �C � �� or �� We collect the �C in avector �� To each edge of G� we want to assign the label �� or �� so thatin the resulting graph G� the real sum of the edge labels of each chord�less cycle C is congruent �mod �� to �C � When such labels can be found�we say that G� is ��balanced� Suppose an ��balanced G� cannot be pro�duced� Then G apparently contains a con�guration of chordless cycles withconicting requirements� In this section� we identify the possible sourcesof such conicts� Speci�cally� we pinpoint three subgraph con�gurationsthat collectively represent the minimal obstacles to ��balancedness� In thelast portion of the section� we specialize that result for the case when �is the zero vector� That specialization leads to a characterization of thebinary matrices that may be signed to become so�called balanced f��greal matrices� From the latter characterization� we deduce a partition ofthe class N of the binary minimal violation matrices of regularity� Thepartition consist of three subclasses� Two of them are well described� andindeed are readily constructed� But the third subclass has a rather unsat�isfactory characterization� One could say that the subsequent sections ofthis chapter are devoted to replacing that unsatisfactory description witha mathematically appealing and practically useful one�

We begin with the detailed technical discussion� We use a number ofterms that we de�ne in the next few paragraphs�

It is convenient for us to consider every graph to be undirected andto have a priori a �� or �� label on each edge� If nothing is said about agraph� then all labels are assumed to be �� Thus� the previously described

�� Characterization of Alpha�Balanced Graphs ��

assignment of f��g labels to G becomes a change of some edge labels� Thatchange we call a signing of G� We scale a star of G by multiplying the edgelabels of that star by �� We scale G by a sequence of scaling steps� eachone involving some star of G� The label sum of a subgraph G is the realsum of the labels of the G edges� That sum is denoted by L�G��

The emphasis on nodes in this section demands that we temporarilyabandon our usual viewpoint where nodes are edge subsets� Thus� we viewnodes as points� and consider edges to be unordered node pairs� Thatapproach is reasonable since we never have to deal with the contractionoperation or with parallel edges� We still view trees� paths� and cycles tobe given by their edge sets� Most subgraphs will be induced by some nodesubset� We use n�subgraph to specify that case� Special instances are n�pathand n�cycle� If G is a subgraph of G but not an n�subgraph� then G hasan edge that is not in G� but both of whose endpoints are in G� Such anedge is a G�chord for G�

Let � be an integer vector whose entries are in one�to�one correspon�dence with the n�cycles of a graph G� Throughout� it is assumed that eachentry of � is �� or �� Then G is ��balanced if for each n�cycle C the la�bel sum L�C� satis�es L�C� � �C�mod �� Note that scaling in G changesL�C� by a multiple of �� and thus does not a�ect ��balancedness� Supposea graph G has only ��s as edge labels and is not ��balanced� Then possi�bly an ��balanced graph G� can be deduced from G by signing� The labelchanges of edges in an n�cycle C� say producing C �� modify L�C� for someinteger k to L�C �� L�C�� k � jCj��k� Thus� if L�C �� C�mod �� isto be achieved at all� then necessarily jCj � �C�mod �� From now on� weassume that any � satis�es this necessary condition�

The relation �is an n�subgraph of� is transitive� In particular� everyn�cycle of an n�subgraph of G is an n�cycle of G� Let � be given for G� Bythe above observation� it makes sense to apply the term ��balanced� notjust to G� but also to n�subgraphs of G� In particular� ��balancedness ofan n�cycle C of G means L�C� � �C�mod �� We also say in that case thatC agrees with ��

One may suspect that signing to achieve ��balancedness is essentiallyunique� The next lemma con�rms this notion� The proof is almost identicalto that of Lemma ��

�� Lemma� Let G and G� be connected ��balanced graphs thatare identical up to edge labels� ThenG� may be obtained fromG by scaling�

Proof� Let T be a tree of G� and T � be the corresponding tree of G��Because of scaling� we may suppose that the labels of G and G� agree onthe edges of T and T �� Suppose the labels of G and G� di�er� say on anedge e of G and on the corresponding edge e� of G�� The edges e and e�

form fundamental cycles C and C � with T and T �� respectively� Select Tand e� and thus T � and e�� so that the cardinality of the cycles is minimum�


Suppose C and C � have chords� By the minimality condition� the label ofany chord of C must agree with that of the corresponding chord of C �� Butthen C and C � do not have minimum cardinality� a contradiction� Thus�C and C � are chordless cycles with labels in agreement except for e and e��But then L�C� and L�C �� di�er by �� and necessarily L�C� �� C�mod ��or L�C �� C�mod �� a contradiction�

In this section� the drawings of graphs follow special rules� A solidstraight line connecting two nodes represents an edge� while a solid line witha short zigzag segment indicates a path where all intermediate nodes havedegree �� A broken line represents a path connecting the two endpointsof the broken line� and two or more such paths may have one or moreintermediate nodes in common� However� the path of a broken line has nointermediate node in common with any node explicitly shown� The labelson edges are always omitted�

Of particular interest are the following graphs�

��

etc.

etc.

Q2

Qk Q1

1

2k

3

Q1 Q2 0mayexist

Q3

jQij � �� odd k � �� oddH� H� H�

Graphs of type H�� H�� and H�

With the aid of the following lemma� it is easy to check whether or not agraph G of type H�� H�� or H� may be signed to become ��balanced for agiven ��

�� Lemma� The following statements are equivalent for a graphG of type H�� H�� or H�� and a given ��

�i� G can be signed so that an even �resp� odd� number of n�cycles do notagree with ��

�ii� G can be signed so that every n�cycle �resp� every n�cycle except adesignated one� agrees with ��

Proof� It is easily seen that one can always sign H�� H�� and H� such thatat most one designated n�cycle does not agree with �� Now every edge ofH�� H�� and H� is part of exactly two n�cycles� and hence every signing ofG produces the same number �mod �� of n�cycles that do not agree with��


The main theorem of this section follows� It establishes the centralrole of n�subgraphs of type H�� H�� and H� when one wants to achieve��balancedness by signing�

�� Theorem� For a given vector �� a graph G may be signed tobecome ��balanced if and only if every n�subgraph of type H�� H�� and H�

can be so signed and �C � jCj�mod�� for every n�cycle C of G�

We prove the theorem in a moment� As an aside� we mention a corol�lary that rephrases the theorem in terms of edge subsets�

�� Corollary� Let � be a f�� g vector whose entries are in one�to�one correspondence with the n�cycles of a graph G� Then there exists asubset F of the edge set of G such that jF �Cj � �C�mod �� for all n�cyclesC of G� if and only if the latter condition is true for all n�subgraphs of typeH�� H�� and H� of G�

Proof� For each n�cycle C of G� de�ne �C � ��C � jCj�mod�� We callthe requirement jF � Cj � �C�mod �� the ��condition� Obviously� we onlyneed to prove the �if� part of the corollary� Thus� we assume that the��condition can be satis�ed for each n�subgraph H of type H�� H�� orH�� say by edge subset FH of H� Sign H so that precisely the edges ofFH receive a �� By the de�nition of � and by the ��condition� for anyn�cycle C of H there are integral lC and kC such that jFH � Cj � �l ��C � ��C � �kC � jCj�� Because of this equation and the signing rulefor H� L�C� � �jFH � Cj � jCj � �C�mod �� Thus� H is ��balanced� ByTheorem �� G can be signed to become ��balanced� Consider G tobe so signed� Let F be the subset of edges of G with �� label� Since forany n�cycle C we have L�C� � �C�mod �� there are integers kC and lCso that jF � Cj � ��C � �kC � jCj�� C � �lC� Thus� F satis�es the��condition�

We accomplish the proof of Theorem �� in two steps� First weprove a rather technical lemma� Let G be a graph without parallel edges�and P� be a path of G where all intermediate nodes have degree �� For agiven vector �� the graph G is almost ��balanced with respect to P� if alln�cycles of G that do not contain P� agree with ��

�� Lemma� Let G be a graph without parallel edges� and let vector� be given� Suppose that every n�subgraph of type H�� H�� or H� of Gcan be signed to become ��balanced� and that G is almost ��balanced withrespect to a given P�� Assume C� and C� are n�cycles of G that include P��and let P� � C� �C� be a path of maximal cardinality satisfying P� � P��If one of the n�cycles C�� C� agrees with �� then the other n�cycle agreeswith � as well� provided one of the following conditions is satis�ed�

�a� jP�j � ��b� jP�j � � and �C� C�� P� is not an n�cycle of G � P��


Proof� The proof is by induction on jP�j� The lemma holds trivially forthe maximal value that jP�j may take on since then C� � C�� Hence� wewill prove validity for jP�j � l assuming that the lemma holds wheneverjP�j � l�� In the nontrivial case� we have C� �� C�� Thus� the endpointsu and v of P� have degree � in G � C�C�� The graph G is depicted below�P�� P�� and P� are the three paths u to v� and for i � �� Ci � Pi P��

��

u

v

P1 P2 P3

Graph G

Note that �C� C�� P� is the set P� P�� Furthermore� jP�j � � sinceC� is an n�cycle of G� Similarly� jP�j � �� It will be convenient to considertwo cases�� jP�j or jP�j � ��

Without loss of generality� suppose jP�j � �� Since C� is an n�cycle�the intermediate node of P�� say w� cannot be a node of P�� Addition ofall edges of G from w to intermediate nodes of P� produces the following

n�subgraph G of G�

��

u

v

wP1 P2 P3etc.

Graph G

If jP�j � �� then w of G has degree of at least � since otherwise �C�C��P�

is an n�cycle of G � P�� Thus� G is a graph of type H� or H�� and all n�

cycles of G are ��balanced except possibly for C� and C�� since G is almost��balanced�

By assumption� G can be signed to become ��balanced� So by Lemma�� C� must agree with � if C� does� and vice versa�


�� Both jP�j� jP�j � ��

For i � �� let ai� bi be the vertices of Pi adjacent to u� v� respectively�Thus� G is as follows�

��

u

v

P1 P2 P3

a1

b1

a2

b2

Graph G

Clearly� u� v� a�� a�� b�� b� are all distinct� Suppose there is a path inG�P� from u to v using only vertices of P� P� and also avoiding a� andb�� Then there exists an n�path P with these properties� and P P� is ann�cycle C of G� Now C �C� contains a path that in turn properly containsP�� Thus� by induction and part �a�� C agrees with � if and only if C�

does� The same conclusion holds for C and C�� so C� and C� both agreewith � or both do not� Hence� we may suppose that there is no such P �Then P� and P� have no vertex in common except for u� v� Also� every�G�P��chord for the cycle P� P� is incident with a� or b�� If there is nosuch chord� we must be in case �a� of the lemma� and C� C� is a graphof type H�� Since P� P� agrees with �� we again have the desired resultby Lemma �� Hence� suppose there exists at least one such chord�Repeat the above argument� but this time try to �nd a P avoiding a� andb�� If again we are unsuccessful� then the �G � P��chords for P� P� arefound only at a� or b�� Thus� there are at most two such chords� one froma� to a�� the other from b� to b�� and C� C� must be a graph of type H��All n�cycles agree with � except at most C� and C�� so Lemma ��produces the desired conclusion�

Proof of Theorem �� For proof of the nontrivial �if� part� it issucient to consider the case where all edges but those incident at somenodem have been signed such that G�fmg is ��balanced� We want to signthe edges of G incident at m so that G becomes ��balanced� Let C� andC� be two n�cycles of G� each containing edges �i�m� and �j�m� for somei and j� Derive G� from G by deleting all neighbors of m not equal to i orj� Clearly� C� and C� are contained in G�� and G� is almost ��balancedwith respect to P� � f�i�m�� j�m�g� Since G� is an n�subgraph of G�every n�subgraph of type H�� H�� or H� of G� can be signed to become��balanced� So by Lemma �� C� and C� agree with � or they bothdo not�


De�ne a graph J � also with f��g arc labels� from G as follows� Eachedge of G incident at node m becomes a node of J � An arc connects twonodes of J if the corresponding edges of G are part of at least one n�cycleC of G� This arc is signed �� if C agrees with �� and �� otherwise� By theprevious argument� the classi�cation of each arc of J is well�de�ned� Forclarity we use �arc� �resp� �edge�� in connection with J �resp� G�� Supposewe change the sign of edge �i�m� in G� In J � we must change the sign onall arcs incident at node �i�m�� so this is a scaling step� Conversely� scalingin J leads to signing of edges in G incident at m� Clearly� it is possible toscale J such that all arcs of a given forest have �� labels� Furthermore�note that a given cycle of J has an even number of �� arcs if and only ifthat is true after scaling�

It is claimed that every cycle of J has an even number of �� arcs� Itis sucient to prove the claim for an n�cycle CJ of J � We will consider twocases depending on k � jCJ j�

�� k � ��

By de�nition of J � the graph G contains the following subgraph G�where each path Qi forms an n�cycle Ci with edges �i�m� and �i � ��m��k � � is interpreted as ��

��

etc.

etc.

Q2

Qk Q1

1

2k

3m

Graph G� case ��

It is claimed that CG �Sk

i��Qi is an n�cycle of G� We �rst show that CG

is a cycle� Suppose Q� and Qj have a node u in common� where u �� ifj � �� and where without loss of generality j �� k� Since Ci is an n�cycle�for any i we have v � Qi� where v �� i� i � � is a neighbor of m in G�Hence� u is not equal to any endpoint of Q� or Qj � If j �� de�ne P tobe composed of the paths from � to u on Q� and u to j on Qj � Clearly� Pcontains no neighbor v of m except for nodes � and j� and we may replaceit by an n�path P observing the same condition� But P and the edges��m�� j�m� form an n�cycle of G� But then� since j �� k� CJ cannot bean n�cycle of J � If j � �� P is composed of paths � to u on Q� and u to �on Q� �� Qj�� Again� we conclude that CJ is not an n�cycle of J � Hence�CG must be a cycle of G� Similar arguments prove that CG is indeed ann�cycle of G� Since G is an n�subgraph of G of type H�� it can be signedto become ��balanced� By Lemma �� an even number of n�cycles of


G do not agree with �� CG cannot be one of these� since it is an n�cycle ofG � fmg� The Ci� i � �� k� constitute the remaining n�cycles of G�The fact that an even number of the Ci do not agree with �� results in aneven number of �� arcs in the n�cycle CJ of J �

�� k � ��

Again� G has G of �� as a subgraph� If CG �S�

i��Qi is an n�cycle ofG� then arguments as for the case k � � yield the desired conclusion�So suppose Q� and Q� prevent CG from being an n�cycle� i�e�� Q� and Q�

have a node i �� in common� or there exists a G�chord for Q� Q��De�ne P� � f��m�g� P� � Q� f��m�g� and P� � Q� f��m�g� Graph

G �S�

i�� Pi is shown below�

��

m

2

P1 P2 P0

1 3

Graph G� case ��

Derive G� from G by deleting all neighbors of m not equal to �� or ��Clearly� Ci � P� Pi� i � �� is contained in G�� Because of scaling inJ � we may suppose that the arc of J connecting nodes ��m� and ��m�has a �� label� This implies that every n�cycle C of G containing edges��m� and ��m� agrees with �� and G� is therefore almost ��balancedwith respect to P�� Furthermore� every n�subgraph of G� of type H�� H��or H� can be signed to become ��balanced� Since P� P� is not an n�cycleof G� � P�� either part �a� or part �b� of Lemma �� holds� and C�

agrees with � if and only if C� does� But this implies that CJ has an evennumber of �� arcs�

The remainder is simple� We scale J so that all arcs of an arbitrarilyselected forest T of J receive �� labels� Then all out�of�forest arcs mustalso have �� labels� since otherwise a cycle with an odd number of �� arcshas been found� Related signing in G results in an ��balanced graph�

One application of Theorem �� is as follows� Call a binary matrixB balancedness�inducing if its �s can be replaced by ��s so that the result�ing real matrix A satis�es the following requirement� For each k � �� eachk � k submatrix of A of the form given by �� below must have realdeterminant �� By Lemma �� the determinant condition is equivalentto the demand that the entries of the submatrix sum to ��mod ��


�� ±1±1 ±1±1 ±1

±1±1±1

.

. ..

Matrix whose bipartite graphis a chordless cycle

The next theorem characterizes the class of balancedness�inducing bi�nary matrices� As we shall see� the proof requires one easy application ofTheorem ��

�� Theorem� A binary matrix B is balancedness�inducing if andonly if B does not have a submatrix B such that BG�B� is one of thegraphs H� or H� below�

��

etc.

etc.

Q2

Qk Q1

1

2k

3

Q1 Q2 0Q3

jQij � �� odd k � �� oddH� H�

Graphs of type H� and H�

for balancedness�inducing matrix case

Proof� Let B be a binary matrix� De�ne G to be the bipartite graphBG�B� with additional �� labels on the edges� Each n�cycle of G corre�sponds precisely to a f�� g submatrix of B that looks like the matrix of�� except for the signs of the entries� Signing B to produce an Awith the desired property is equivalent to signing G so that each n�cycleC of the resulting graph has L�C� � ��mod �� Put di�erently� an appro�priate signing of B is possible if and only if G can be signed to become a��balanced graph� say G�� We call G� as well as the related f��g signedreal version of B balanced� By Lemma �� G� is unique up to scaling�The latter operation corresponds to scaling of some rows and columns ofA by �� factors� Thus� up to such scaling� the matrix A is unique� Theproof of Lemma �� implies a simple scheme to e�ect the signing if itis possible at all�

By Theorem �� G� and thus A exist if and only if every n�subgraph of G of type H�� H�� orH� �see �� can be signed to become


balanced� The graph H� is not bipartite� so it cannot be present in thebipartite G� For the same reason� the paths Q�� Q�� and Q� of any H�

n�subgraph must have the same parity� One readily sees that H� with a ��label on each edge has an odd number of circuits C with L�C� � ��mod ��if and only if for i � �� jQij is odd and at least �� By Lemma ��the latter condition characterizes the H� graphs that cannot be signed tobecome balanced� Corresponding arguments for H� show that H� cannotbe signed to become balanced if and only if k� which is the number ofspokes of the wheel�like graph� is odd and at least ��

Balancedness�inducing matrices are related to regular ones as follows�

�� Lemma� Let B be a regular matrix� Then B is balancedness�inducing� Furthermore� any balanced f��g real matrix derived from Bby signing is totally unimodular�

Proof� Regularity implies that a totally unimodular matrix A can be de�duced from B by signing� By Lemma �� every k�k� k � �� submatrixA equal to the matrix of �� must have its entries sum to ��mod ��Thus� A is balanced� As observed above� any balanced matrix A� derivedfrom B by signing must be obtainable from A by scaling� Thus� A� istotally unimodular�

Suppose we want to characterize the class N of binary minimal viola�tion matrices of regularity� Theorem �� and Lemma �� bringus quite close to that goal� as follows�

�� Theorem� Let N be the class of binary minimal violationmatrices of regularity� Then N has a partition into the following threesubclasses N�� N�� N��

�a� N� �resp� N�� is the set of binary matrices B for which BG�B� is agraph of type H� �resp� H�� of ��

�b� N� is the set of binary balancedness�inducing matrices B satisfyingthe following condition� Any balanced f��g real matrix A derivedfrom B by signing is a minimal violation matrix of total unimodularitywith at least three ��s in some row and in some column�

Proof� Let B be any matrix in N � Thus� B is not regular� but everyproper submatrix of B is regular� Suppose B is not balancedness�inducing�By Theorem �� BG�B� has as n�subgraph a graph H of type H� orH� of �� Since H cannot be signed to become balanced� by Lemma�� we must have BG�B� � H� Thus� B �N� N��

Now supposeB to be balancedness�inducing� Let A be any real f��gbalanced matrix obtained from B by signing� Since B is not regular� A can�not be totally unimodular� Since every proper submatrix of B is regular� byLemma �� every proper submatrix of A is totally unimodular� Thus�A is a minimal violation matrix of total unimodularity� The nonregularity


of B implies that B is not graphic or cographic� Thus� B has a row and acolumn with at least three �s� Then A has at least three ��s in some rowor column� We conclude that B N��

So far we have shown that N � N� N� N�� It is easy to see by�� that any B �N� N�� is a minimal nonregular matrix� Thus�N� N� � N � Let B N�� By de�nition� every proper submatrix of B isregular� If B is regular� then by Lemma �� any balanced matrix Ainduced by B is totally unimodular� But this violates the de�nition of N��Thus� N� � N �

At this point� we know N � N� N� N�� By Theorem ��N� N� consist of the minimal binary matrices that are not balancedness�inducing� The matrices of N� are by de�nition balancedness�inducing�Thus� N� N� and N� are disjoint� By �� N� and N� are disjointas well� Thus� N�� N�� and N� partition N as claimed�

Theorem �� is a substantial step toward the goal of understand�ing the class N of binary minimal violation matrices of regularity� Indeed�the subclasses N� and N� of such matrices have a simple and appealingdescription� But the characterization of the remaining class N� in termsof minimal violation matrices of total unimodularity has little value unlesswe understand the latter matrices� In the next section� we take small butnevertheless useful steps toward understanding those matrices�

�� Several Matrix Classes

We de�ne four classes of matrices� Each of them is connected in some waywith the binary minimal violation matrices of regularity� We point out therelationships as the classes are introduced one by one� Their signi�cancewill become apparent in Sections �� and ��

Complement Totally Unimodular Matrices

We begin with f�� g real matrices� For such a matrix U � we may de�ne thefollowing complement operations� Let k index a row of U � Then the row

k complement of U is the real f�� g matrix U � derived from U as follows�For all column indices j with Ukj � � and for all row indices i �� k� replacethe entry Uij by its complement� i�e�� by �� and � by �� Let l index acolumn of U � Then the column l complement of U is the transpose of therow l complement of U t�


For example� if U is the �� identity with indices k and l as shown�

��U =

1kl

0 0 00 1 0 00 0 1 00 0 0 1

Matrix U

then the row k complement of U is the matrix U � below�

�� U' =

1 0 0 0111

1 0 00 1 00 0 1

Row k complement of U

The column l complement of U � is the following matrix U ��

�� U'' =

11 011

1

11

1101

1110

Column l complement of U �

How many di�erent matrices� up to indices� may be derived from an m�nf�� g matrix U in a sequence of complement operations� The answer relieson two readily veri�ed claims� For k �� l� let U � be the row k complementof U � and let U �� be the row l complement of U � Then up to a change ofindices� the row l complement of U can be seen to be U �� Suppose oneobtains a matrix from U by a row complement step followed by a columncomplement step� Then the same matrix results if one reverses the orderof the two steps� By these two claims� any matrix obtainable from a givenm � n f�� g matrix U in a sequence of complement operations may upto indices be produced by at most one row complement step and�or onecolumn complement step� Thus� at most �m��n�� numerically di�erentmatrices may be deduced from U by repeated complement steps�

With respect to total unimodularity� the complement operation can bevery destructive� For example� U of �� is the �� identity� and thustotally unimodular� Consider the matrix U �� of �� deduced from U bytwo complement steps� In the right�hand corner� U �� has a �� submatrixwith real determinant �� Thus� U �� is not totally unimodular�

We de�ne a real f�� g matrix U to be complement totally unimodular

if U and all matrices derivable from U by possibly repeated complement op�erations are totally unimodular� Clearly� complement total unimodularity


is maintained under submatrix�taking and complement operations� We col�lect the complement totally unimodular matrices in a set U � By the aboveobservation and examination of a few additional examples� one quicklysees that complement total unimodularity is a very demanding property�In fact� one is inclined to believe that there are only a few structurallydi�erent complement totally unimodular matrices� But that notion turnsout to be mistaken� as we shall see in Section ��

Complement total unimodularity of U implies that certain extensionsof U are totally unimodular as follows�

�� Lemma� Let U be a real f�� g matrix that is complementtotally unimodular� Enlarge U to a matrix U � by adjoining a row or columncontaining only �s� Then U � is totally unimodular�

Proof� Let U be any square submatrix of U �� We must show that detIR � �or �� In the nontrivial case� U intersects the vector of �s adjoined to U � Areal pivot on any � of that vector followed by deletion of the pivot row and

column produces a matrix U � One readily con�rms that� up to scaling by

f��g factors� U is a submatrix of a row or column complement of U � Thus�

U is totally unimodular� and hence detIRU � � or �� as desired�

Almost Representative Matrices

We change �elds and consider matrices over GF�� or GF�� Let A be amatrix over GF�� and B its support matrix� View B to be over GF��Let M be the ternary matroid represented by A over GF�� and N be thebinary matroid represented by B over GF�� By the derivation of B fromA� certain sets are bases of both M and N � In fact� the two matroids maybe identical� Suppose they are not� If there is just one set that is a base ofone of the matroids and not of the other one� then we say that A almost

represents N over GF�� and that B almost represents M over GF�� Onemay re�express the assumption as follows� The GF��determinant of eachsquare submatrix of A is nonzero if and only if the GF��determinantof the corresponding square submatrix of B is nonzero� except for onesquare submatrix A of A and for the corresponding submatrix B of B� ByCorollary �� we know that det�A �� and det�B � �� The argumentsto come will con�rm this fact�

Suppose we perform a GF��pivot on a �� of A in A� We also carryout the corresponding GF��pivot on the related � of B in B� In the ma�trices so deduced from A and B� the di�erence betweenM and N manifests

itself by submatrices A and B that are smaller than A and B� Indeed� hadwe carried out the respective pivots just in A and B and deleted the pivot

row and column� we would have obtained A and B� Because of this reduc�tion possibility� we may assume A and B to be of order �� A simple case


analysis of the � � � matrices proves that the di�erence in determinantsbetween A and B can be produced in essentially one way� That is� we musthave� up to scaling in A�

��11 1

1B =

-11 1

1A = ;

Submatrices A and B proving M �� N

Thus� det�A �� and det�B � �� as predicted by Corollary �� Ev�idently� the same relationship must have held prior to any pivots� Thereader who has covered Section �� surely recognizes the similarity of theabove arguments to those of the proof of Theorem �� and Corollary�� Here we have a GF��matrix A instead of the abstract matrix ofSection ��

We continue with A and B given by �� We perform one moreGF��pivot on a �� of A in A� and also carry out the related GF��pivot

in B� This change e�ectively reduces A to a � � � matrix A� and B to a

� � � matrix B� for which det�A �� and det�B � �� Thus� B � � � ��

Because of scaling of A� we may assume A � ��

Before going on� we record the insight attained so far in the followinglemma�

�� Lemma� Let A be a matrix over GF�� View the supportmatrix B of A to be over GF�� Assume that the GF��determinant ofeach square submatrix of A is nonzero if and only if the GF��determinantof the corresponding submatrix of B is nonzero� with the exception of justone submatrix A in A and the related submatrix B in B� both of orderk � �� Then by GF��pivots within the submatrix A and scaling� and bycorrespondingGF��pivots in B� the matricesA andB can be transformedto matrices with determinants agreeing analogously to A and B� except for

a submatrix A � �� and B � � � ��

For notational convenience� we now rede�ne A and B to be the ma�

trices produced by the pivots� Thus� A � �� is a submatrix of A� and

B � � � � is the related submatrix of B� We want to analyze the structure

of A and B� To this end� let y be the row index of A and of B� and letx be the column index� Assume that X �resp� Y � is the index set of theremaining rows �resp� columns� of A and B� Also assume that A has nozero rows or columns� Recall that � denotes a column vector containingonly �s� We claim that up to row and column scaling of A� the matrices Aand B are of the form given by �� below� where U is a f�� g matrixviewed to be over GF�� or GF�� as needed�


��0y

;-1x Y

y

XA =

U1

1tx Y

XB =

U1

1t

Matrix A over GF�� for M �and matrix B over GF�� for N

The claim plus additional facts make up the next theorem�

�� Theorem��a� Let A be a matrix over GF�� without zero rows or columns� and let

B be a matrix over GF�� of the same size and with the same row andcolumn indices� Assume that the GF��determinant of each squaresubmatrix of A is nonzero if and only if the GF��determinant ofthe corresponding submatrix of B is nonzero� except for one � � �

submatrix A � �� of A and the corresponding submatrix B � � � � ofB� say with row index y and column index x� Let the remaining rowsof A be indexed by X� and the remaining columns by Y � Then up to ascaling of rows and columns of A� the matrices A and B are given by�� where U is a f�� g matrix to be viewed over GF�� or GF��as needed� When U is considered to be real� then it is complementtotally unimodular�

�b� Let A and B be the matrices of �� The column x and the rowy must be present� but X or Y may be empty� Assume that thesubmatrix U of either matrix is a f�� g matrix that is complementtotally unimodular when considered real� View A to be over GF��and B to be over GF�� Then the GF��determinant of each squaresubmatrix of A is nonzero if and only if the GF��determinant ofthe corresponding submatrix of B is nonzero� except for the � � �

submatrix A � �� of A indexed by x and y� and the corresponding

submatrix B � � � � of B�

Proof� We establish part �a�� A and B of �� correctly display A and

B� Suppose column x of A contains a �� say in row i X� Since A has nozero rows� there is a j Y with Aij � �� From the rows x� i and fromthe columns y� j of A and B� we extract the submatrices

�� ;-1xy j

0i

γA =

±1 ix 0

y j

0δ

B =1

Submatrices A and B of counterexample


We ignore the entries � and �� Indeed� for any � and �� we have det�A �� and det�B � �� which contradicts the presumed agreement of determi�nants� Thus� by scaling in A� we may assume that column x of A and

column x of B contain only �s except for the entry of A or B in row y� Bysymmetry� we also may assume that the row y of A and the row y of Bcontain only �s except for the entry in column x�

It remains for us to prove that U is a f�� g matrix that occurs inboth A and B� and that U � when viewed as real� is complement totallyunimodular� If U of A contains a �� say in row i and column j� then rowsx� i and columns y� j of A de�ne a � � � GF��singular submatrix of A�But in B� these rows and columns specify a GF��nonsingular submatrix�a contradiction� Thus� U in A is a f�� g matrix� Because of the agreementof determinants of A and B� the matrix U must also occur in B� this timeconsidered over GF�� of course� By Theorem �� the agreement ofdeterminants on U when viewed over GF�� and GF�� implies that U asreal matrix is totally unimodular�

Suppose we perform a GF��pivot in column x� row i of B of ��The latter matrix is like B except that the indices x and i have tradedplaces and U has been replaced by its row i complement U �� We performthe analogous GF��pivot in A� That pivot plus some scaling with f��gfactors produces a matrix A� that is like A except that the indices x andi have switched and U has become U �� The determinants of A� and B�

agree analogously to A and B� except for the �� submatrix in row y andcolumn i of A� and the corresponding � � � submatrix of B� By the precedingdiscussion� U � must be totally unimodular� Using additional pivots� we seethat U as real matrix is complement totally unimodular�

We turn to part �b�� Let C be a square submatrix of A� and let D bethe corresponding submatrix of B� We must show that det� C is nonzero

if and only if det�D is nonzero� with the single exception of C � A and

D � B� Suppose C intersects at most one of column x and row y of A� Bythe complement total unimodularity of U and Lemma �� the matrixC is totally unimodular� By Theorem �� the determinants of C andD agree as desired� This leaves the case where C and D properly include

A and B� respectively� Carry out a GF��pivot on any � in column x ofC� Then delete the pivot row and column� Let a matrix C � result� Cor�respondingly� reduce D by a GF��pivot to D�� Up to scaling by f��gfactors in C �� both reduction steps produce a submatrix of a row comple�ment of U with an additional row of �s adjoined� By the above discussion�the determinants of C � and D�� and hence of C and D� agree�

The above proof contains an observation that we want to record as alemma for future reference�

�� Lemma� Let B be the matrix over GF�� of �� Then


a GF��pivot in column x or row y of B transforms B to a matrix B�

structured like B� except that a row index and a column index have tradedplaces and U has been replaced by a row or column complement of U �

Collect in sets A and B the possible cases of A and B� respectively�of part �b� of Theorem �� That is� each A A and each B B isgiven by �� and the f�� g submatrix U of either matrix when viewedas real is complement totally unimodular� We permit the extreme caseswhere U is trivial or empty� Thus� one of the index sets X and Y � or evenboth of them� may be empty�

Let N be the matroid represented by some B B over GF�� By part�b� of Theorem �� the corresponding A A almost represents Nover GF�� For this reason� we call A a collection of almost representative

matrices over GF�� Analogously� the matroidM represented by anA Aover GF�� is almost represented by the corresponding B B over GF��Thus� B is a collection of almost representative matrices over GF��

Minimal Violation Matrices of Total Unimodularity

So far in this section� we have de�ned three matrix classes� U � A� and B�We need one additional class V� which contains the real minimal violationmatrices of total unimodularity� Thus� every V V is not totally unimod�ular� but this is so for every proper submatrix of V � Clearly� each V Vis square� To avoid uninteresting instances� we exclude from V the cases Vof order �� Thus� each V V is for some k � �� a k � k f��g matrix�Let W be the support matrix of V � Consider V to be real or over GF��as needed below� and W to be over GF�� We have the following result�

�� Theorem� Let V be any matrix of V� and let W be its binarysupport matrix� Then by GF��pivots in V and scaling with f��g factors�and by corresponding GF��pivots in W � the matrices V and W can betransformed to matrices A A and B B� respectively� of order at least��

Conversely� suppose in A A and B B of order at least �� weperform GF��pivots and related GF��pivots� respectively� so that A�

and B� result that satisfy the following condition� Let X be the subset ofX fxg indexing columns of A� and B�� and Y be the subset of Y fygindexing rows� The condition is that jXj � � or jY j � �� Then thesubmatrix A of A� indexed by X and Y is in V� and the correspondingsubmatrix B of B� is the support of A�

Proof� We start with the �rst part� where V and W are given� Let V bea proper submatrix of V � Then V as real matrix is totally unimodular�Thus� by Theorem �� V as GF�� matrix has det� V nonzero if and


only if the corresponding W of W has det�W nonzero� By analogousarguments� exactly one of det� V and det�W is nonzero� Thus� V andW constitute a pair of matrices to which Lemma �� can be applied�Accordingly� V and W can by pivots and scaling be transformed to A overGF�� and B over GF�� respectively� with agreeing determinants� except

for a submatrix A � �� of A and B � � � � of B� By part �a� of Theorem�� we have A A and B B�

Reversal of the above arguments essentially proves the converse part�We only need to show that the matrices A of A� and B of B� deducedfrom A A and B B are the ones with disagreeing determinants� LetM �resp� N� be the matroid represented by A over GF�� resp� B overGF�� In the matroid M � the set X fxg is a base� But that set is not abase in N � By assumption� the set X with jXj � � is the subset of X fxgindexing columns of A�� and Y is the subset of Y fyg indexing rows� SinceX fxg is a base of M but not of N � the submatrix A of A� indexed by Xand Y has det�A nonzero� while the corresponding submatrix B of B� hasdet�B � ��

Several simple but useful results follow from Theorem �� Recallfrom Section �� that a f��g matrix is Eulerian if in every row and everycolumn the entries sum to ��mod �� Equivalently� each row and columnmust have an even number of nonzeros�

�� Corollary��a� A square f��g matrix of order at least � is in V if and only if the

following holds� V can be scaled with f��g factors to become� forsome square� nonsingular� complement totally unimodular matrix U �the matrix

�� b = U-1 α = U-1 − 2. 1.1t

a = U-1

;U-1

aα

b. 1

.1t

Matrix V up to scaling

�b� For every matrix V V� the real inverse of V contains only �

�entries�

jdetIR V j � �� V is Eulerian� and the real sum of the entries of V inIR is congruent to ��mod ��

Proof� We start with part �a�� By Theorem �� we may deduce V �up to scaling with f��g factors� from a square GF��nonsingular A Aby GF��pivots� Rede�ne V so that it is the appropriately scaled versionof the original V � The matrix A is given by �� We may recreateA from V by performing the GF��pivots in reverse order� Suppose weuse real pivots instead� All intermediate real matrices must be numerically


identical to the GF��matrices in the original GF��pivot sequence� byvirtue of the fact that each �� submatrix of each such real matrix is totallyunimodular� A di�erent conclusion applies to the last pivot� If performedin GF�� it would produce A� But carried out in IR� it produces a matrix Acontaining detIR V as entry where A has the �� All other entries of A mustagree numerically with those of A� Now detIR V is the real determinant ofa � � � submatrix of the predecessor matrix of the real pivot sequence�Since detIR V �� we must have jdetIR V j � �� We know that the �nalpivot� when done in GF�� produces the �� of A instead of detIR V � Thus�detIR V is congruent to ��mod �� and accordingly detIR V � ��

We just have proved that A is the matrix

��y 2

x Y

X U1 A =~

1t

Matrix A derived by real pivots from V

The submatrix U is complement totally unimodular� By the existence ofthe real pivot sequence� detIRU must be nonzero� Thus� jdetIRU j � ��

Suppose we employ the following well�known method for computingthe inverse of V � We begin with the real matrix �I j V �� Then we carryout elementary row operations until the submatrix V of �I j V � has becomean identity matrix� At that time� the submatrix I of �I j V � has becomeV �� We claim that the matrix A of �� contains in compact formthe results of most of these row operations� For a proof� we �rst note thateach one of the real pivots deducing A of �� from a scaled versionof V involves a �� as pivot element� Then by the relationship betweenelementary row operations and pivots described in Section �� the pivotsproducing A correspond to row operations and scaling steps with f��gfactors in �I j V � that convert the submatrix I of �I j V � to the matrix

��1t1

0 U V1 =~

Matrix V � derived from �I j V �by row operations and scaling

and that change the submatrix V of �I j V � to the matrix

�� 02

11

1 .. V2 =~

Matrix V � derived from Vby row operations and scaling


Observe that the columns of A save the �rst one occur in V �� and thatthe �rst column of A is also the �rst column of V �� as is implied by thediscussion of Section ��

We now perform row operations in � V � j V �� that convert V � to anidentity matrix� Correspondingly� V � becomes the inverse matrix of ascaled version of V � Up to scaling� that inverse matrix is

��1t

U = 2U − 1 .-1

1 ; U~ V =

~ 12

~1t

Inverse matrix V of scaled version of V

A multiplication check veri�es that the matrix of �� is V �� and thusis V up to scaling� This fact proves part �a��

Part �b� is now easily shown� We already know jdetIR V j � �� ThatV �� is a f��

�g matrix is evident from the scaled version V of V �� given

by �� The matrix V is Eulerian� since this is clearly so for thescaled version given by �� The latter matrix has its entries sumto ��mod �� Scaling does not a�ect that result� so the same conclusionapplies to V �

Let us summarize the main results of this section for the matrix classesU � A� B� and V� The class U contains the real f�� g complement totallyunimodular matrices� A and B contain the almost representative matricesover GF�� and GF�� respectively� as depicted by �� From any oneof the three classes� the remaining two are obtained by trivial operations�as is evident from �� The class V contains the real minimal violationmatrices of total unimodularity of order at least �� By �� eachV V can be readily computed from a square IR�nonsingularU U � Withequal ease� we can derive V from A or B� But note that we do not knowhow to deduce the entire class U � or A or B� from V� It turns out thatthis is not possible by the matrix operations of scaling by f��g factors�pivots� submatrix�taking� and change of �elds from IR to GF�� or GF��We prove this fact in Section �� There we construct U � A� B� and V�taking the following viewpoint� First we construct U by a process yet tobe described� Then we deduce A� B� and V as just mentioned�

Recall that Theorem �� establishes a partition of the class N ofbinary minimal violation matrices of regularity� Two subclasses labeled N�

and N� are well described by that theorem� But the third class N� is notwell explained� Indeed� each B N� is the support matrix of a minimalviolation matrix V of total unimodularity with at least three nonzeros insome row and some column� Thus� V V� As an aside� the just�mentionedbound of � on the number of nonzeros in some row and some column ofV can now be strengthened to � by part �b� of Corollary �� At


any rate� the construction of V via U gives a construction of N�� So theyet�to�be�described construction process for U e�ectively brings to an endthe quest for an understanding of the structure of N �

The promised construction of U relies on some matroid results that weintroduce in the next section�

�� De�nition and Construction of

Almost Regular Matroids

As argued in the introductory section of this chapter� one is tempted toclaim that matroids are not suitable for investigations of minimal violationmatrices� There it is also claimed that this argument is awed� Here weshow why� by providing a general method for a matroid�based investigationof the minimal violation matrices of certain matrix properties� We special�ize the method to a particular instance� In doing so� we de�ne and analyzethe matroids that we called almost regular in Section �� In particular�we establish the construction for almost regular matroids that was alreadylisted in Section �� We use that construction to obtain a construction forthe class U of the complement totally unimodular matrices�

We begin with a general discussion about matrix properties and ma�troids� Let P be a property de�ned for the matrices over a �eld F � whereF must be GF�� or GF�� Technically� one may consider P to be asubset of the matrices over F � The property is to be maintained undersubmatrix�taking� row and column permutations� scaling with f��g fac�tors� and F�pivots� and when a row or column unit vector is adjoined�

Examples for P are regularity� graphicness� cographicness� graphic�or�cographicness� and planarity� all de�ned for F � GF�� Also qualifying isthe following property for F � GF�� A f��g matrix has the propertywhen over IR it is totally unimodular�

Suppose we want to understand the matrices over F that are minimalviolation matrices of P� For any matrix A over F � de�ne M�A� to be thematroid represented by A over F � M�A� may be representable over F bya number of di�erent matrices A�� If F � GF�� then all such A� areobtainable from A by GF��pivots� By assumption� both A� and A haveP or they do not� Thus� it is well de�ned when we declare M�A� to haveP if A has P� The same conclusion applies when F � GF�� This timeany A� representing M�A� over GF�� may� by a slightly modi�ed proofof Lemma �� be obtained from A by GF��pivots and scaling withf��g factors� By the assumptions on P� thus A� and A have P or theyboth do not�

Note that series or parallel extensions ofM�A� maintain P� for such anextension corresponds to at most one F�pivot in A followed by adjoining of

�� Definition and Construction of Almost Regular Matroids ��

a row or column unit vector� By assumption� the latter operations maintainP�

We describe a �ve�step process that leads to insight into the minimalviolation matrices of P over F �

Step �� Let W be the class of minimal matrices over F that do not haveP� Assign to the indices of each matrix W W the following additionallabels� If an index z labels a row �resp� column� of W � then assign the label�con� �resp� �del�� For example� if the rows of W are indexed by X andthe columns by Y � then W with the labels is the matrix of �� below�From now on� we assume each W W to be so labeled� We also assumethat the elements of the matroid M�W � are labeled correspondingly�

��

del

del

W

Y

Xcon

con

...

. . .

Minimal violation matrix W with labels

The unusual �con� and �del� labels of W and M�W � tell the following� Ifelement z of the matroid M�W � has a �con� label� then z X� By theminimality ofW � deletion of row z X fromW results in a matrixW � withP� Correspondingly� M�W �� which is M�W ��z� has P� The �con� labelon z allows us to predict this outcome for M�W ��z� That is� �con�tractionof an element of M�W � with a �con� label produces a minor having P�Similarly� �del�etion of an element with a �del� label results in a minorhaving P� Note that we cannot tell from the labels whether for an elementz with �con� label the minor Mnz has P� Similarly� when z has a �del�label� we are ignorant about M�z having or not having P�

Collect in a setM� the matroidsM�W � with W W� For brevity� wecall an element with �con� �resp� �del�� label simply a �con� �resp� �del��element�

Step �� Establish elementary facts about the matrices W W� If possible�translate some of these facts into matroid language so that they apply tothe matroids of M�� Let E be the collection of such matroid facts�

Step �� At this time� we reverse the sequence of arguments� We use certainnecessary conditions satis�ed by the matroids of M� to de�ne a class M�

of matroids representable over F � The conditions for membership in M�

are as follows� Each M M� must not have P� Each one of its elementsmust be labeled �con� or �del� in such a way that a �con� �resp� �del��label on an element z of M implies M�z �resp� Mnz� to have P� Finally�M must satisfy the conditions of E� Clearly� M� is a subset of M��


Step �� EnlargeM� by adding for each member M all proper minors� Theelements of these minors are labeled in agreement with the labels of M �By de�nition� M� is now closed under minor�taking�

Step �� Analyze the structure of the matroids of M�� Specialize the con�clusions for M� to M�� Finally� translate the latter results to statementsabout the matrices W W�

The above procedure is of course more of a recipe than an algorithm�Let us demonstrate its use by applying it to the case where F is GF�� andP is regularity�

Step �� By Theorem �� we have a good understanding of a portionof the class N of minimal violation matrices of regularity� Indeed� at thispoint� only the subclass N� is poorly characterized� That class consistsof the binary support matrices of the minimal violation matrices of totalunimodularity with at least three �by Corollary �� at least four�nonzeros in some row and some column� Thus� we take W to be N�� Weassign �con� and �del� labels to the rows and columns� respectively� of thematrices of W� Then we de�ne M� to be the set of matroids M�W � withW W�

Step �� By Corollary �� each matrix of N� is Eulerian� That is�each W W as given by �� has an even number of �s in each rowand each column� This latter fact has a convenient translation into matroidlanguage� Each M M� has a base such that each fundamental circuit�resp� cocircuit� has an even number of �con� �resp� �del�� elements� Noweach circuit �resp� cocircuit� of M is the symmetric di�erence of some ofthese fundamental circuits �resp� cocircuits�� We conclude that each circuit�resp� cocircuit� has an even number of �con� �resp� �del�� elements� An�other fact� trivial yet important� is that M has at least one �con� elementand at least one �del� element� The preceding conditions on circuits� co�circuits� and labels we declare to be the collection E of matroid facts aboutM��

Step �� We reverse the arguments and de�neM� from facts known forM��Speci�cally� M� is the class of binary matroids M satisfying the followingconditions� First� M must be nonregular� Second� each element z of Mmust be labeled �con� or �del�� The �con� �resp� �del�� label must implythat M�z �resp� Mnz� is regular� Third� the circuits and cocircuits of Mmust obey the following parity condition� Each circuit �resp� cocircuit� isto have an even number of �con� �resp� �del�� elements� Fourth and last�the following existence condition must be satis�ed� There is to be at leastone �con� element and at least one �del� element� The matroids of M� sode�ned we call almost regular�

Step �� We enlargeM� by adding all possible minors� Each minor assumesthe labels of the matroid producing it� We claim that each minor so added


toM� is regular or almost regular� By duality and induction� we only needto consider a ��element deletion of an element z in an almost regular M �If Mnz is regular� we are done� So assume Mnz to be nonregular� A �con��resp� �del�� label on an element w �� z of M impliesM�w �resp� Mnw� tobe regular� Thus� for such w the minor �Mnz��w �resp� �Mnz�nw� of Mnzis regular as well� Since Mnz is nonregular� z must have a �con� label� By�� each circuit of Mnz is a circuit of M � and each cocircuit of Mnz isa minimal member of the collection fC��fzg j C� � cocircuit of Mg� Bythe parity condition� each circuit �resp� cocircuit� of M has an even num�ber of �con� �resp� �del�� elements� Since z has a �con� label� the sameconclusion applies to the circuits and cocircuits of Mnz� The parity andexistence conditions for M imply that M has at least two �con� elementsand at least two �del� elements� Thus� Mnz satis�es the existence con�dition� These arguments establish Mnz to be almost regular� They alsoprove that the enlarged M� is the class of almost regular matroids plustheir regular minors� As desired� M� is now closed under minor�taking�

Step �� We must analyze M�� Then we must specialize the results to M��Finally� we must express the latter results in matrix terminology to obtainconclusions about W� and hence about N��

In the remainder of this section� we carry out the tasks mandatedby Step �� We begin with some elementary facts about almost regularmatroids� Let M be such a matroid� We dualize M in the usual way�but also switch �con� �resp� �del�� labels to �del� �resp� �con�� labels�As usual� we denote that dual matroid of M by M�� The next lemmaestablishes M� to be almost regular and provides some additional factsabout M � A hyperplane of a matroid M is a maximal set of M with rankequal to the rank of M minus �� A cohyperplane is a hyperplane in thedual matroid M� of M �

�� Lemma� Let M be an almost regular matroid� Then the fol�lowing holds�

�a� The dual matroid M� of M is almost regular��b� Every nonregular minor of M is almost regular��c� The set of con� �resp� del�� elements of M is a cocircuit and a

cohyperplane �resp� a circuit and a hyperplane� of M �

Proof� �a� This is proved by routine checking of the de�nition of almostregular matroids given in Step � above��b� The proof is given under Step � above��c� Since M is nonregular� it has by Theorem �� an F� or F �

� minor�By duality and part �a�� we may assume presence of F�� The �con� and�del� labels for F� are unique up to isomorphism� as may be checked bya simple case analysis� Indeed� the matrix B� given by �� below is arepresentation matrix of F� with correct �con� and �del� labels�


�� condeldel

del

con

con

con

B7 =0 1 1 11 1 0 11 0 1 1

Matrix B� for matroid F�with �con� and �del� labels

Take B to be any representation matrix of M that displays an F� minor inagreement with B� of �� We claim that B can be partitioned as

��condeldel

del

del

con

con

con

con

B =

0 1 1 11 1 0 11 0 1 1

X1

Y1 Y2

X2 b

a

0/1

Matrix B displaying F� minor

Note that up to indices� B� is displayed in the upper left corner of B� Foreach x X�� the minor M�x is nonregular� Thus� x must have a �del�label� Similarly� for each y Y�� nonregularity of Mny implies a �con�label for y� So far� we have justi�ed all labels of B�

We claim that the subvectors a and b of B contain only �s� If thesubvector b has a �� say in row x X�� then the �s in that row of Bcorrespond to a cocircuit of M with exactly one �del� element� in violationof the parity condition� Similarly� a � in the subvector a contradicts theparity condition for a fundamental circuit of M �

At this point� we have shown that the �rst column and the �rst rowof B contain only �s except for the � in the �� position� This fact� plusthe given assignments of labels to B� implies that the elements with �con��resp� �del�� labels form a cocircuit and cohyperplane �resp� circuit andhyperplane� of M � as is easily con�rmed by direct checking�

We now review the construction of the almost regular matroids asdescribed in Section �� The starting point for the construction of analmost regular matroid is either F� or a ��element extension of R�� bothwith appropriate �con� and �del� labels� The representation matrix of F�we have already seen� It is B� of �� The matrix for the extension ofR�� we call B�� It is derived from the matrix of �� as follows� Wepermute the rows of the matrix of �� so that the last row becomes the


�rst one� To the resulting matrix� we adjoin a new leftmost column� Thenwe assign appropriate �con� and �del� labels� The matrices B� and B��

have already been listed under �� We repeat them here for readyreference�

��

del11

1

0condeldel

del

100

0 1 1 1 1 11

1

del

con

con

con

con

con

110

011

0 10 11 1

0 1 1B11 =

1 01 0

condeldel

0 1 1 11

del

con

con

con

1 11B7 = ;

Labeled matrices B� and B��

By now� the reader has acquired enough machinery� in particular that ofgraphs plus T sets of Section �� that he�she can quickly verify the ma�troid of B�� to be almost regular� Thus� we omit details of that check�

As stated above� one of the two matroids represented by B� or B�� isthe starting point for the construction of an almost regular matroid� Theconstruction itself consists of a sequence of series or parallel extension stepsand of triangle�to�triad and triad�to�triangle exchanges� The extensionsteps we call SP steps� and the exchanges steps� �Y exchanges� Theseoperations are controlled by rules that may be summarized as follows� Aparallel �resp� series� extension is permitted only if the involved elementz has a �con� �resp� �del�� label� The new element receives the samelabel as z� The �Y exchanges are depicted in terms of representationmatrices by �� The conditions on labels are summarized by�� Instead of covering all possibilities for the �Y exchange as done inSection �� here we show just one instance based on �� All other casescan be reduced by GF��pivots to the one displayed� In �� below�the matrix on the left represents the matroid with the triangle fe� f� gg�and the one on the right the matroid with the triad fx� y� zg�

��

1a

b b

con

gcon

e

0delf

replace triangleby triad

replace triadby triangle

delzdely a

b

a 10

con

x

BB

�Y exchange rule for almost regular matroids

In Section �� we de�ne a sequence of SP extensions and �Y exchangesunder the preceding rules to be a restricted �Y extension sequence� Thenext result justi�es our reliance on such sequences�

� Chapter �� Almost Regular Matroids

�� Lemma� Let a matroid M � be created from an almost regularmatroid by a restricted�Y extension sequence� ThenM � is almost regular�

Proof� By induction� we may assumeM � to be derived fromM in a singleSP extension or �Y exchange step� Consider the �rst case� Clearly� M �

has a minor isomorphic to M � Thus� M � is nonregular since M is almostregular� The parity condition for M � is readily veri�ed because of therestriction that a parallel �resp� series� extension is only permitted for a�con� �resp� �del�� element of M � For the same reason� for each �con��resp� �del�� element of M �� we have M ��z �resp� M �nz� regular� Clearly�both �con� and �del� labels occur in M �� Thus� M � is almost regular�

The �Y exchange is almost as easily proved� By Theorem ��such an exchange maintains regularity� Thus� by contradiction� M � is non�regular� The remaining conditions are readily veri�ed with the matrices of�� Thus� M � is almost regular�

The surprising fact is that restricted �Y extension sequences createall almost regular matroids from the two matroids given by B� and B��

of �� The precise statement is given in Theorem �� which werepeat here�

�� Theorem� The class of almost regular matroids has a partitioninto two subclasses� One of the subclasses consists of the almost regularmatroids producible by �Y extension sequences from the matroid repre�sented by B� of �� The other subclass is analogously generated byB�� of �� There is a polynomial algorithm that obtains an appropri�ate �Y extension sequence for creating any almost regular matroid fromthe matroid of B� or B�� whichever applies�

Proof� The existing proof is so long that we cannot include it here� Nev�ertheless� we sketch the main arguments� since they involve interesting ma�troid decomposition ideas involving the matroids R�� and R�� of Sections�� and ��

We take M to be a minimal almost regular matroid that cannot beproduced by restricted �Y extension sequences from any one of the twomatroids given by �� Put di�erently� M cannot be reduced to one ofthe two matroids by restricted �Y extension sequences� For short� we saythat M is not reducible�

By the minimality�M obviously cannot have series or parallel elements�In addition� it turns out that M cannot have a ��separation with at leastfour elements on each side� Indeed� such a ��separation implies a ��sumdecomposition ofM where one component is a wheel� That ��sum is readilyproved to be reducible� a contradiction of the minimality of M �

SupposeM does not have R�� or R�� minors� First� one can show thatM must have a �del� element z such that Mnz is graphic� Thus� M canbe represented by a graph plus T set� Rather complicated arguments then

�� Definition and Construction of Almost Regular Matroids �

prove M to be isomorphic to F� or F �

�with appropriate labels� A single

�Y exchange transforms the F �

�case to F��

Next� we assume that M has an R�� minor� Rather easily� we reachthe conclusion that M is isomorphic to the matroid of B�� or its dual� Thesecond case can be transformed to the �rst one by one �Y exchange� Theproof relies on the graph plus T set representation of Section �� Thatapproach also produces the insight� at present irrelevant but later useful�that any almost regular matroid with an R�� minor must be labeled in sucha way that the �con� elements do not form a base and the �del� elementsdo not form a cobase�

One case remains� where M has an R�� minor but no R�� minors�By duality and the symmetry of R�� we may assume that M has a �del�element z such that Mnz has an R�� minor� To investigate this case� weapply the recursive decomposition algorithm of Section �� starting withH � fR��g� The class of matroids under consideration is M�� de�ned inSteps � and � at the beginning of this section� In the �rst iteration of thedecomposition algorithm� we use the ��separation of R�� given by ��We �nd exactly two matroids that prevent induced ��separations� They areduals of each other� Let V�� be one of them� Thus� after the �rst iteration�we have H� � fV�� V �

��g� which is the set H for the next iteration� Therewe see that both V�� and V �

��induce certain ��sum decompositions� We

conclude the second iteration with H� � �� and stop the decompositionalgorithm�

We utilize the output of the decomposition algorithm as follows� Asargued at the beginning of the proof� M cannot have a ��separation withat least four elements on each side� In particular� the ��separation of anyR�� minor� as given by �� cannot induce a ��separation of M � Bythe results of the decomposition algorithm�M then has a V�� or V �

�� minor�

By duality� we only need to pursue the case where M has a V�� minor�We prove that one component of a certain ��sum induced by V�� is almostregular and is represented by a graph plus T set with special structure� Infact� that graph can be created by a particular �Y extension sequence� Forthat sequence� the last SP extension step plus the subsequent �Y exchangescan be viewed as �nal steps of a construction of M � thus contradicting theminimality of M �

Later we need the following observation made in the preceding proof�

�� Lemma� Let M be an almost regular matroid� Then M hasan R�� minor if and only if it is produced from the almost regular matroidof B�� by some restricted �Y extension sequence� Furthermore� if M hasan R�� minor� then the set of con� elements does not form a base of M �and the set of del� elements does not form a cobase�

At this point� it probably is not apparent how the matrix classes U �


A� B� V of Section �� are related to the class of almost regular matroids�The next section will quickly change that situation�

�� Matrix Constructions

Let us pause for a moment to assess our position� In Section �� we havede�ned the matrix classes U � A� B� and V� We have seen that knowledgeof just one of the classes U � A� or B permits easy construction of all others�In Section �� we have established an elementary procedure for creatingthe almost regular matroids from two initial matroids given by B� and B��

of �� In this section� we use these results to determine elementaryconstructions for U � A� B� and V� From B or V� we obtain the class N�

of Theorem �� Thus� we solve the characterization problem of theminimal binary nonregular matrices�

We could proceed in several ways� Particularly appealing appears tobe the following route� We �rst tie a subclass of U to the almost regularmatroids� Then we identify the remaining members of U � Finally� weconstruct U � A� B� and V � in that order� The �rst step is accomplished bythe following lemma�

�� Lemma� A binary matroid M is almost regular if and only ifM is represented by a binary nonregular matrix of the form

�� 1t

con

0

Y

con

del

y

X U1 B =~

xdel

Matrix B for almost regular matroid M

where U when viewed over IR is complement totally unimodular�

Proof� We start with the �only if� part� Let M be almost regular� Ac�cording to some arbitrary choice� partition the set of �del� elements of Minto a singleton set fxg and a remainder X� Similarly� partition the set of�con� elements into fyg and Y � By Lemma �� X fxg is a circuitand hyperplane of M � and Y fyg is a cocircuit and cohyperplane� Thesefacts implyXfyg to be a base ofM � and the corresponding representationmatrix to be B of �� for some f�� g matrix U �

We must show U to be complement totally unimodular� Since elementx of M has a �del� label� Mnx is regular� Thus� the column submatrixof B indexed by Y is regular� Indeed� because of the �s in row y of thatsubmatrix� the matrix U when viewed as real must be totally unimodular�

�� Matrix Constructions ��

By GF��pivots in column x or row y of B� we can transform U to allmatrices obtainable from U by complement operations� As just argued�each such matrix when viewed over IR must be totally unimodular� Thus�U is complement totally unimodular�

For proof of the �if� part� we reverse the above arguments� Thus�according to the nonregular matrix B and the complement totally unimod�ular matrix U � the matroid M is nonregular� and each of its �con� �resp��del�� labels indicates regularity upon a contraction �resp� deletion�� Theparity condition is easily con�rmed for the fundamental circuits and co�circuits displayed by B� Thus� that condition holds for all circuits andcocircuits� Since B is nonregular� it must have at least three rows andat least three columns� Thus� the existence condition on �con� and �del�labels is satis�ed� We conclude that M is almost regular�

Recall from Section �� that the class B consists of the binary matricesB of the form

��1t0

x Yy

XB =

U1

Matrix B of class B

where U when viewed over IR is complement totally unimodular� Lemma�� thus implies the following result for B�

�� Corollary� The representation matrices B of �� of thealmost regular matroids become upon removal of labels precisely the non�regular matrices of B�

Proof� This follows directly from Lemma �� and a comparison of Bof �� with B of ��

By Lemma �� and Corollary �� we have a characterizationof the subclass of U whose members produce the nonregular matrices of B�The next lemma establishes the structure of the remaining members of U �i�e�� those generating the regular matrices of B� To this end� we de�ne anynonempty square triangular matrix U satisfying for all i � j� Uij � �� tobe solid triangular� When we add parallel or zero vectors any number oftimes to such a matrix� we get a solid staircase matrix� A typical exampleof such a matrix is given below�

��0

1s..

Solid staircase matrix

We need an auxiliary result about solid staircase matrices�


�� Lemma� A f�� g matrix is a solid staircase matrix if and onlyif it has no �� identity as submatrix�

Proof� The �only if� part is elementary� The �if� part is proved by astraightforward inductive argument� One removes a row with maximumnumber of �s� invokes induction� then adds that row again for the desiredconclusion�

Here is the promised characterization of the regular matrices of B�

�� Lemma� A matrix B of B as given by �� is regular if andonly if the submatrix U of B is a zero matrix or solid staircase matrix� orbecomes a matrix of the latter type by some complement steps�

Proof� We start with the �if� part� We must show that any B of ��with U as speci�ed is regular� Evidently� this is so when U is a zeromatrix� So assume that by some complement steps� U becomes a solidstaircase matrix� According to the proof of Lemma �� any such stepscorrespond to GF��pivots in B� Thus� they maintain regularity� Seriesand parallel extensions of a binary matroid also retain regularity� Thus� wemay further assume that B has no parallel or unit vectors� By �� Uthen has no parallel or zero vectors� Thus� U is solid triangular� and B is

��1t0y

x Y

XB =

11

11s...

Matrix B with solid triangular U

We claim that B represents the graphic matroid of a wheel� with X fxgas set of rim edges� and Y fyg as set of spokes� The edges x and y aresuch that X fyg is a path� The claim is easily veri�ed� One only checksthat the fundamental cycles for X fyg are displayed by B of �� Weconclude that B is regular�

For proof of the �only if� part� we assume B of �� to be regular�Since B is in B� the submatrix U when viewed over IR is complementtotally unimodular� We are done if U is a zero matrix� So assume U tobe nonzero� We may assume that U has no zero or parallel vectors� Thus�we must show U or a matrix obtained from U by complement steps� to besolid triangular�

If U has a �� identity submatrix� say indexed byX � X and Y � Y �then B contains the matrix of �� below� A pivot on any � of the ��identity submatrix� plus a pivot in the resulting matrix on the � in the�x� y� position� produces a matrix that displays an F� minor� But then B


is not regular� a contradiction�

��1

0 01 10yx

11

1

00

10

1

01

Y

X

Nonregular submatrix of B inducedby a �� identity submatrix of U

Suppose U has a �� identity submatrix� If U is connected� then U

or its transpose contains the submatrixh � � ��

i� say indexed by X � X

and Y � Y � By duality� we may assume the former case� Then B containsthe matrix

�� 10 11 1

0yx

1

1

0 1

1

1

Y

X

Nonregular submatrix of B inducedby a certain �� submatrix of U

which represents an F � minor� and once more we have a contradiction�Still assume that U has a �� identity submatrix� As argued above� U

does not contain a �� identity submatrix and is not connected� Thus� Uhas exactly two connected blocks� neither of which contains a �� identitysubmatrix� By Lemma �� and by the absence of zero or parallelvectors from U � each block must be solid triangular� Let z index a row ofU with maximum number of �s� The row z complement of U is then solidtriangular� as desired�

Finally� assume that U has no � � � identity submatrix� By Lemma�� and by the absence of zero or parallel vectors� U is then solidtriangular�

We are ready to state and validate the following construction of thecomplement totally unimodular matrices� The construction relies on fourseemingly strange matrices� Their origin will become clear shortly�

�� Construction of U � De�ne real f�� g matrices U�� U�� U��and U�� as follows�

�� 0 ;U0= 1 ;U1= U11=

11 1

000

00 0

111

01

11

10

11

010 1 1

1U7= ;

Complement totally unimodularmatrices U�� U�� U�� and U��


Starting with U � U�� U�� U�� or U�� apply a sequence of operations eachof which is one of �i�� ii�� or �iii� below�

�i� Perform a row or column complement operation��ii� Add a zero or parallel row or column vector��iii� ��Y exchange� If U is one of the two matrices below� replace U by

the other matrix�

��1a

b b

0

U

a

b

a 10

U

�Y exchange for complementtotally unimodular matrices

Collect in sets U�� U�� U�� and U�� the matrices that can be so deducedfrom U�� U�� U�� and U�� respectively� These sets form a partition of U �

An example matrix constructed from U� is given by �� below�

Proof of Construction �� It is easy to see that the steps�i��iii� produce from U� � � � � the class of zero matrices� which thusconstitutes U�� Indeed� steps �i� and �iii� never apply� For validation ofthe remaining cases� we assign a �con� �resp� �del�� label to each column�resp� row� of any U produced by Construction �� from U�� U�� orU�� For the moment� these labels are purely formal� Next� we embed Uinto the matrix B of �� which we repeat here�

�� 1t

con

0

Y

con

del

y

X U1 B =~

xdel

Matrix B derived from matrix U

Steps �i��iii� of Construction �� are then equivalent to the follow�ing operations on B� The complement�taking of step �i� corresponds to aGF��pivot in row y or column x of B� The addition of zero or paral�lel vectors of step �ii� is the addition of parallel vectors or of unit vectorswith � in row y or column x� Finally� the exchange of step �iii� is the �Yexchange depicted by ��

We take these observations one step further� Let M be the labeledmatroid represented by B� We claim that the operations just speci�ed for B are equivalent to the operations of restricted �Y extension sequences


in M � Indeed� the addition of parallel vectors or of unit vectors with � inrow y or column x of B becomes the restricted SP extension for M � TheGF��pivot in B plus the exchange given by �� are equivalent to arestricted �Y exchange in M �

We interpret the matrices U of U�� U�� and U�� in terms of the relatedmatroid M � We start with U�� The matrix B of �� produced byU � U� � � � � is

��01 1

1

del

con

con

del

x

y

Y

X

Matrix B derived from U � U� � � � �

The correspondingM is the graphic matroid of the wheel with two spokes�Indeed� the spokes are labeled �con� and form the set Y fyg� The rimedges are labeled �del� and constitute the set X fxg� We interpret re�stricted �Y extension sequences for M as sequences of graph operationsapplied to the preceding graph� Each such operation is either a subdivisionof a �del� edge into two �del� edges� or an addition of a �con� edge parallelto a �con� edge� or an exchange of a triangle by a ��star� or an exchange ofa ��star by a triangle� In the latter two operations� the labels are assignedaccording to �� which we repeat below�

�� xcon

econ

gcon

f del

y del z del

Triangle and ��star with labels

We leave it to the reader to verify that any graph produced by these op�erations is obtainable from some wheel graph with at least two spokes bysubdivision of rim edges and addition of edges parallel to spokes� In termsof M � the rim edges of such an extended wheel form the set X fxg� andthe spokes constitute Y fyg� We interpret this result in terms of U as inthe proof of Lemma �� Thus� we see that U or some matrix obtainedfrom U by complement operations� is a solid staircase matrix�

We turn to the cases of U� and U�� The matrix U� �resp� U��produces as B the matrix B� �resp� B�� of �� In either case� Bis nonregular� and M is an almost regular matroid� By Theorem ��restricted �Y extension sequences produce all almost regularmatroids from


the two matroids represented by B� and B�� By Lemma �� andCorollary �� the sets U� and U�� thus contain the complement totallyunimodular matrices that produce the nonregular members of the class B�

By Lemma �� and the above characterization of U� and U�� theclass U�U� contains precisely the complement totally unimodular matricesU that produce the regular matrices B of B� We have also seen that theclass U� U�� generates the nonregular matrices B of B� Thus� U� U� U�U�� is equal to U � The sets U�U� and U�U�� are necessarily disjoint�Evidently� this is also so for U� and U�� Theorem �� says that anyalmost regular matroid can be generated from exactly one of the matroidsrepresented by B� and B�� Thus� U� and U�� are disjoint� We concludethat U�� U�� U�� and U�� form a partition of U as claimed by Construction��

Construction �� has two interesting corollaries�

�� Corollary� Let U be a nonempty complement totally uni�modular matrix without zero or parallel vectors� Then U � or some matrixobtainable from U by complement operations� contains a unit vector�

Proof� We use the notation of the proof of Construction �� By theassumptions� U is in U� U� U�� Let U de�ne B of �� and Mbe the labeled matroid represented by B� Now U has no zero or parallelvectors� Thus� M has no series or parallel elements� M is obtained fromthe graphic matroid of �� or from the almost regular matroid of B�

or B�� by a restricted �Y extension sequence� In all three cases� the initialmatroid has a triangle with two �con� labels and one �del� label� and the�nal matroid has a triangle with two �con� labels or triad with two �del�labels� By duality� we may assume M to have a triangle C with two �con�labels� In the notation for B of �� assume x � C and y C� Thus�y is one of the two �con� elements of C� The second �con� element of Cmust be in Y � The third element of C� with �del� label� must be in X�Then column z Y of B has exactly two �s� one of which is in row y�Thus� column z of U is a unit vector� If x C or y � C� we can achievethe desired con�guration by GF��pivots in column x or row y of B� Thepivots correspond to complement operations for U � Thus� the corollaryholds in all cases�

�� Corollary� If U U is IR�nonsingular� then U is in U� or U��but not in U� or U��

Proof� Candidate classes are the claimed ones and U�� Let U U��and let M be the matroid de�ned via B of �� By the proof ofConstruction �� M is an almost regular matroid produced by somerestricted �Y extension sequence from the matroid represented by B��Lemma �� says that the set Y fyg of �con� elements of M doesnot form a base� and the set X fxg of �del� elements does not form a


cobase� By assumption� U is IR�nonsingular� Since U is complement totallyunimodular� U when viewed as binary is GF��nonsingular as well� Butthen� by B of �� the set Y fyg is a base ofM � a contradiction�

At this point� it is a simple matter to construct the classes A� B� andV� For completeness� we include details� We start with A and B� Recallthat A �resp� B� is the class of almost representative matrices over GF��resp� GF��

�� Construction of A and B� The classes A and B are deducedfrom U as follows� Each matrix of A �resp� B� is precisely a matrix of theform

��1tα

U1

Matrix of A or B

For A �resp� B�� the matrix is over GF�� resp� GF�� with � � ��resp� � � �� The submatrix U is a f�� g matrix that� when consideredover IR� is in U �

Proof� Validity of the construction follows directly from the de�nition ofA and B following Lemma ��

Next we give a construction for V� the class of minimal violation ma�trices of total unimodularity�

�� Construction of V� The class V is deduced from U as follows�Each matrix V of V is up to scaling by f��g factors of the form

�� b = U-1 α = U-1 − 2. 1.1t

a = U-1

;U-1

aα

b. 1

.1t

Matrix V up to scaling

The matrix U is a square IR�nonsingular matrix of U� or U�� If U U��then V has exactly two nonzeros in each row and in each column indeed�the bipartite graph BG�V � is a cycle� If U U�� then V has at least fournonzeros in some row and in some column�

Proof� Validity of the construction follows from Corollaries �� and��

Corollary �� and Construction �� yield an interesting re�sult�

� Chapter �� Almost Regular Matroids

�� Corollary� Every matrix V V has a row or column withexactly two nonzeros�

Proof� The inverse of V is� up to scaling by f��g factors� given by Vof �� The U�� occurring in V of �� is the inverse of the Ude�ning V of �� A simple check con�rms that replacement of Uby a matrix deduced from U by complement operations e�ectively may beviewed as scaling in V � Thus� up to scaling� U and all matrices obtainablefrom U by complement operations produce the same V � Since U is nonsin�gular� it cannot contain zero or parallel vectors� By Corollary �� U �or some matrix deduced by complement operations from U � contains a unitvector� Let U � be that matrix� The inverse of U � contains a unit vector aswell� By �� and the above discussion� V must have a row or columnwith exactly two nonzeros�

At long last� we can �ll the gap left by Theorem �� and com�plete the characterization of minimal nonregular submatrices� The desiredtheorem is as follows�

�� Theorem� Let N be the class of binary minimal violationmatrices of regularity� Then N has a partition into three subclasses N��N�� N� as follows�

�a� N� �resp� N�� is the set of binary matrices B for which BG�B� is agraph of type H� �resp� H�� of ��

�b� N� is the set of binary support matrices of V V produced via�� from the IR�nonsingular matrices U U��

Proof� Part �a� is taken from Theorem �� Part �b� follows fromthat theorem and Construction �� of V�

The above constructions are readily performed by hand� That wayone can rapidly produce structurally interesting matrices� An example of acomplement totally unimodular matrix U and a minimal violation matrixV of total unimodularity generated that way is as follows� We �rst obtainby Construction �� the following complement totally unimodularmatrix U �

��

1aghjkd

1 1 1

U =

fbiz e c1 1

0 1 0 1 1 10 0 11 1 1

0 0 0 0 1 11 0 11 0 1

1 0 01 0 1

Complement totally unimodular matrix Urepresenting R��

�� Matrix Constructions �

The labels may be used to verify that U represents the regular matroidR�� of �� One only needs to check via the graph plus T set of�� the fundamental circuits for the base fa� d� g� h� j� kg� which is theindex set of the rows of U � The indicated partition of U corresponds tothe ��separation �fa� b� g� h� i� zg� fc� d� e� f� j� kg� of R�� Straightforwardcomputations con�rm that U is IR�nonsingular� and that up to scaling byf��g factors and row and column exchanges� U is its own inverse� Thus�for this special case� we are tempted to use U instead of U�� in the formulafor V of �� But U�� is a scaled and permuted version of U � Becauseof the scaling� the formula �� cannot be used� But we may rely onpart �b� of Corollary �� According to that result� V is Eulerian andits entries sum in IR to ��mod �� Using these two facts� we determine thefollowing V from U �

��

111

00000

0 0 0 0 01 1 1a

ghjkd

V =

fbiz e c

1 10 1 0 1 1 10 0 11 1 1

0 0 0 0 1 11 0 11 0 1

11 0 01 0 1

Matrix V deduced from U of ��

The above example supports the following result�

�� Lemma� The regular matroidR�� has a real nonsingular f�� grepresentation matrix that is complement totally unimodular�

Construction �� and the matrix V of �� permit us torelate V to the regular matroids R�� and R�� of �� and �� asfollows�

�� Lemma� Let M be the binary matroid represented by thebinary support matrix W of a minimal non�totally unimodular matrix V V� Then M does not have R�� minors� but may have an R�� minor�

Proof� By Theorem �� M is represented not just by W as stated�but also by some B B given by �� Indeed� the submatrix U�� ofV of �� is the inverse of the matrix U de�ning B� By Construction�� for V� we have U U� U�� Thus� U � U�� By the proof of Con�struction �� for U � U � U�� implies that M has no R�� minor� Thematrix V of �� demonstrates that M may have an R�� minor�

Lemma �� implies the claim made toward the end of Section ��that none of the classes U � A� and B can be produced from V by the matrixoperations of scaling by f��g factors� pivots� submatrix taking� and change


of �elds from IR to GF�� or GF�� In particular� the matrix U�� of�� cannot be obtained that way�

In the last section� we discuss applications and extensions� and includereferences�

�� Applications Extensions and

References

The entire Section �� is taken from Truemper ��b�� Variations of thegraph signing problem of Section �� are treated in Zaslavsky ��a��b�� The notion of ��balancedness ex�tends the concept of f�� g balanced matrices covered extensively elsewhere�see Berge �� Fulkerson� Ho�man� and Oppenheim�� and Anstee and Farber �� Profound decomposition results forbalanced matrices are proved in Conforti and Rao �� a��d��Conforti and Cornu!ejols �� and Conforti� Cornu!ejols� and Rao ��In Truemper and Chandrasekaran �� balancedness of f�� g matricesis tied to total unimodularity by the exclusion of certain minimal violationmatrices of total unimodularity� That result motivated the search for aconstruction of V�

A f�� g matrix is perfect if the polyhedron fx j A x � �� x � �ghas only integer vertices� A graph is perfect if its clique�node incidencematrix is perfect� Any f�� g balanced matrix is perfect� but the conversedoes not hold� The so�called strong perfect graph conjecture says that theminimal nonperfect graphs� and thus the minimal nonperfect matrices� havea certain simple structure �see� e�g�� Padberg �� There are numerouspartial results with regard to that conjecture� Approaches based on graphdecomposition have been cited in Section �� Padberg �� contains avery interesting matrix�based attack on the problem�

The concept of complement total unimodularity of Section �� is de��ned in Truemper ��b�� Almost representative matrices are character�ized in Truemper ��b�� A number of papers explicitly or implicitly in�clude properties of the class V �see Ghouila�Houri �� Camion �� a�� b�� Chandrasekaran �� Gondran �� Padberg �� Tamir �� Kress and Tamir �� Truemper �� b�� b�� and de Werra �� But none of the results capturesthe complexity of V� Indeed� the cited results do not contain a single clueto how one might construct even a small subset of structurally di�erentmatrices of V�

Section �� relies on Truemper ��a�� b�� The analysis tech�nique applies not just to the cited properties� but also to certain repre�sentability questions� An important case is treated in Truemper ��b��

�� Applications Extensions and References ��

where the minimal violation matrices of abstract matrices not representableover GF�� or GF�� are characterized�

Section �� is entirely based on Truemper ��b�� That referencealso contains alternate constructions using pivots� The constructions donot have descriptions as brief as the ones included here� but they are wellsuited for hand calculations� Truemper ��b� also includes a proof thatevery ��connected almost regular matroid di�erent from F� and F �

�has a

binary representation matrix that is balanced� An example is the matrixV of �� The proof relies on an extension of an ecient algorithm ofFonlupt and Raco �� that proves existence of a balanced binary repre�sentation matrix for any regular matroid� The algorithm is a modi�cationof a scheme due to Camion �� where the latter existence result was�rst established�

Chapter ��

Max�Flow Min�Cut Matroids

�� Overview

Concepts� theorems� or algorithms in one area of mathematics often inspirenew approaches in another area� In turn� the ensuing new developments inthe latter area may lead to new ideas in the former one� In this chapter�we describe an interesting instance of this cyclic process�

We start with the max �ow problem for undirected graphs� which maybe de�ned as follows� Given is a connected and undirected graph G� Oneof the edges of G� say l� is declared to be special� To each edge e of G otherthan l� a nonnegative integer he is assigned and called the capacity of e�De�ne G to have �ow value F if there are F cycles� not necessarily distinct�that satisfy the following two conditions� Each cycle of the collection mustcontain the special edge l� and any other edge e of G is allowed to occuraltogether in at most he of the cycles� The max �ow problem demands thatone solve the problem maxF � The solution value must be accompanied bya collection of cycles producing that value�

A companion of the max �ow problem is the following min cut problem�For any cocycle D of G containing the special edge l� de�ne the capacity ofD to be the sum of the capacities of the edges in D other than l� Denote thecapacity of D by h�D�� The min cut problem asks one to solve minh�D��The solution value must be accompanied by a cocycle D containing theedge l and having the solution value as capacity�

The famous max��ow min�cut theorem for graphs says that maxF minh�D� no matter how the nonnegative integral edge capacities are se�lected�

��


The max �ow and min cut problems for graphs have obvious matroidtranslations� In the above description� one replaces the graph G by a ma�troid M and speci�es elements instead of edges� and circuits and cocircuitsinstead of cycles and cocycles� One might conjecture that the max��owmin�cut theorem still holds in the expanded setting� But this is not so ingeneral� Counterexamples can be produced with small matroids� in partic�ular with U�

�� the rank uniform matroid on four elements� and with F �

��

the Fano dual matroid� Indeed� because of the symmetry of U�

�as well as

of F �

�� the conjecture is false for these matroids no matter which element

is declared to be special� In general� there are matroids where the equalitymaxF minh�D� does or does not hold for all capacity vectors h� depend�ing on the selection of the special element� Thus� we are motivated to de�nea matroid M with a special element l to have the max��ow min�cut prop�

erty� or to be a max��ow min�cut matroid� if maxF minh�D� no matterwhich nonnegative integral values are assigned as capacities� Absence orpresence of the max��ow min�cut property for M is evidently governed bythe connected component of M containing the element l� Thus� for the pur�poses of characterizing the max��ow min�cut matroids� we might as wellrestrict ourselves to connected matroids�

Using an ingenious but complicated induction hypothesis� Seymourproved in a long paper that the connected max��ow min�cut matroids areprecisely the connected binary matroids where the special element l is notcontained in any F �

�minor� In this chapter� we prove Seymour�s result using

a quite di�erent approach� Let us de�ne M to be the class of connectedbinary matroids where each matroid has a special element l such that l isnot contained in any F �

�minor� In Section �� we establish � and ��sum

decomposition theorems for the matroids of M� In Section �� we usethese theorems to prove Seymour�s result that M is precisely the class ofconnected max��ow min�cut matroids� In Section �� we use a part of theproof to validate a construction for M that involves certain � and ��sums�We show that the construction can be determined for any matroid of Min polynomial time� and thus conclude that one can test for the max��owmin�cut property of binary matroids in polynomial time� We also describepolynomial algorithms for the following problems involving the matroidsof M� the max �ow problem� the min cut problem� and a certain shortestcircuit problem�

In Section �� we examine an interesting graph application of theabove results for M� Let H be an undirected graph each of whose edgesis declared to be odd or even� Recall that K� is the complete graph onfour vertices� Declare a K� minor of H to be an odd�K� minor if it hasa certain property that is de�ned via the relative position of its even andodd edges� Then let G be the class of �connected graphs without odd�K�

minors� The graphs of G have pleasant properties� so an understandingof their structure is desirable� It turns out that the graphs of G can be

�� Chapter �� Max�Flow Min�Cut Matroids

linked to the class M� As a result� we can apply the cited constructionfor M to understand the structure of the graphs of G� In that way� weobtain a construction for the graphs of G� At the same time� we produce apolynomial test for membership in G� Evidently� we have moved from themax��ow min�cut property of graphs to the max��ow min�cut matroids�and then to the graphs without odd�K� minors� Thus� we have an instanceof the cyclic process mentioned in the introductory paragraph�

The �nal section� �� contains additional applications� extensions�and references�

The chapter makes use of Chapters � �� and �� It is also assumedthat the reader has a basic knowledge of linear programming� Relevantreferences are included in Section ��

�� Sum and Delta�Sum Decompositions

Recall that M is the class of connected binary matroids where each matroidhas a special element l such that l is not contained in any F �

�minor� In

this section� we �rst show that any �separable matroid of M has a certain�sum decomposition where both component matroids are also in M� Thenwe establish that any ��connected nonregular matroid of M has a particular��sum decomposition where again the components are in M� These tworesults will be used in the next section to show that M is precisely theclass of max��ow min�cut matroids� Before proceeding� the reader maywant to review brie�y the results for � and ��sums in Sections �� and�� respectively�

We begin with some de�nitions� By symmetry� there is essentially justone way to declare an element of the Fano matroid F� or of its dual F �

�to

be the special element l� When this is done� we get the matroid F� with lor F �

�with l� Suppose two binary matroids M and M � contain the element

l� If an isomorphism exists between M and M � that takes the element lof one of the matroids to l of the other one� then the two matroids arel�isomorphic� Finally� we emphasize that l is always the special element ofany matroid in M� Note that F� with l is in M� while F �

�with l is not�

We are ready for the detailed discussion of the � and ��sum resultsfor M�

��Sum Decomposition

The structure theorem for the �sum case is as follows�

�� Theorem� Any �separable matroid M � M on a set E hasa �sum decomposition where both components M� and M� are connected


minors ofM � contain l� and thus are inM� In addition�M� has an elementy �� l so that any set C � E is a circuit of M with l if and only if �i� or �ii�below holds�

�i� C is a circuit of M� with l but not y��ii� C �C� � flg� � �C� � fyg� where C� is a circuit of M� with l� and

where C� is a circuit of M� with both l and y�

Proof� Using the results of Section �� for �separations and �sums� aswell as the path shortening technique of Chapter �� one readily shows thatany �separable M � M has a representation matrix B of the form

� ��

Y1y

A2

B =

A1

Y2

l

X1

X2

0

0

1all1s

Matrix B of M � M with exact �separation

Note the position of the element l in X�� and the indicated element y � Y��By Section �� M is a �sum where both components M� and M� areconnected minors of M � have the element l� and are represented by thematrices B� and B� of � �� below� Routine arguments using B� B�

and B� con�rm the claims of the theorem concerning the circuits of M �M�� and M��

� ��A2B2 =

y Y2l

X210

Y1y

B1 = A1

l

X1

0 1 1

1

Matrices B� and B� of �sum decomposition of M

Delta�Sum Decomposition

We turn to the much more challenging case where M is ��connected� InSection �� it will be shown that any regular matroid of M has the max��ow min�cut property� Thus� we concentrate here on the situation whereM is not regular� For that case� one can prove M either to be isomorphicto F�� or to have a particular ��sum decomposition� The precise statementis as follows�


�� Theorem� Any ��connected nonregular matroid M � M on aset E is isomorphic to the Fano matroid F�� or has a ��sum decompositionwhere the components M� and M� are connected minors of M � contain l�and thus are inM� In the ��sum decomposition� both connecting trianglesof M� and M� contain l� Let the remaining elements of the connectingtriangle of M� �resp� M�� be a and b �resp� v and w�� Then any set C � Eis a circuit of M with l if and only if �i�� ii�� iii�� or �iv� below holds�

�i� C is a circuit C� of M� with l but without a and b��ii� C is a circuit C� of M� with l but without v and w��iii� C �Ca � fag� � �Cv � fvg� where Ca is a circuit of M� with l and

a but without b� and where Cv is a circuit of M� with l and v butwithout w�

�iv� C �Cb � fbg� � �Cw � fwg� where Cb is a circuit of M� with l andb but without a� and where Cw is a circuit of M� with l and w butwithout v�

The proof of Theorem � �� takes up the remainder of this section�We proceed as follows� First� we show that any ��connected nonregularM � M is isomorphic to F� or has a minor that is l�isomorphic to thematroid N� of M de�ned by the matrix B� below�

� ��

X2

11u

0

v wY2Y1

0

1bc

da

1

1

11

l

01

1

10

01

B8=

X1

Matrix B� for the matroid N�

Evidently� �X��Y��X��Y�� is a ��separation of N�� Note that Y� containsjust the element l� It is easy to con�rm that N� is indeed in M�

Next� we establish that M has a minor N with the following prop�erties� N contains l and is l�isomorphic to N�� and a ��separation of Ncorresponding to �X� �Y��X� � Y�� of N� under one of the l�isomorphismsinduces a ��separation of M � From that induced ��separation of M � we�nally derive the ��sum decomposition claimed in Theorem � ��

Induced ��Separation

We begin the detailed discussion� The �rst lemma deals with the presenceof N� minors in M �

�� Lemma� Let M be a ��connected nonregular matroid of M�Then M is isomorphic to the Fano matroid F�� or has a minor that isl�isomorphic to N��


Proof� If M has seven elements� then it must be isomorphic to F� sincemembership in M rules out F �

�� So assume that M has at least eight

elements� If M does not have any F �

�minors� then by the splitter result of

Lemma � �� M has an F� minor and is �separable� or is regular� Theassumptions made here rule out these cases� Thus� M has an F �

�minor�

say N � Obviously� the minor N does not contain the element l�According to Lemma �� M has a connected N � minor that is a

�element extension of N by l� If in N � the element l is parallel to or inseries with some other element� then clearly l is part of an F �

�minor of

N � and thus of M � a contradiction� Thus� N � is ��connected� Now F �

�has

no ��connected �element expansion� so N � is obtained from N by additionof l� Routine calculations con�rm that up to l�isomorphism� there is justone addition case that does not have an F �

�minor with l� That case is

represented by the matrix of � ��

From now on� we assume that M is a ��connected nonregular matroidof M with at least eight elements� By Lemma � �� M has a minor thatis l�isomorphic to N�� For one such minor� we will exhibit a ��separationthat corresponds to the ��separation �X� � Y��X� � Y�� of N� under one ofthe l�isomorphisms� and that induces a ��separation of M � For this task�we invoke Corollary �� and Theorem �� Recall that Corollary�� contains su�cient conditions for induced k�separations� while The�orem �� extends those conditions to the case where M has a specialsubset L of elements� For the case at hand� we de�ne L to be the setcontaining just l� We combine Corollary �� and Theorem �� forthis special set L to the following theorem�

�� Theorem� Suppose a ��connected N given by the matrix BN

� ��

Y1 Y2

X1BN =

A1

DX2 A2

0

Partitioned matrix BN for N

is in M and has l � �X� � Y�� Assume that �X� � Y��X� � Y�� is a k�separation of N � Furthermore� assume that N��X� � Y�� has no loops andthat Nn�X� � Y�� has no coloops� Finally� assume for every ��connected �element extension of N in M� say by an element z� that the pair �X� �Y��X� � Y� � fzg� is a k�separation of that extension� Then for any ��connected matroid M � M with a minor l�isomorphic to N � the followingholds� Any k�separation of any such minor that corresponds to �X� �Y��X� � Y�� of N under one of the l�isomorphisms induces a k�separationof M �

� Chapter �� Max�Flow Min�Cut Matroids

We use Theorem � �� plus the recursive decomposition scheme ex�plained in Section �� to establish ��separations for the nonregular ��connected matroids of M� We do not repeat here details of the decom�position scheme� so the reader may want to review that material beforeproceeding� The �rst iteration of that scheme is e�ectively accomplishedby the following lemma�

�� Lemma� Let M be a ��connected matroid of M with N� asminor� Then the ��separation �X� �Y��X� �Y�� of N� as given by � ��induces a ��separation of M � or M has a minor that is l�isomorphic to thematroid N� represented by B� below�

� ��

l

11u

0

v w zY2Y1

0

1bc

da

1

1

11

01

1

10 0

0

0 111

B9=

X1

X2

Matrix B� for the matroid N�

Proof� We assume absence of the speci�ed N� minors in M � and applyTheorem � �� The matroid N� plays the role of N of that theorem�Since inN� we have Y� flg� we have l � �X��Y�� It also is easily checkedthat N� satis�es the conditions of Theorem � �� involving N��X� �Y�� and Nn�X� � Y�� For veri�cation of the remaining conditions of thetheorem� we must compute the ��connected �element extensions N � of N��say by an element z� such that in N � the pair �X� � Y��X� � Y� � fzg� isnot a ��separation� We are done by contradiction once we show that anysuch case of N � is l�isomorphic to N� or is not in M�

We �rst consider the addition of z� For this� we adjoin to B� of� �� a column �g�h� representing the element z� The partition of �g�h�corresponds to that of B�� By assumption� �X� � Y��X� � Y� � fzg� is nota ��separation of N �� Thus� we must have g � By the ��connectednessof N �� the vector h must have one or three s� Routine checking con�rmsthat all such instances are l�isomorphic to N�� We turn to the expansionby z� We adjoin to B� of � �� a row �e j f � representing the element z�The partition of �e j f � corresponds to that of B�� The conditions imposedon z demand that e has one or three s and that f � In each case�the element l of N � can be placed into an F �

�minor� and thus is not in

M�

In the terminology of the decomposition scheme of Section �� wehave just completed the �rst iteration� We begin the second iteration byanalyzing N� for ��separations� That matroid has several such separations�some of them useful for our purposes� and others not� Indeed� in the initial


investigation into the class M� a ��separation of N� was selected that led toa large number of iterations of the decomposition scheme� But eventuallya better choice was found� It is the ��separation �fa� b� d� ug� fc� l� v� w� zg��It can be displayed in the accustomed format once we compute the repre�sentation matrix for N� corresponding to the base fb� c� v� wg� That matrixfor N� is as follows�

� ��

l

11a

0

d u z

0

1vc

bw

1

1

11

01

1

10 1

0

0 011

Matrix for N� corresponding to base fb� c� v� wg

The second iteration of the decomposition scheme is accomplished by thefollowing lemma�

�� Lemma� Let M be a ��connected matroid of M with N� asminor� Then the ��separation �fa� b� d� ug� fc� l� v� w� zg� of N� induces a��separation of M �

Proof� The arguments are virtually identical to those for Lemma � ��except that this time each ��connected �element extension satis�es theconditions of Theorem � �� or is not in M� We leave it to the readerto �ll in the details�

Lemmas � �� and � �� imply the following theorem�which thus is the result of two iterations of the decomposition scheme�

�� Theorem� Let M be a ��connected nonregular matroid of Mwith at least eight elements� Then M has a minor N with the followingproperties� N contains the element l and is l�isomorphic to N�� and a ��separation of N corresponding to the ��separation �X� �Y��X� �Y�� of N�

under one of the l�isomorphisms induces a ��separation of M �

Proof� By Lemma � �� M has a minor that contains l and that isl�isomorphic to the matroid N�� We may suppose that N� itself is thatminor� Then by Lemma � �� M has a ��separation induced by the��separation �X� � Y��X� � Y�� of N�� or M has a minor with l that isl�isomorphic to the matroid N�� In the former case� we are done� In thelatter case� we apply Lemma � �� Accordingly� M has a ��separationinduced by the ��separation �fa� b� d� ug� fc� l� v� w� zg� of N�� From thematrix of � �� for N�� we now delete the last column� Evidently� up toindices other than l� a matrix for N� results� The matroid N representedby that matrix is thus l�isomorphic to N�� Furthermore� by the derivationof N from N�� the ��separation �fa� b� d� ug� fc� l� v� wg� of N also induces

� Chapter �� Max�Flow Min�Cut Matroids

the ��separation of M derived earlier from N�� But that ��separation of Ncorresponds to the ��separation �X� � Y��X� � Y�� of N�� Thus� the caseinvolving N� also leads to the desired conclusion�

From ��Separation to Delta�Sum

The reader probably anticipates that the conversion of the just�proved ��separation of M to the ��sum decomposition claimed in Theorem � ��is straightforward� Unfortunately� the situation is not quite as simple� Inparticular� we must obtain some insight into the position of l relative tothe ��separation of M before we can proceed to the ��sum decomposition�We obtain this insight next�

By � �� the minor N N��c of N� is l�isomorphic to the Fanomatroid F� with l� Indeed� the groundset of N is fa� b� d� l� u� v�wg� andthe ��separation �fd� u� v�wg� fa� b� lg� of N induces the same ��separationin M that N� induces�

From now on� we do not need the matroid N� any more� Thus� we canswitch notation� and can utilize the sets X�� X�� Y�� and Y� so far employedfor N� to denote induced ��separations of M � We do this next in a repre�sentation matrix B of M � That matrix displays the just�de�ned F� minorN � with indices a� b� d� u� v� w� l and ��separation �fd� u� v�wg� fa� b� lg��

� ��

u v

11

A2D

B =

A1

Y1 Y2w l

ad

b

X1

X2

0

1 110 111 01

0

Matrix B of M displaying Nand induced ��separation �X� � Y��X� � Y��

To capture the role of l in the ��separation �X� � Y��X� � Y�� of M � wede�ne l to straddle the ��separation if a shift of l from Y� to Y� results inanother ��separation of M � The next theorem says that l must straddlethe ��separation� This fact will be essential for the ��sum decompositionto come�

�� Theorem� Let a ��connected matroidM � M be representedby the matrix B of � �� where �X� � Y��X� � Y�� is a ��separation ofM � Then the element l straddles that ��separation�

Proof� Evidently� the element l straddles the ��separation of M if andonly if column l of the submatrix A� of B and column u of the submatrix


D of B are parallel� The former column vector we call gl� and the latterone gu� Note that gu is the sum �in GF�� of the columns v and w of D�

Assume that l does not straddle the ��separation� Thus� gl and gu

are not parallel� Equivalently� one of these vectors contains a in a rowx � X� where the other one has a �� Thus� two cases are possible�

In the �rst case� glx � and gux � Then the submatrix of B indexedby a� b� d� x and l� u� v� w is either

� ��

x

l

11u

0

v w

1

1b

da

1

1

10

01

1

10

01

Submatrix of B

or is obtained from the matrix of � �� by exchanging the row indicesa and b� and the column indices v and w� Thus� we may assume that thesituation depicted by � �� is at hand� Let M be the minor representedby the matrix of � �� The minor Mnb of M turns out to be an F �

�

minor with l� and thus M �� M� a contradiction�In the second case� glx and gux �� If row x of D is nonzero� or

equivalently� if row x has s in both columns v and w� then rows a� b� d� xand columns l� v� w prove l to be in an F �

�minor� a contradiction� Thus�

row x of D is zero� By this fact and by the completion of the �rst case� wecan narrow down the second case as follows� There is a nonempty subset

X� � X� such that for all x � X�� we have glx while row x of D is

equal to �� Furthermore� for all x � �X� �X�� we have glx gux �Let X� be the index set of the nonzero rows of D� By the above

de�nition of X�� we know X� � �X��X�� We now redraw B of � �� by enlarging and repartitioning the submatrices A� and D� The revised Bis as follows�

� �� column≠ 0, g= 0 or

= g10 1

01

B =

A1

l

X1

Y1

Y2

X2

0

0

00 1

111

X2

Y2

g g

1

1

...

0

0

...

X2

Y2

0 1

eachcolumn

each

Repartitioned matrix B


The partition of the rows of B of � �� is induced by the sets X�

and X� introduced above� The partition of Y� into the sets l� Y �� and Y �

is based on the information given in the submatrices indexed by X�� Y �

and X�� Y �� Thus� each column of the former submatrix is zero or parallelto the subvector g of column l� By the de�nition of X�� the subvector galso occurs in D� as shown�

Suppose Y � �� Then it is readily checked via B of � �� that X��Y��X� is one side of a �separation of the ��connected M � a contradiction�

Thus� Y � � ��Derive a matrix B from B by deleting all s in positions �x� y� where

x � X� and y � �flg � Y �� Consider the bipartite graph BG�B�� If that

graph does not have a path from l to Y �� then arguments analogous tothose of the proof of Lemma �� prove M to be �separable� Thus�

a path exists from l to some y � Y �� Because of path�shortening pivots�we may assume that the path has exactly two arcs� Thus� we can extract

from B the following submatrix B�

� ��

u

10 1

hab

v w l y

00

0

01

11

111

X2g g

Submatrix B of B

where h � � and h � g� ReduceX� to a minimal set containing a and b suchthat the just�mentioned condition is still satis�ed by the correspondinglyreduced g and h� Thus� jX�j or �� Since g has two s in rows a andb� we have jX�j � if and only if h has either two or no �s in rows aand b� If h is a unit vector� then a pivot on the of h plus deletion ofthe pivot column produces a previously resolved instance where glx � guxand where a row x of D is nonzero� Otherwise� jX�j �� and h has two s in rows a and b� In each one of the three possible cases� a pivot in hplus deletion of the pivot column and of one row proves that l is in an F �

�

minor� a contradiction�

As described in Section �� a ��sum decomposition is derived from a��sum decomposition via a certain �Y exchange� We �rst carry out the��sum decomposition of M using the matrix of � �� except that thistime we ignore the column index u and the details of that column� Weare guided in that decomposition by the matrices of �� and �� Below� we list the matrix B for M � as well as the representation matricesB� and �B� for the components M� and �M� of the ��sum decomposition ofM �


� ��

l

11

A2D

B =

A1

Y1 Y2v w

ad

b

X1

X2

0

1 10 11 0

0

Matrix B for M

� ��

l

11

B1 =A1

Y1v w

ad

b

X1

1 10 11 0

0

0

0 1

11

A2

lY2

v w

ad

bX2

01 10 11 0

0

B2 =~

0 1

Matrices B� and �B� of ��sum decomposition of M

The ��sum decomposition has M� and a second matroid� say M�� as com�ponents� whereM� is derived from �M� by the exchange of the triad fd� v�wgwith a triangle� say Z� according to the rule of �� For the particularcase at hand� we modify that exchange using the following arguments�

By Theorem � �� the element l straddles the ��separation �X� �Y��X� � Y�� of M � Thus� in B� the columns v and w of the submatrix Dspan the column l of the submatrix A�� This implies that one element ofM� in the triangle Z is parallel to l� Thus� we can delete that element fromM�� and still have enough information to compose M� and the reducedM� to M again� Accordingly� we rede�ne M� to be that reduced matroid�From �B� of � �� we see that the just�de�ned matroid M� may betaken to be �M��d� Note that M� may have just six elements� in which caseit is isomorphic to the wheel matroid M�W��

For later reference� we include the matrices B� and B� of the ��sumdecomposition of M into M� and M� in � �� below� We have simpli��ed the matrices by omitting indices that are of no consequence for thesubsequent discussion� With B� and B� of � �� one readily con�rmsthe statements �i��iv� of Theorem � �� about certain circuits of M �M� and M�� We leave the simple calculations to the reader� Thus� we havecompleted the proof of that theorem�


� ��

l

B1 =A1

Y1

ab

X1

0 11 0

11

0

0 1

11

A2X2

lY2

v w0 11 0

B2 =0 1

Matrices B� and B� of ��sum decomposition of M

In the next section� we use Theorems � �� and � �� to character�ize the max��ow min�cut matroids with special element l by the exclusionof U�

�minors and of F �

�minors with l�

�� Characterization of Max�Flow

Min�Cut Matroids

Recall from the introduction that a max��ow min�cut matroid is a con�nected matroid with a special element l such that for any nonnegativeintegral edge capacity vector h� we have maxF minh�D�� Here maxFis the optimal value of the max �ow problem� where one must �nd a max�imum number of cycles with l such that each element e � l is containedin at most he of these cycles� Then minh�D� is the optimal value of themin cut problem� where one must determine a cocircuit D with l such thath�D�� de�ned to be the sum of the values he of the elements e � �D � l��is minimum�

In this section� we show that a connected matroid with a special el�ement l has the max��ow min�cut property if and only if M has no U�

�

minors and has no F �

�minors with l� By Theorem �� a matroid is

binary if and only if it has no U�

�minors� Thus� the claimed characteriza�

tion is equivalent to the statement that M� the class of connected binarymatroids having no F �

�minors with l� is the class of connected max��ow

min�cut matroids�The characterization is summarized by the following theorem and

corollary�

�� Theorem� A connected matroid M with a special element l isa max��ow min�cut matroid if and only if M has no U�

�minors and has no

F �

�minors with l�

�� Corollary� M is the class of connected max��ow min�cut ma�troids�


We begin the proof of Theorem � �� by noting that the resultis trivially true for matroids whose groundset contains only the specialelement l� Thus� we assume from now on that all matroids M examinedbelow have at least one additional element besides l� We �rst formulatethe max �ow problem and the min cut problem as linear programs with anintegrality condition�

Max Flow Problem

Let M be a connected but not necessarily binary matroid whose groundsetE contains a special element l� We construct the following matrix H fromM � The rows of H correspond to the elements of E other than l� and thecolumns to the circuits C of M containing l� The entry of H in row e andcolumn C is then if e occurs in circuit C� and � otherwise� Considerthe following linear program involving H and an arbitrary nonnegativeintegral vector h� Recall that is a vector containing only s� All vectorsare assumed to be column vectors of appropriate dimension� Below� theabbreviation �s� t�� stands for �subject to��

� ��max t � rs� t� H � r h

r �

We call this problem the fractional max �ow problem� since it becomesthe max �ow problem with capacity vector h when we require the solutionvector r to be integral� Indeed� for any integral solution vector r� the entryin position C speci�es the number of times the circuit C is to be selected�

Min Cut Problem

The linear programming dual of � �� is

� ��min ht � ss� t� Ht � s

s �

We call � �� the fractional min cut problem� We justify the term next�Suppose that we require the solution vector s of � �� to be integral�According to the constraints of � �� any optimal solution s can thenbe assumed to be a f�� g vector� We do this from now on when we imposethe integrality condition on s of � �� By the constraint Ht � s of� �� the vector s is thus the incidence vector of a subset Z of E thatintersects every circuit encoded by a column of H� Let C� Z � flg� We


claim that C� contains a cocircuit of M with l� For a proof� select a baseX of M that contains l and that avoids the set Z as much as possible�Collect in a set Y each element y � �E�X� whose fundamental circuit Cy

with X contains l� By this de�nition� Y � flg is the fundamental cocircuitof M that l forms with �E�X�� Furthermore� by the derivation of Z fromthe vector s� Z intersects� for each y � Y � the fundamental circuit Cy insome element z di�erent from l� Suppose no such z is equal to y� Then forsome z � Z� �X �fzg��fyg is a base� which proves that X does not avoidZ as much as possible� a contradiction� Thus� Y � Z� and C� containsa cocircuit of M with l as claimed� Since the vector h is nonnegative� wemay assume C� to be that cocircuit�

By Lemma �� any circuit of M with l and any cocircuit of Mwith l cannot intersect just on the element l� Thus� any cocircuit of Mwith l is a candidate for producing an integral solution for � �� and thebest such candidate corresponds to a f�� g vector s solving � �� withintegrality condition� Thus� that problem represents the min cut problem�and we are justi�ed in calling � �� without the integrality requirement�the fractional min cut problem for M �

Necessity of Excluded Minors Condition

We are ready to show the �only if� part of Theorem � �� which saysthat a connected max��ow min�cut matroid cannot have U�

�minors� or

F �

�minors with l� The proof is based on two reductions� The �rst one

is accomplished by the following lemma and corollary� We omit the proofof the lemma� since it is just a particular version of the so�called dualitytheorem of linear programming�

�� Lemma� For any feasible vectors r and s of � �� and� �� respectively� we have t � r h � s� with equality holding if andonly if both vectors r and s are optimal�

�� Corollary� If every optimal solution vector r of � �� isnonintegral� then the matroid de�ning that problem is not a max��owmin�cut matroid�

Proof� Let r and s be optimal solutions for � �� and � �� respec�tively� Because of the assumptions and Lemma � �� we must havemaxF � t � r ht � s h�D�� Thus� maxF � minh�D�� and the matroidde�ning � �� cannot be a max��ow min�cut matroid�

For the second reduction� we rewrite Lemma �� to get the followingresult�

�� Lemma� Let M be a connected matroid with an element l� IfM has U�

�minors� then M has a U�

�minor with l�


By the preceding reductions� we may prove the necessity of the ex�cluded minors condition of Theorem � �� by producing� for any con�nected matroid with a U�

�minor with l or with an F �

�minor with l� a

nonnegative integral capacity vector h such that any optimal solution vec�tor for � �� is nonintegral� The next lemma says that such a vector hcan always be found�

�� Lemma� Let a connected matroid M with an element l havea U�

�minor with l or an F �

�minor with l� Then there is a nonnegative

integral vector h such that all optimal solution vectors r for � �� withthat h are nonintegral�

Proof� Let N be a minor of M with l and isomorphic to U�

�or F �

�� By

the discussion of Section �� we may assume that N M�UnW � whereU does not contain any cycle of M � and W does not contain any cocycle�Let M have n elements� De�ne the vector h by he for each element ofN except l� he n for each e � U � and he � for each e � W � Routinecalculations show that any optimal solution vector r for � �� with thish is nonintegral� We leave the veri�cation to the reader�

Su�ciency of Excluded Minors Condition

We prove the �if� part of Theorem � �� which says that any connectedmatroid M with a special element l� without U�

�minors� and without F �

�

minors with l is a max��ow min�cut matroid�We invoke a result of polyhedral combinatorics to simplify the proof�

That result concerns a certain integrality property of linear programs calledtotal dual integrality and is due to Edmonds and Giles�

�� Theorem� Suppose the matrix A and the vector c of the linearprogram

� �� max ct � fs� t� A � f b

f �

are integral and permit a feasible solution for the dual linear program of� �� which is

� �� min bt � gs� t� At � g c

g �

Furthermore� assume that the linear program � �� has an integraloptimal solution for every integral vector b for which it has a feasible solu�tion� Then all extreme point solutions for the dual linear program � �� are integral�


We apply Theorem � �� as follows� We view the fractional max �owproblem � �� as an instance of � �� Then the linear programmingdual of � �� which is the fractional min cut problem � �� is theproblem � �� Note that � �� has a feasible solution if and only ifh �� Suppose for a given M with l� and for any integral h �� we canshow that the fractional max �ow problem has an optimal solution vectorthat is integral� By Theorem � �� the fractional min cut problem thenhas� for any integral h �� an optimal solution that is integral� By Lemma� �� the optimal objective function values of the two problems agree�and thus we have maxF minh�D�� We conclude that M is a max��owmin�cut matroid�

By the above arguments� we may establish the max��ow min�cut prop�erty for M by showing that the fractional max �ow problem � �� has�for any integral h �� an optimal solution vector that is integral� Wedivide the latter task into two parts� In the �rst one� we assume M to beregular� In the second one� M is assumed to be nonregular�

Regular Matroid Case

We begin with the �rst part� So assume that a connected matroid M on aset E and with special element l is regular� Let h be any nonnegative inte�gral capacity vector� By Theorem �� M has a real totally unimodularrepresentation matrix B� Adjoin an identity matrix I to B� getting a realmatrix A �I j B� whose columns are indexed by E� Consider the fol�lowing linear program� where the solution vector f is a real column vectorindexed by E�

� ��

max fls� t� A � f �

f �hf h

The linear program is clearly bounded and has the zero vector as solution�Thus� there is an optimal solution vector� say �f � By linear programmingresults� �f may be assumed to be the solution to a system of equations ofthe form �A � f �h� where �A is a square nonsingular matrix consisting ofsome rows of A plus some unit vector rows� and where �h contains zeros andsome entries of h and �h� Evidently� �A is totally unimodular� By Lemma�� A�� is totally unimodular as well� and thus �f �A�� h is integral�We may suppose �f � since any negative entry �fe of �f can be transformedto a positive one by a clearly permissible scaling of column e of A by � �Thus� �f is nonnegative and integral�

Until stated otherwise� we assume that �fl � �� From �f � we derive�fl circuits that we later show to correspond to an optimal solution of the


fractional max �ow problem � �� We obtain these circuits by repeat�edly solving a certain linear inequality system� In the �rst iteration� thatsystem is

� ��

A � g �

ge � if e � l and �fe � �

ge � if e � l and �fe �

gl

The vector g �f� �fl is feasible for � �� Arguing analogously to thecase involving � �� we are thus assured of an integral nonnegativesolution �g for � �� Indeed� this time the vector playing the role ofthe earlier �h is a unit vector� and �g may be taken to be the characteristicvector of a circuit of M with l�

We now replace �f in � �� by �f � �f � �g and deduce from theso�modi�ed system � �� another circuit of M with l� We repeat thisiterative derivation of circuits until �fl has been reduced to �� At that time�we have �fl circuits� each containing the element l� By the derivation� anyelement e of M occurs in at most he of these circuits�

Now assume that �fl �� Then we do not select any circuit at all� Thediscussion to follow applies to this case as well as to the one where �fl � �and where we do select circuits�

De�ne a vector �r from the circuits of M with l that we just haveselected by setting �rC equal to the number of times the circuit C occursin the collection� Thus� t � �r �fl� We claim that �r solves the fractionalmax �ow problem � �� By the construction� �r is feasible for � ��We prove optimality as follows� Take any feasible solution r for � ��From each nonzero entry rC of r� we derive a nonzero solution fC forthe equation A � f � as follows� First� we obtain a nontrivial f�� gsolution for A � f � such that the support vector of that solution is thecharacteristic vector of C� This can clearly be done� since A is totallyunimodular� Next� we scale that f�� g solution by rC to get a vectorfC � We de�ne f to be the sum of the vectors fC if fl � �� and to be thezero vector otherwise� By the derivation� f is a solution for � �� withobjective function value fl t � r� Recall that the vector �f is optimal for� �� Thus� t � �r �fl fl t � r� which proves �r to be optimal for� �� We conclude that M is a max��ow min�cut matroid�

Nonregular Matroid Case

We turn to the second part� where M is not regular� We divide the proofinto three subcases� First� we assume M to be F� with l second� to haveat least eight elements and to be �separable and third� to have at leasteight elements and to be ��connected�


Fano Matroid Subcase

Routine calculations prove that F� with l is a max��ow min�cut matroid�To assist the reader with the checking� we list the matrix H of � �� forF� with l below� but otherwise omit all details� We have partitioned H toexhibit its structure�

� ��

0

1

0

1

00

0

0010

10

1001

0

0011

0101

1 00 1

1110

00

1000

11

Matrix H of � �� for F� with l

For both the second and third subcase of the proof� we introduce the fol�lowing terminology in connection with the fractional max �ow problem� �� For any matroid under consideration� we de�ne a collection of

weighted circuits to be a collection of the circuits with l where a real non�negative weight has been assigned to each circuit� We say that a collectionof weighted circuits is feasible or optimal if the vector r with the weightsof the circuits as entries is feasible or optimal for � �� The collectionhas �ow value � if the corresponding objective function value of � �� is�� Finally� we say that a collection of weighted circuits uses an edge e �times if the weights of the circuits containing the element e sum to ��

The arguments of the second and third subcase use induction� Thesmallest instance involves the already treated F�� Thus� for some n ��we assume the desired conclusion for matroids with at most n elements andtake M to have n! elements� By the excluded minors condition� we knowthat M is in M� Thus� the decomposition results of Section �� apply�

��Sum Decomposition Subcase

We use the decomposition Theorem � �� which we repeat below�

�� Theorem� Any �separable matroid M � M on a set E hasa �sum decomposition where both components M� and M� are connectedminors ofM � contain l� and thus are inM� In addition�M� has an elementy �� l so that any set C � E is a circuit of M with l if and only if �i� or �ii�below holds�

�i� C is a circuit of M� with l but not y��ii� C �C� � flg� � �C� � fyg� where C� is a circuit of M� with l� and

where C� is a circuit of M� with both l and y�


The notation below is that of Theorem � �� In particular� M�

and M� are the two components of a �sum decomposition of M �First we �nd a collection of weighted circuits C that solves the frac�

tional max �ow problem for M for an arbitrary nonnegative integral vectorh� Using parts �i� and �ii� of Theorem � �� we derive from C twocollections C� and C� of weighted circuits for M� and M�� as follows� LetC be a circuit of C of M with positive weight� By Theorem � �� C isa circuit of M� with l but not y� or C �C� � flg� � �C� � fyg� where C�

is a circuit of M� with l� and where C� is a circuit of M� with both l andy� In the �rst case� we assign the weight of C to that circuit of M�� In thesecond case� we assign the weight of C to both the circuit C� of M� andthe circuit C� of M�� By this construction� C� has the same �ow value asC� Furthermore� C� uses the element l of M� just as often as C� of M� usesthe element y� say � times�

Round up � to the next integer� getting� say� �� Declare �� to be thecapacity of the element y of M�� Except for the element l of M� and forthe elements l and y of M�� assign the entries of the capacity vector h ascapacities to the elements of M� and M��

Solve the fractional max �ow problem for M�� By induction and theearlier proof for regular matroids� we may assume that a collection C�

�of

circuits with integral weights is found� Since C� produces a feasible solutionwith �ow value � for that problem� the new collection C�

�must have a �ow

value of at least ��Solve the fractional max �ow problem for M�� Once more� we may

suppose that a collection C��

of circuits with integral weights is found� The�ow value of C�

�must be at least as large as that of C�� which in turn we

know to be equal to that of C�Suppose C�

�uses the element y �� times� for some �� Derive a

collection C��

from C��

by arbitrarily reducing weights of some circuits untilthe element l is used exactly �� times� We combine C��

�with C�

�via Theorem

� �� to a collection of circuits for M with integral weights and with�ow value at least as large as that of C� as desired�

Delta�Sum Decomposition Subcase

We turn to the �nal subcase� where the nonregular M has at least eightelements and is ��connected� We use Theorem � �� which we list againbelow�

�� Theorem� Any �connected nonregular matroidM � M on aset E is isomorphic to the Fano matroid F�� or has a ��sum decompositionwhere the components M� and M� are connected minors of M � contain l�and thus are inM� In the ��sum decomposition� both connecting trianglesof M� and M� contain l� Let the remaining elements of the connecting


triangle of M� �resp� M�� be a and b �resp� v and w�� Then any set C � Eis a circuit of M with l if and only if �i�� ii�� iii�� or �iv� below holds�

�i� C is a circuit C� of M� with l but without a and b��ii� C is a circuit C� of M� with l but without v and w��iii� C �Ca � fag� � �Cv � fvg� where Ca is a circuit of M� with l and

a but without b� and where Cv is a circuit of M� with l and v butwithout w�

�iv� C �Cb � fbg� � �Cw � fwg� where Cb is a circuit of M� with l andb but without a� and where Cw is a circuit of M� with l and w butwithout v�

The proof to come is similar to that for the �sum case� But thereare subtle di�erences� as we shall see� We start again with a collection Cof weighted circuits for M that solves the fractional max �ow problem foran arbitrary nonnegative integral vector h� By Theorem � �� M hasa ��sum decomposition into two matroids M� and M�� We derive from Ctwo collections C� and C� of weighted circuits for M� and M� by processingeach circuit C of C as follows�

Suppose C is a circuit of case �i� or �ii� of Theorem � �� ThenC occurs in M� or M�� and we assign the same weight to that circuit inthe corresponding collection C� or C�� Suppose C is a circuit of case �iii�of Theorem � �� Thus� C �Ca � fag� � �Cv � fvg�� where Ca is acircuit of M� with l and a but without b� and Cv is a circuit of M� with land v but without w� Then we assign to both Ca and Cv the weight of C�Finally� case �iv� of Theorem � �� is handled analogously to case �iii��

Suppose the collection C� uses the element a �resp� b� � times �resp� �times�� By the above derivation� C� uses the element v �resp� element w�also � times �resp� � times�� If there are several choices for C� we preferone that minimizes �! ��

We claim that � � or � �� Suppose both � and � are positive�Thus� there are� in the notation of parts �iii� and �iv� of Theorem � �� four circuits Ca� Cb� Cv� and Cw with positive weights� where Ca and Cb

occur in M�� and Cv and Cw in M�� Let be the minimum of thesefour weights� Recall that fa� b� lg is a triangle of M�� Lemma �� saysthat the symmetric di�erence of two disjoint unions of circuits of a binarymatroid is a disjoint union of circuits� By two applications of this result�we see that the elements of Ca �Cb � fa� b� lg not contained in exactly twoof the circuits Ca� Cb� and fa� b� lg of M� form a disjoint union of circuitsthat contains l but not a or b� Let C � be the circuit of that disjoint unioncontaining l�

Derive two collections C��

and C��

from C� and C� as follows� First� add to the weight of the circuit C � in C�� Then reduce the weights of the fourcircuits Ca� Cb� Cv� and Cw by � Derive a collection C� for M from C�

�and

C��

of M� and M� in the by now obvious way� By the construction� C and

�� Construction of Max�Flow Min�Cut Matroids ��

C� have the same �ow value� De�ne �� and �� for C� analogously to � and� of C� By the construction� �� ! �� ! �� a contradiction�

We thus may assume without loss of generality that � �� In thenotation of Theorem � �� C produces circuits of type C� and Ca forC�� and of type C� and Cv for C�� Round up � to the next integer� getting�� Assign �� as capacity to the element a of M� �resp� element v of M��and declare the element b of M� �resp� element w of M�� to have capacityzero� To all other elements of M� and M�� assign the capacity accordingto the vector h for M �

Solve the fractional max �ow problem for M�� By induction and theearlier proof for regular matroids� we may assume that a collection C�

�of

circuits with integral weights is found� Suppose that collection does not usethe element a exactly �� times� Then� analogously to the earlier situationwhere both �� one easily shows that C does not minimize �! ��

Now solve the fractional max �ow problem for M�� We may assumethat a collection C�

�of circuits with integral weights is found that uses the

element v exactly �� times� It is a simple matter to combine C��

and C��

of M� and M� to a collection C� of circuits with integral weights for M �The latter collection is readily seen to have a �ow value that is at least aslarge as that of C� Thus� we have completed the last step in the proof ofthe excluded minors characterization of the connected max��ow min�cutmatroids of Theorem � ��

In the next section� we utilize a part of the preceding proof of Theo�rem � �� to validate a construction of the connected max��ow min�cutmatroids� We also describe polynomial algorithms for several problemsinvolving these matroids�

�� Construction of Max�Flow

Min�Cut Matroids and

Polynomial Algorithms

In this section� we establish a construction of the connected max��ow min�cut matroids� We utilize the �sum and ��sum of Section �� and rely onthe construction of the regular matroids given by Theorem � �� Fromthe proof of the construction� we derive a polynomial algorithm for decidingwhether a binary matroid has the max��ow min�cut property� Finally� wedescribe polynomial algorithms that� for any connected max��ow min�cutmatroid� solve the max �ow problem� the min cut problem� and a certainshortest circuit problem� We begin with the construction of the connectedmax��ow min�cut matroids�


Construction of Max�Flow Min�Cut Matroids

We use the same terminology as for the construction of the regular matroidsdescribed in Theorem � �� In particular� the two initial matroids ofany construction sequence� as well as all matroids that recursively are com�posed with the matroid already on hand� are the building blocks� Amongthe building blocks is the by now familiar matroid R�� The two represen�tation matrices for that matroid are given by � �� and � ��

We split the description of the construction into two cases� dependingon whether the resulting max��ow min�cut matroid is to be regular� Webegin with the case where this is so�

�� Theorem� Any connected regular max��ow min�cut matroidwith special element l is graphic� cographic� or isomorphic to R�� or maybe constructed recursively by �sums and ��sums using as building blocksgraphic matroids� cographic matroids� or matroids isomorphic to R�� Inthe case of the recursive construction� one of the two initial building blockscontains the special element l� and no other building block contains thatelement�

Proof� The theorem is nothing but Theorem � �� except for the oc�currence of the special element l� It is a trivial matter to adapt the proofof Theorem � �� to account for that element� We leave it to the readerto �ll in the details�

Next we deal with the nonregular case�

�� Theorem� Any connected nonregular max��ow min�cut ma�troid with special element l is isomorphic to F�� or may be recursivelyconstructed by �sums given by � �� and by ��sums given by� �� In the case of the recursive construction� one of thetwo initial building blocks is isomorphic to F� and contains l� Furthermore�each additional building block also contains l� and is isomorphic to F� oris a connected regular max��ow min�cut matroid�

A key result for the proof of Theorem � �� is the following compo�sition theorem�

�� Theorem� Let M� andM� be two connected max��ow min�cutmatroids that are represented by the matrices B� and B� of � �� or� �� Then the �sum or ��sum M of M� and M� as represented bythe matrix B of � �� or � �� is a max��ow min�cut matroid�

Proof� The part of the proof of Theorem � �� concerned with �sumand ��sum decompositions proves the result�

Proof of Theorem �� By Theorem � �� the compositionsspeci�ed in Theorem � �� maintain the max��ow min�cut property if


it is present in the components� Except for this observation� the remain�ing arguments are analogous to those proving Theorem � �� Indeed�the ��sum situation is easier to handle than the ��sum case of Theorem� �� since the element l straddles any ��separation induced by an N�

minor� Thus� the reader should have no di�culties in �lling in the de�tails�

Polynomial Test for Max�Flow Min�Cut Property

The proofs of Theorems � �� and � �� just discussed are easily trans�lated to polynomial algorithms that �nd the constructions claimed by thesetheorems� Any such method may be used to test for the max��ow min�cutproperty in binary matroids� The next theorem and corollary record thesefacts�

�� Theorem� There is a polynomial algorithm that for any con�nected max��ow min�cut matroid �nds an applicable construction as de�scribed by Theorems � �� and � ��

�� Corollary� There is a polynomial algorithm for determiningwhether an arbitrary binary connected matroid with a special element lhas the max��ow min�cut property�

Next we devise polynomial algorithms for the max �ow problem� themin cut problem� and a certain shortest circuit problem� In each case� weassume the matroid to have the max��ow min�cut property� In general�these problems seem to be di�cult for arbitrary binary matroids� Forexample� the min cut problem and the as�yet�unspeci�ed shortest circuitproblem can be shown to be NP�hard�

Polynomial Algorithm for Shortest Circuit Problem

The shortest circuit problem is as follows� For each element e of a connectedbinary matroid M with special element l� a nonnegative rational distancede is given� Then one must �nd a circuit C of M containing l such thatthe sum of the distances de� e � C� called the length of C� is minimal� Thisproblem is well solved for the max��ow min�cut matroids� as follows�

�� Theorem� There is a polynomial algorithm for the shortestcircuit problem of the max��ow min�cut matroids�

Proof� Let M be the given connected max��ow min�cut matroid� withgiven nonnegative distance vector d� We �rst identify a construction asspeci�ed by Theorems � �� and � �� If M is isomorphic to F��then enumeration solves the problem� If M is regular� then we representM by a real totally unimodular matrix B� de�ne A �I j B�� and solve


the following linear program using any one of several known polynomialalgorithms for linear programs�

� ��

min dt � f ! dt � gs� t� A � f �A � g �

f �g �fl gl �

Arguing as in Section �� the linear program � �� must havef�� g solution vectors f and g� Indeed� the set fe j fe � or ge � gmay be assumed to be the desired shortest circuit of M with l�

In the remaining case� M is not regular� Then by Theorem � ��M is a �sum or ��sum where one component is regular or isomorphic toF�� We discuss the ��sum case� and leave the easier �sum situation for thereader� Let M� and M� be the components of the ��sum decompositionof M � In agreement with the discussion of Section �� let fa� b� lg andfv�w� lg be the triangles of M� and M� created by the decomposition�

We may assume that M� is isomorphic to F� or is regular� Exceptfor a� b� and l� we assign the appropriate distance values of the vector dto the elements of M�� To the former elements� we assign � as distance�Temporarily� we delete the element b fromM�� and solve the shortest circuitproblem for that minor of M�� Let Ca be the circuit so found� with lengthma� Similarly� by deletion of a from M�� we obtain Cb and mb� Finally�we delete both a and b� and get Cl and ml� Note that ml ma ! mb�as is readily proved by comparing ml with the sum of the distances of thesymmetric di�erence of Ca� Cb� and fa� b� lg�

Until stated otherwise� we assume that both a � Ca and b � Cb� ByTheorem � �� we conclude the following� M has Cl as shortest circuit�or M has a shortest circuit that intersects M� in Ca or Cb� or M has ashortest circuit that is contained in M�� To decide which case applies� weassign ma and mb as distances to the elements v and w� respectively� of M�

and recursively solve the shortest circuit problem in that matroid using theconstruction we already have on hand for it�

Let C be the circuit so found for M�� with length m�� If m� ml�then Cl is the desired shortest circuit of M � Otherwise� C contains justv� or just w� or none of v and w� or both of v and w� In the �rst threecases� the respective shortest circuit for M is �Ca � fag� � �C � fvg�� or�Cb � fbg� � �C � fwg�� or C� In the fourth case� the previously derivedinequality ml ma!mb implies that m� ml� a situation already treated�

We now discuss the case for M� where a �� Ca or b �� Cb� Thus� theshortest circuit of M� using a or b is at least as long as Cl� and hence weare justi�ed to assign a distance larger than ma or mb to v or w of M� to


get the correct conclusion by the preceding algorithm� It is easy to see thatthis method is polynomial�

Polynomial Algorithm for Max Flow and Min Cut

Problems

We turn to the max �ow problem and the min cut problem� The reader hassurely recognized that the min cut problem is the shortest circuit problemfor the dual matroid of the given max��ow min�cut matroid� Thus� wecould adapt the preceding proof procedure to directly solve the min cutproblem� We will not do so� Instead� we invoke Theorem � �� and aresult of linear programming to achieve a brief presentation� The latterresult is concerned with the e�cient solution of linear programs and is dueto Gr"otschel� Lov#asz� and Schrijver�

�� Theorem� Suppose there is a polynomial algorithm with thefollowing features� For any linear program of a given class and any rationalvector� the algorithm decides whether the vector is feasible for the linearprogram� If the answer is negative� the algorithm also obtains a violatedconstraint� Then there is a polynomial algorithm for the linear programsof the class and their duals� If any such linear program or its dual hasextreme points� then an extreme point solution will be produced for thatlinear program�

The next theorem contains the result that the max �ow problem andthe min cut problem can be solved in polynomial time�

�� Theorem� There is a polynomial algorithm that for any con�nected max��ow min�cut matroid and for any nonnegative integral capacityvector solves the max �ow problem and the min cut problem�

Proof� Let M be a connected max��ow min�cut matroid� and let h be agiven vector of nonnegative integral capacities for the max �ow problemand the min cut problem�

For the solution of the max �ow problem� we �rst �nd a constructionas described in Theorem � �� Note that the decompositions of theconstruction always involve at least one component that is isomorphic to F�or regular� Thus� we can carry out the proof procedure of Theorem � �� knowing that we only need to solve max �ow subproblems for matroids thatare isomorphic to F� or regular� The F� case is straightforward� For regularcomponents� linear programs of the form � �� and inequality systemsgiven by � �� must be solved� These tasks are handled e�cientlyby any polynomial algorithm for linear programs� Thus� the entire proofprocedure for Theorem � �� can be converted to a polynomial algorithmfor the max �ow problem�


We turn to the min cut case� By the de�nition of the max��ow min�cutproperty� any max �ow solution also solves the fractional max �ow prob�lem � �� Thus� by Theorem � �� the fractional min cut problem� �� which is the dual of the fractional max �ow problem � �� hasonly integral extreme solutions� Thus� by Theorem � �� a polynomialsolution algorithm for the min cut problem exists if we have a polyno�mial algorithm that decides whether a given rational vector s is feasible for� �� The latter algorithm must also identify a violated constraint of� �� if s is not feasible�

The feasibility test for s is trivial if that vector has any negative en�tries� So assume s to be nonnegative� View the entries of s as distances forthe elements of M di�erent from l� and assign � as distance to the latter el�ement� Then deciding whether s is feasible for � �� is clearly equivalentto deciding whether the length of a shortest circuit of M with l is greaterthan or equal to � To answer the latter question� we compute a shortestcircuit C of M using the polynomial algorithm of Theorem � �� If thelength of C is at most � then the vector s is feasible for � �� Other�wise� the characteristic vector of C de�nes an inequality of � �� that isviolated by s�

In the next section� we meet an important graph application of themax��ow min�cut matroids�

�� Graphs without Odd�K� Minors

In this section� we transform a graph problem involving certain signedgraphs into a matroid problem involving max��ow min�cut matroids� Asa result� we can apply the construction and polynomial testing algorithmfor max��ow min�cut matroids to obtain a construction and polynomialtesting algorithm for the signed graphs� We begin with an application thatgives rise to the signed graphs� Let P be a bounded polyhedron of the formfx � IRn j A � x a x �g� We are to determine an inequality systemD �x d such that the polyhedronQ fx j A�x a D �x d x �g hasonly integral vertices and contains all integral points of P � The polyhedronQ is usually called the integer hull of P � The arrays D and d can havecomplex structure even for small arrays A and a� Thus� determinationof the structure of D and d for special cases of A and a� either by somecombinatorial description or in terms of a constructive scheme� has becomeone of the basic problems of polyhedral combinatorics�

An equally important converse problem is as follows� One postulatessome construction for D and d and characterizes the cases of A and afor which the construction does work� For the description of one such con�struction� we need a rounding operation for rational vectors� The operation

�� Graphs without Odd�K� Minors ��

rounds down each entry of a given vector b to the next integer� We denotethe resulting vector by bbc� The construction is as follows� Each row Di

of D is obtained from some linear combination of the rows of A by round�ing down the entries� Thus� for some rational row vector i �� we haveDi bi � Ac� The corresponding value di of the vector d is bi � bc� Wecall this construction simple rounding�

Just a few classes of combinatorially interesting polyhedra are knownfor which simple rounding does produce the desired D and d� The mostimportant case is due to Edmonds and has as A any f�� g matrix withtwo � s in each column� The vector a may contain any integers� Thepolyhedron P fx � IRn j A � x a x �g for this case is known as thefractional matching polyhedron� On the other hand� it is well known thatsimple rounding generally does not work for the polyhedra produced byf�� g matrices A and integral vectors a where A has two � s in each rowinstead of each column� Polyhedra of the latter variety are very important�since they arise from a number of combinatorial problems� An example isthe vertex cover problem� where one must cover the edges of an undirectedgraph by vertices� The matrix A is then the real and negated version of thetranspose of the node�edge incidence matrix of the graph� and the vectora contains only � s�

There are special cases� though� where simple rounding does work forthe latter polyhedra� To describe one such class� we derive a special graphG�A� from any f�� g matrix A with two � s in each row� Each columnof A corresponds to a node of G�A�� and each row to an edge� Speci�cally�if a row of A has two entries of opposite �resp� same� sign� say in columnj and k� then an edge connects the nodes j and k of G�A� and is declaredto be even �resp� odd�� We call G�A� a signed graph�

We de�ne a cycle C of a signed graph to be odd if it contains an oddnumber of odd edges� If we scale a column j of A by � � getting� say�A�� then the graph G�A�� may be obtained from G�A� by declaring eacheven �resp� odd� edge incident at node j to be odd �resp� even�� We alsosay that G�A�� is obtained from G�A� by scaling� Note that any cycle ofG�A� is even or odd if and only if this is so for the corresponding cycleof G�A�� Any graph obtained from G�A� by repeated column scaling wecall equivalent to G�A�� As expected� �is equivalent to� is an equivalencerelation�

We derive minors from a signed graph by a sequence of deletions andcontractions interspersed with scaling steps� with the restriction that anycontraction may involve even edges only� It is easily checked that twominors produced by the same deletions and contractions� in any order�are equivalent� provided the reductions obey the cycle�cocycle condition ofSection ��

Consider the signed graph G obtained from K�� the complete graphon four vertices� by declaring the edges of one triangle to be odd and the


remaining three edges to even� Evidently� the triangles of G are preciselyits odd circuits� Declare any signed graph that is equivalent to the just�speci�ed one to be an odd K��

The signed graphs G�A� without odd�K� minors have very pleasantproperties� Indeed� the preceding simple rounding construction does pro�duce the desired integer hull for a variant of the previously speci�ed poly�hedron involving the matrix A� The result� due to Gerards and Schrijver�is as follows�

�� Theorem� Let A be a f�� g matrix with two � s in eachrow� and let P be the polyhedron fx � IRn j a� A �x a� c� x c�g�Then simple rounding can derive� for all integer vectors a�� a�� c�� and c��a description of the integer hull of P if and only if G�A� has no odd�K�

minors�

At this point� we have a nice theorem of polyhedral combinatoricsinvolving the graphs without odd�K� minors� but actually have no cluewhat these graphs look like� Nor do we know how to decide e�cientlywhether a signed graph has odd�K� minors�

To gain the desired insight� we move from signed graphs G to cer�tain binary matroids M � Let G be given� De�ne the edge set of G plusan additional element l to be the groundset of M � Next we produce arepresentation matrix for M � We begin with the usual binary node�edgeincidence matrix F for G� and record whether edges are even or odd byappending to F an additional row having �s �resp� s� in the columns cor�responding to the even �resp� odd� edges of G� We adjoin a column unitvector with index l� having its in the new row� Let the resulting matrixbe F �� Finally� we declare the circuits of M to be the index sets of theGF��mindependent column subsets of F �� Since scaling in G does nota�ect evenness or oddness of cycles� all graphs equivalent to G produce thesame matroid M �

From F �� we obtain a matrix �I j B� by elementary row operations �inGF�� ThusB is a representation matrix for M � The subsequent analysisis simpli�ed if we let l index a row of B� or equivalently� if we accept theunit vector of F � indexed by l as part of the identity I of �I j B�� Indeed� inthat case we only need to choose a tree T of G and scale G so that all treeedges become even� and �nally use the fundamental cycles of the new Gto write down the columns of B� By the derivation� any such fundamentalcycle is even �resp� odd� if the out�of�tree edge producing that cycle is even�resp� odd�� It is easy to see that M is connected if G is �connected andhas at least one odd cycle� Also� every �connected minor of G with at leastone odd cycle corresponds to a minor of M with l� as expected� But whatproperty does M have when G has no odd�K� minors$ The next lemma�also due to Gerards and Schrijver� gives the surprising answer�

�� Lemma� Let G be a �connected signed graph� and let M


be the corresponding binary connected matroid� Then G has no odd�K�

minors if and only if M has no F �

�minors with l� and thus� if and only if

M is a max��ow min�cut matroid�

Proof� Assume that G has an odd�K� minor� say G� Because of scaling�we may presume that a triangle of that minor has all edges odd� and thatthe remaining edges� which form a ��star� are even� Derive a representationmatrix B for the corresponding minor M of M as described above� usingthe ��star as tree� Then one readily sees that M is an F �

�minor of M

with l� By reversing the arguments� we see that any F �

�minor of M with

l corresponds to an odd�K� minor of G�

By Lemma � �� and Corollary � �� we already have a polyno�mial algorithm for deciding whether a signed graph has no odd�K� minors�We also have� indirectly� a complete analysis of the graphs without odd�K�

minors� in the form of the decomposition Theorems � �� and � ��and the construction Theorems � �� and � �� for the max��ow min�cut matroids� To understand the structure of the graphs without odd�K�

minors� we only need to translate these results into graph language� Wecarry out this task next�

Construction of Graphs without Odd�K� Minors

Let M be the connected max��ow min�cut matroid corresponding to a �connected signed graph G without odd�K� minors� We assume that G hasat least one odd circuit� As before� l is the special element of M � De�ne G�

to be the unsigned version of G� Since M�l is the graphic matroid of the �connected G�� that matroid is connected� We will work out a description ofG that involves scaling� That way� a particularly simple description results�We divide the discussion into �ve cases� depending on whether G or M hasa �separation� and whether M is graphic� cographic� regular nongraphicand noncographic� or nonregular�

��Separation Cases

Suppose �E� � flg� E�� is a �separation of M � Suppose that jE�j � sayE� feg� Recall that M�l is connected and graphic� Thus� e and l mustbe two series elements of M � Accordingly� G can be scaled so that e is theonly odd edge� Let G be that scaled graph� We call a signed graph with allodd edges incident at one vertex� and thus G� a graph with one partially

odd vertex� Later� we will see another instance of such a graph�Suppose jE�j � Then �E�� E�� is a �separation of G� Select a base

X� � X� of M where X� � E� and l � X�� and where X� is a maximalindependent subset of E�� In the corresponding representation matrix for


M � the row l has only �s in the columns of E� �X�� Correspondingly� Ghas� up to scaling� odd edges only in E�� Assume such scaling� We derivefrom G the �sum components G� and G� as depicted in �� where theexplicitly shown connecting edges are declared to be even� Note that G� hasonly even edges� Later� we will encounter another �sum decomposition� sowe de�ne the present one to be a �sum of type �

As a result of the preceding discussion� we may assume from now onthat M is ��connected� This implies that any parallel class of edges of Gcontains at most two edges� one even and one odd�

Suppose the deletion of parallel edges from G creates a �separablegraph� Then G has a �separation �E�� E�� where neither E� nor E� isjust a set of parallel edges� Assume jE�j � Since E� is not just aset of parallel edges� the two edges of E� must be incident at a degree node� But by scaling� both edges can be made even� so �E�� E��flg� is a �separation of M contrary to the assumption that M is ��connected� Hence�jE�j� jE�j �� Clearly� �E� � flg� E�� and �E�� E� � flg� are ��separationsof M � In the terminology of Section �� the element l straddles the ��separation �E�� E� � flg� of M � Furthermore� that ��separation induces a��sum decomposition of M into M� and M�� as depicted by the matricesB� B�� and B� of � �� and � �� Since l straddles the ��separationof M � the column l of the submatrix A� of B of � �� is spanned by thecolumns of the submatrix D of B� So if we GF��pivot in column l of Band subsequently delete the pivot row� then the matrix D of B is reducedto a matrix with GF��rank � We rely on this fact next when we interpretthe ��sum decomposition of M in terms of a decomposition of G�

Suppose in each one of the matrices B� B�� and B� of � �� and� �� we perform a GF��pivot on the in row a and column l� Let �B��B�� and �B� result� Delete row l from �B� �B�� and �B� � getting B�� B�� andB�� Clearly� the matrix B� corresponds to G�� the unsigned version of G�Since l straddles the ��separation of M and B� the matrices B�� and B�� arereadily seen to correspond to a somewhat unusual �sum decomposition ofG� Indeed� the component graphs are as depicted in �� once we replacethe connecting edge x of G� in �� by two parallel edges a and b� and theconnecting edge y of G� in �� by two parallel edges v and w� Assumesuch a replacement has been done� and denote by G�

�and G�

�the graphs

corresponding to B�� and B�� Because of scaling� we may suppose thatthe signature of G agrees with that given by row l of �B� Then accordingto row l of �B� and �B�� we may sign G�

�and G�

�to obtain graphs for the

matroids M� and M� of the ��sum decomposition� as follows� In G�

��resp�

G�

�� we declare the edge a �resp� w� to be odd� and the edge b �resp� v� to

be even� The remaining edges of G�

�and G�

�are signed in agreement with

the signature of the edges of G� Let G� and G� be the graphs so obtainedfor M� and M�� We call this special �sum decomposition a �sum of type

�


Theorem � �� Corollary � �� and Theorem � �� whichcover the composition of regular matroids and of max��ow min�cut ma�troids� imply that the above �sum decompositions of type and arereversible� That is� if both components G� and G� correspond to regularmatroids �resp� max��ow min�cut matroids�� then the �sum G of type or also corresponds to a regular matroid �resp� max��ow min�cut matroid��

We have completed the �separation cases� From now on� we assumethat M is ��connected� and that G is ��connected up to parallel edges� Infact� any parallel class of edges of G consists of at most two edges� one evenand one odd�

Graphic Matroid Case

We assume M to be graphic� Thus� there is a graph H so that M is thegraphic matroid M�H� of H� Recall that G� is the unsigned version ofG� Since M�l is the graphic matroid of H�l as well as of G�� the graphsH�l and G� must be �isomorphic� We know that G is ��connected up toparallel edges� so by Theorem �� H�l G�� The graph H may thusbe derived from G� by splitting some vertex v into two nodes� which arethen connected by the edge l� Pick a spanning tree X � of G� that has thenode v as tip node� Then X X � � flg is a tree of H� The representationmatrix of M corresponding to the base X has l as row index� The s inthat row correspond to a subset E of the edges of G incident at v� Thus� upto scaling� the odd edges of G are precisely the edges of E� Assuming suchscaling� G is therefore a graph with one partially odd vertex� By reversingthe construction� we see that any signed graph G with one partially oddvertex produces a graphic M �

Cographic Matroid Case

We assume M to be cographic but not graphic� Hence� M� is graphic� Asbefore� G� represents M�l� so both M�l and �M�l�� M�nl are planar�We conclude that G� and G may be taken to be ��connected plane graphsplus parallel edges� and that there is a graph H such that M�H� M�

and G� �Hnl�� In H� let v� and v� be the endpoints of the edge l�Pick a tree of H that does not contain the edge l� In the correspondingrepresentation matrix for M�� the s of column l represent a path P fromv� to v�� Thus� in Hnl� each vertex other than v� and v� has an evennumber of edges of P incident� By scaling in the plane graph G� the edgesof P become precisely the odd edges� Thus� up to scaling� exactly the twofaces of G corresponding to v� and v� of Hnl have an odd number of oddedges incident� The latter property is invariant under scaling� so the sameproperty holds for G� We call a planar graph with this property� and thusG� a graph with two odd faces�


Regular Nongraphic and Noncographic Matroid Case

We turn to the case where the ��connected matroid M is regular but notgraphic and not cographic� By Theorem � �� M is isomorphic to R��or is a ��sum with components M� and M� where one component does notcontain l� The R�� case is not possible� since M�l is graphic� and since thecontraction of any element of R�� produces an M�K�� minor� as shownin the proof of Lemma � �� Thus� we only need to examine the case ofa ��sum� Let �E� � flg� E�� be the corresponding ��separation of M � Bythe derivation of Theorem � �� via Theorem � �� we may assumethat jE� � flgj� jE�j �� The pair �E�� E�� cannot be a �separation of Gsince otherwise jE�j� jE�j � implies that the earlier discussed �separationcase applies�

Suppose �E�� E��flg� is also a ��separation of M � In the terminologyof Section �� the element l straddles the ��separation �E� �flg� E�� LetB of � �� be a representation matrix for M exhibiting the ��separation�E�� E� � flg�� except that we ignore in B the explicitly shown �s and sin the submatrix indexed by fa� b� dg and fl� v� wg� Since l straddles the��separation� the submatrix D of B spans the column l of A�� Then aGF��pivot on any in column l of A�� followed by the deletion of thepivot row and pivot column� reduces D� which has GF��rank D � to amatrix with GF��rank � We conclude that �E�� E�� is a �separation ofM�l� and hence of G� contrary to assumption� Hence� l does not straddlethe ��separation� Thus� analogously to the case of a �sum of type � thereis a base X� � X� for M where X� � E� and l � X�� and where X� is amaximal independent subset of E�� Furthermore� up to scaling� all edgesof G in E� are even� For the remainder of this case� we suppose G to be ofthis form�

Assume that �E�� E�� is a graph ��separation of G� Arguing as in the�sum case� G has a ��sum decomposition where we declare all edges ofthe connecting triangles to be even� and where the component de�ned fromE� has even edges only� Without chance of confusion� we call this processa ��sum decomposition of G� By the composition Corollary � �� forregular matroids� the ��sum decomposition of G is reversible�

So suppose �E�� E�� is not a graph ��separation of G� This is possible�by Theorem �� Recall that each parallel class of G has at mosttwo edges� one even and one odd� Since the edges of E� are even� allparallel edges must occur in E�� De�ne G�� to be G minus all odd paralleledges� Correspondingly� de�ne E��

�and E��

�from E� and E�� respectively�

Since jE�j� jE�j �� we have jE��

�j� jE��

�j �� Let M �� be the minor of M

corresponding to G�� Then �E��

�� E��

�� is a ��separation of M �� that induces

the ��separation �E� � flg� E�� of M � If �E��

�� E��

�� is a graph ��separation

of G�� then �E�� E�� is a graph ��separation of G contrary to assumption�Thus� by Theorem �� E��

�is a cutset of cardinality �� and removal


of that cutset from G produces two connected graphs H�� and H�� whoseedge sets form a partition of E�� Furthermore� each one of the latter graphscontains a cycle�

If H�� or H�� has at least four edges� then a graph ��separation of Gexists where one side contains only even edges� and the earlier case applies�Thus� each one of H�� and H�� is just a triangle� and G consists of a pairof triangles whose nodes are pairwise joined by three edges� say e� f � and g�plus edges parallel to the latter edges� Then G corresponds to a cographicmatroid case unless G is of the form

� ��

e

g

f

Exceptional signed graph case

where e� f � and g are odd� and where all other edges are even� The matroidM for this graph is regular� as is easily checked�

At this point� we want to summarize the above analysis for the situa�tion where M is regular� For that summary� in Theorem � �� below� weneed a characterization of the regularity of M in terms of signed graphs�To this end� we de�ne an odd double triangle to be the following signedgraph� We start with a triangle� all of whose edges are even� Then we addto each edge one parallel odd edge� Using a representation matrix for F��for example the matrix of � �� one readily con�rms that M has an F�minor with l if and only if G has an odd double triangle minor� It is easyto check that M is regular if and only if M has no F� minors with l andno F �

�minors with l� Indeed� this claim follows almost immediately from

the discussion of Section �� We thus have the following theorem for thecase where M is regular�

�� Theorem� Let G be a �connected signed graph without odd�K� minors and without odd double triangle minors� but with an odd circuit�Then up to scaling� G has exactly one partially odd vertex� or is planar andhas exactly two odd faces� or is the graph of � �� or may be constructedrecursively by �sums of type or � and by ��sums� Up to scaling� thebuilding blocks are the cited graphs and graphs having only even edges�

Proof� The above analysis and a simple inductive argument prove theresult�

We turn to the �nal case� where M is nonregular�


Nonregular Matroid Case

SupposeG produces a nonregular max��ow min�cut matroidM � The trans�lation of the construction Theorem � �� forM gives the following result�

�� Theorem� Let G be a �connected signed graph without odd�K� minors� but with an odd double triangle minor� Then up to scaling�G is an odd double triangle� or may be recursively constructed by �sumsof type or � In the case of the recursive construction� one of the initialbuilding blocks is an odd double triangle� Furthermore� each additionalbuilding block is also an odd double triangle� or corresponds to a signedgraph having no odd�K� minors and no double triangle minors�

Proof� As shown above� matroid �sums and matroid ��sums with l asstraddling element correspond to graph �sums of type or � Thus� thetheorem follows from Theorem � ��

In the next section� we include applications� extensions� and references�

�� Applications Extensions and

References

The �rst comprehensive treatment of the max �ow problem for graphs iscontained in Ford and Fulkerson � �� The de�nition and characteriza�tion of max��ow min�cut matroids by excluded minors is due to Seymour� ��a�� Related is a characterization of matroid ports in Seymour � �� b�� The decomposition of the max��ow min�cut matroids is describedin Tseng and Truemper � �� That reference proves a decomposition re�sult slightly stronger than Theorem � �� Any N� minor of a max��owmin�cut matroid M has a ��separation that up to indices other than l isgiven by � �� and that induces a ��separation of M � A short proof ofTheorem � �� is given in Bixby and Rajan � ��

Basic linear programming results may be found in any standard text�for example in Dantzig � �� Chv#atal � �� or Schrijver � �� Theconcept of total dual integrality and the related Theorem � �� are dueto Edmonds and Giles � ��

The existence of polynomial algorithms for the various problems ofSection �� is established in Truemper � ��a�� That reference provesthe min cut problem and the shortest circuit problem to be NP�hard� andalso shows that the max �ow problem is solved by an integer extremepoint solution of the linear program � �� not just an integer solutionas proved here� Theorem � �� is due to Gr"otschel� Lov#asz� and Schrij�ver � �� Actually� simpler machinery su�ces to prove the results of

�� Applications Extensions and References ��

Section �� as shown in Truemper � ��a�� The recognition problemof max��ow min�cut path matrices is treated in Hartvigsen and Wagner� ��

The simple rounding scheme is analyzed in Chv#atal � �� The mostimportant case treatable by simple rounding is the matching problem�Its solution is due to Edmonds � ��c�� d� see Lov#asz and Plum�mer � �� for an excellent exposition of this result and of many relatedones� Theorem � �� and Lemma � �� are from Gerards and Schrijver� �� General results for packing and covering problems involving ma�troid circuits are described in Seymour � ��c�� Additional decompositiontheorems and other results for signed graphs are given in Gerards � �� b�� Gerards� Lov#asz� Schrijver� Seymour� and Truemper � �� sum�marize various decomposition results for signed graphs�

In Section �� it is shown that the fractional min cut problem � ��has only integer extreme point solutions� provided max��ow min�cut ma�troids are involved� The latter condition is su�cient� but not necessary�Thus� one may want to characterize when precisely � �� has only inte�ger extreme point solutions� Much progress has been made on this di�cultproblem �see Lehman � �� Seymour � �� and Cornu#ejols and Novick� �� but �nding a complete characterization seems to be very di�cult�

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Robertson� N�� and Seymour� P� D� ��c� Graph minors� IX� Disjointcrossed paths� Journal of Combinatorial Theory �B� ��

Robertson� N�� and Seymour� P� D� �� Graph minors� X� Obstructionsto tree�decomposition� Journal of Combinatorial Theory �B� � ��

Robertson� N�� and Seymour� P� D� �� Graph minors� XI� Circuits ona surface� Journal of Combinatorial Theory �B� ��

Robertson� N�� and Seymour� P� D� �� a� Graph minors XII� Distanceon a surface� Journal of Combinatorial Theory �B� ��

Robertson� N�� and Seymour� P� D� �� b� Graph minors� XIII� Thedisjoint paths problem� Journal of Combinatorial Theory �B� � ��

Robertson� N�� and Seymour� P� D� �� c� Graphminors� XIV� Extendingan embedding� Journal of Combinatorial Theory �B� � ��

Robertson� N�� and Seymour� P� D� �� Graph minors� XV� Giant steps�Journal of Combinatorial Theory �B� ��

Robertson� N�� and Seymour� P� D� ��a� Graphminors� XVI� Excludinga non�planar graph� working paper� ��

Robertson� N�� and Seymour� P� D� ��b� Graph minors� XVII� Taminga vortex� working paper� ��

Robertson� N�� and Seymour� P� D� ��c� Graph minors� XVIII� Tree�decompositions and well�quasi�ordering� working paper� ��

Robertson� N�� and Seymour� P� D� ��d� Graph minors� XIX� Well�quasi�ordering on a surface� working paper� ��

Robertson� N�� and Seymour� P� D� ��e� Graph minors� XX� Wagner�sconjecture� working paper� ��

Robertson� N�� and Seymour� P� D� ��f� Graph minors� XXI� Graphswith unique linkages� working paper� ��

Robertson� N�� and Seymour� P� D� ��g� Graph minors� XXII� Irrele�vant vertices in linkage problems� working paper� ��

Robertson� N�� and Seymour� P� D� ��h� Graph Minors� XXIII� Nash�Williams�s immersion conjecture� working paper� ��

Robinson� G� C�� and Welsh� D� J� A� �� The computational complex�ity of matroid properties� Mathematical Proceedings of the CambridgePhilosophical Society ��

Rosenthal� A�� and Frisque� D� �� Transformations for simplifying net�work reliability calculations� Networks � ��

Rota� G��C� �� Combinatorial theory� old and new� Actes du Congr�esInternational des Math�ematiciens �Nice� Vol� � Gauthier�Villars� Paris�� pp� ��

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Roudne�� J��P�� and Wagowski� M� �� Characterizations of ternarymatroids in terms of circuit signatures� Journal of Combinatorial Theory�B� ��

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Satyanarayana� A�� and Wood� R� K� �� A linear�time algorithm forcomputingK�terminal reliability in series�parallel networks� SIAM Jour�nal on Computing ��

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Tseng� F� T�� and Truemper� K� �� A decomposition of the matroidswith the max��ow min�cut property� Discrete Applied Mathematics � �� Also� Addendum� Discrete Applied Mathematics ��

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Tutte� W� T� �� A homotopy theorem for matroids I� II� Transactionsof the American Mathematical Society ��

Tutte� W� T� �� Matroids and graphs� Transactions of the AmericanMathematical Society ��

References ��

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Zaslavsky� T� �� Signed graphs�Discrete Applied Mathematics � ��

References ��

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Zaslavsky� T� �� Biased graphs� II� The three matroids� Journal ofCombinatorial Theory �B� � ��

Ziegler� G� M� �� Matroid representations and free arrangements�Transactions of the American Mathematical Society ��

Ziegler� G� M� �� Binary supersolvable matroids and modular con�structions� Proceedings of the American Mathematical Society ��

Author Index

A

Agrawal� A��

Ahuja� R� K��

Aigner� M��

Akers� S� B�� Jr��

Akgul� M��

Anstee� R� P��

Appel� K��

Arnborg� S��

Asano� T��

Auslander� L��

B

Bachem� A��

Barahona� F��

Barnette� D� W��

Baum� S��

Berge� C��

Bern� M� W��

Bixby� R� E��

� ��

��

Bjorner� A��

Bland� R� G��

Bock� F� C��

Bondy� J� A��

Boulala� M��

Brown� D� P��

Bruno� J��

Brylawski� T� H��

��

Burlet� M��

C

Camion� P��

Cederbaum� I��

Chandrasekaran� R��

Chandru� V��

Chartrand� G��

Chu� Y��J��

Chv�atal� V��

Colbourn� C� J��

Commoner� F� G��

Conforti� M��

Cornu�ejols� G��

Coullard� C� R��

��

Crama� Y��

Crapo� H� H��

Cunningham�W� H��

��

��

Author Index ��

D

Dantzig� G� B��

Dirac� G� A��

Dorfman� Y� G��

Duchamp� A��

Du�n� R� J��

E

Edmonds� J��

��

El�Mallah� E� S��

Epifanov� G� V��

F

Faddeev� D� K��

Faddeeva� V� N��

Farber� M��

Farley� A� M��

Fellows� M� R��

Feo� T� A��

Folkman� J��

Fonlupt� J��

Ford� L� R�� Jr��

Fournier� J��C��

Frank� A��

Frisque� D��

Fujishige� S��

Fulkerson� D� R��

��

G

Gabor� C� P��

Gardner� L� L��

Garey� M� R��

Geelen� J� F� ��

Gerards� A� M� H��

��

Ghouila�Houri� A��

Giles� R��

Gitler� I��

Gondran� M��

Gould� R��

Grotschel� M��

Grunbaum� B��

H

Hadlock� F��

Hadwiger� H��

Haken� W��

Halin� R��

Hammer� P� L��

Harary� F��

Hartvigsen� D� B��

Hassin� R��

Heller� I��

Hempel� L��

Herrmann� U��

Hoang� C� T��

Ho�man� A� J��

Holzer� W��

Hopcroft� J� E��

Hsu� W��L��

I

Ibaraki� T��

Ingleton� A� W��

Inukai� T��

Iri� M��

Ishizaki� Y��

J

Jensen� P� M��

Johnson� D� S��

Johnson� E� L��

Junger� M��

K

Kahn� J��

Kainen� P� C��

Kajitani� Y��

�� Author Index

Kapoor� A��

Karp� R� M��

Kelmans� A� K��

Kern� W��

Kishi� G��

Koch� J��

Konig� D��

Korte� B��

Kress� M��

Krogdahl� S��

Kruskal� J� B��

Kuhn� H� W��

Kung� J� P� S��

Kuratowski� K��

L

Lancaster� P��

Las Vergnas� M��

Laurent� M��

Lawler� E� L��

Lawrence� J��

Lazarson� T��

Lehman� A��

��

Lemos� M��

Lenhart� W� J��

Liu� T��H��

Lofgren� L��

Lov�asz� L��

��

Lucas� D��

M

MacLane� S��

Magnanti� T� L��

Mahjoub� A� R��

Martel� C� U��

Maurras� J� F��

Menger� K��

Minty� G� J��

Monma� C� L��

Moore� E� F��

Motzkin� T� S��

Murota� K��

Murty� U� S� R��

N

Naddef� D��

Nakamura� M��

Narayanan� H��

Nash�Williams� C� St� J� A��

��

Negami� S��

Nemhauser� G��

Nishizeki� T��

Novick� B��

O

Ohtsuki� T��

Oppenheim� R��

Ore� O��

Orlin� J� B��

Orlova� G� I��

Oxley� J� G��

� � � ��

Ozawa� T��

P

Padberg� M� W��

Plummer� M� D��

Politof� T��

Proskurowski� A��

Provan� J� S��

Pulleyblank� W� R��

R

Raco� M��

Rado� R��

Rajan� R��

Rao� M� R��

Ratli�� H� D��

Rebman� K� R��

Author Index ��

Recski� A��

Reid� R��

Reid� T� J��

Reinelt� G��

Robertson� N��

Robinson� G� C��

Rosenthal� A��

Rosenthal� A� S��

Rota� G��C��

Roudne�� J��P��

Row� D��

S

Saaty� T� L��

Saito� N��

Satyanarayana� A��

Sbihi� N��

Schrijver� A��

��

Seymour� P� D��

��

��

��

Shannon� C� E��

Shirali� S��

Sidney� J� B��

Soun� Y��

Strang� G��

Sturmfels� B��

Supowit� K� J��

Swaminathan� R��

T

Takamizawa� K��

Tamir� A��

Tan� J� J��M��

Tarjan� R� E��

Thomassen� C��

Tismenetsky� M��

Tomizawa� N��

Tompkins� C� B��

Trent� H� M��

Trotter� L� E�� Jr��

Truemper� K��

��

��

��

��

Tseng� F� T��

Tsuchiya� T��

Tutte� W� T��

� � ��

U

Uhry� J� P��

V

Vamos� P��

Vartak� N��

Veinott� A� F�� Jr��

W

Wagner� D� K��

��

Wagner� K��

��

Wagowski� M��

Wald� J� A��

Walton� P� N��

Watanabe� H��

Weinberg� L��

Welsh� D� J� A��

de Werra� D��

Wetzel� R��

White� N��

Whitesides� S� H��

Whitney� H��

Wilson� R� J��

Wolsey� L��

Wong� A� L��

Wood� R� K��

�� Author Index

Y

Yannakakis� M��

Young� H� P��

Z

Zaslavsky� T��

Ziegler� G� M��

Subject Index

A

Abstract determinant� ��Abstract matrix� �� Abstract pivot� ��Additionbinary matroid� �dual of binary matroid� �graph� ��graphic matroid� �matroid� ��

AG�� a ne geometry� ��Agrees with �� Algorithm� see Polynomial algo�

rithmAlmost ��balanced� with respect

to a path� ��Almost regularmatroid� ��

see also Construction� Exis�tence condition� Parity con�dition� Restricted

Almost representative matrix�� see also Construc�tion

Applications� ��

Arc� �

Articulation point� ��Axiomsabstract matrix� ��matroid� ��

��balanced graph� �� see

also Characterization

B

B� matrix� �� see also N� ma�troid

B�� B�� matrix� ��B�� matrix� �� B� matrix� �� see also N� ma�

troidB� matrix� ��

�� see also F� matroidwith labels� ��

see also F� matroid with la�bels

B�� matrix� �� see also R�� ma�troid

B�� B�� matrix� �� see also

R�� matroidB�� matrix� �� see

also R�� matroidcomponent matrices� ��

��

�� Subject Index

totally unimodular version��

Balancedgraph� �matrix� ��

Balancedness�inducing matrix��

Basebinary matroid� �� matroid� �� minor of binary matroid� ��

Binary� see also Characterization�Construction

�eld �� GF�� see Fieldmatroid� �� space� ��

Bipartite graph� ��Black box� ��Bridge� �Building block� �� condition �graph��

C

C�

�graph� ��

Capacitycutset of graph� ��edge� ��

Cardinalityintersection problem� ��set� �

Census� binary matroids� �� Characteristiccolumn vector� ��row vector� ��vector� ��

Characterization��balanced graph� ��binary matroid� ��cographic matroid� �conjecture for �Y graphs� ��

cycle polytope� ��graph �isomorphic to a sus�

pended tree� ��graphic matroid� �graphic matroid of suspendedtree� ��

matroid representable overevery �eld �� regular ma�troid��

GF�� GF�� ternary matroid��

GF�� binary matroid��

max��ow min�cut matroid��

outerplanar graph� �planar graph� ��planar matroid� �� regular matroid� ��SP graph� ��SP matroid� �� ternary matroid� ��

Chord� �� see also G�chordChromatic number� �Circuitbinary matroid� �� matroid� �� minor of binary matroid� ��

Circuit�cocircuit condition� ��Clique� �Clique�node incidence matrix�

�� Cobasebinary matroid� �� matroid� �minor of binary matroid� ��

Cocircuitbinary matroid� �� matroid� �minor of binary matroid� ��

Cocycle� �graphic matroid� minor of graph� �minor of graphic matroid� �

Cographic� see also Characteri�

Subject Index ��

zation� Planar� Regularmatrix� �matroid� ��

Cohyperplane� ��Co�independencebinary matroid� �graphic matroid� �

Collection of weighted circuits�

Coloopbinary matroid� ��graph� �matroid� ��

Colorable with n colors� ��Columncomplement of matrix� ��Eulerian matrix� �� submatrix� ��vector� terminology� ��

Common vertex� �Complement totally unimodular

matrix� �� see also Con�struction

Completebipartite graph Km�n� �� see

also K�� K�� K�� graphgraphKn� �� see also K� K��K graph

Complexity of algorithms� �� seealso Polynomial algorithm

Component matrices of B�� seeB�� matrix

Components of sum� see �� k�� Y�sum

Composition� see �� k�� Y�sum

Conjecture� nonisomorphic min�imal matroids for inheritedproperties� ��

Connectedbinary matroid� �� block� � ��components of graph� �graph� �

graphic matroid� �in cycle fashion� �in tree fashion� �matrix� matroid� ��

Connectinggraph� ��matroid� ��minor� �� nodes� ��triad� ��triangle� ��vertex� �

Connectivity� see also k�connec�tivity

function� �� types� graph� ��

Constructionalmost regular matroid� ��

almost representative matrix�over GF�� or GF��

binary representation matrix��

complement totally unimodu�lar matrix� ��

complement totally unimodu�lar matrix� example� ��

graph without G�� minors��

graph without K minors� �graph without Kny minors��

graph without odd�K� minors��

max��ow min�cut matroid��

minimal violation matrix ofregularity �� minimal non�regular matrix��

minimal violation matrix of to�tal unimodularity �� mini�mal non�totally unimodularmatrix��


regular matroid� ��sequence of nested �connectedminors� ��

SP graph� ��SP matroid� �� totally unimodular f�� g ma�trix� ��

totally unimodular f��gmatrix� ��

�Y graph� ��Y matroid� ��

Contractionbinary matroid� ��dual of binary matroid� �� graph� ��graph with T nodes� �� graphic matroid� �matroid� ��minor� binary matroid� ��minor� graph� ��minor� matroid� ��

Coparallelclass� graph� ��class� graphic matroid� �edges� graph� ��edges� graphic matroid� �elements� binary matroid� ��elements� matroid� ��

Corank gap� ��Corner node� ��Cosimple graph� ��Cotree� �graphic matroid�

Cubic node� �Cycle� �agrees with �� connectivity� ��graphic matroid� k�connected� ��k�separable� ��k�separation� ��length� �minor of graph� �� minor of graphic matroid� �

polytope� binary matroid� ��

Cycle�cocycle condition� ��

D

Decomposition� see also �� k�� Y�sum

circle graph� �combinatorial structure� �graph into �connected compo�nents� �

minimally �connected graph��

optimization problem� �perfect graph� �

Degree �vertex�� Deletionbinary matroid� ��dual of binary matroid� �� graph� ��graph with T nodes� ��graphic matroid� �matroid� ��minor� binary matroid� �minor� graph� �minor� matroid� �

Delta�sum� see ��sumDelta�wye� see �YDependent set� matroid� �det �abstract matrix�� detF �F�determinant�� det� �GF��determinant�� det� �GF��determinant�� Determinant� ��Di�erence of two sets� �Displayconvention� matrix� �minor� ��

Double triangle� see C�

�graph

Dualbinary matroid� �graph� �graphic matroid� �


linear program� ��matroid� �� plane graph� ��

Duality theorem� see Linear pro�gram

��sumbinary matroid� ��components� ��composition� binary matroid��

composition� graph� ��decomposition� binary ma�troid� ��

decomposition� graph� ��graph� �� max��ow min�cut matroid��

regular matroid� �� signed graph� ��

��to�Y graph� ��Y� see also Characterization�

Constructionexchange� almost regular ma�troid� ��

exchange� binary matroid��

exchange� graph� �� exchange� regular matroid��

exchange� restricted� �� extension sequence� almost reg�ular matroid� ��

extension sequence� binary ma�troid� ��

extension sequence� restricted��

graph� ��matroid� �� reducibility� binary matroid��

reducibility� graph� ��reducibility� planar graph� ��reduction sequence� binary ma�troid� ��

reduction sequence� graph� ��extension sequence� binary ma�troid� ��

extension sequence� graph� ��

E

Ear decomposition� �Edge� �return edge �SP graph�� see SPsubdivision� see Subdivision

Element� �Emptygraph� �matrix� �� matroid� �set� �

Endpointsedge� �path� �

Equality� matrices� �� Equivalence �signed graph�� see

Signed graphEulerian matrix� �� Even edge� see Signed graphExact k�separationbinary matrix� �binary matroid� �inducing another k�separation��

in terms of matroid rank func�tion� ��

Existence condition� almost reg�ular matroid� ��

Expansionbinary matroid� �dual of binary matroid� �� graph� ��graphic matroid� �matroid� ��

Extensionbinary matroid� �graph� ��matroid ��


F

F� matroid� �� see also B� matrix

graph with T nodes� ��with labels� �� see

also B� matrix with labelswith special element l� �� see

also Odd double trianglesplitter� ��

F �

�matroid� ��

graph with T nodes� ��with special element l� �� see also Odd K� graph

splitter� ��Face� plane graph� ��Fano matroid� see F�� F �

�ma�

troidF�determinant �� detF �� Fieldbinary �� GF�� F �arbitrary �eld�� GF�� GF�� GF�� IR �real numbers�� ternary �� GF��

Finding �� sums� see Poly�nomial algorithm

Flow valuecollection of weighted circuits�

max �ow problem� ��Forest� �Four�colorconjecture� ��theorem� ��

F�pivot� �� properties� �

Fractionalmatching polyhedron� ��max �ow problem� �

min cut problem� �F�rank� ��Fundamentalcircuit� binary matroid� ��circuit� matroid� �� cocircuit� binary matroid� ��cocircuit� matroid� ��cocycle� graphic matroid� cycle� graphic matroid�

G

G� graph� ��splitter�

Gi�

graph� �� Gi��

graph� �� G� graph� �G�� graph� �� G�� graph� �� Gap� ��G�chord� ��Generalized network� recognition

problem� �Geometric viewpoint� ��GF�� see also FieldGF�� see also Field�determinant �� det�� pivot� see F�pivot�rank� see F�rank

GF�� see also Field�dependence� ��determinant �� det�� mindependence� ��pivot� �� rank� see F�rank

Graph� �� see also Character�ization� Construction

addition� see Additioncoloop� see Coloopcontraction� see Contractiondeletion� see Deletionexpansion� see Expansionextension� see Extensionwithout G�� minors� �


without K minors� ��without Kny minors� �k�connectivity� see k�connec�tivity

k�separation� see k�separationloop� see Loopmax cut problem� see Max cutproblem

max �ow problem� see Max�ow problem

min cut problem� see Min cutproblem

without odd�K� minors� ��

reduction� see Reductionsubdivision� see Subdivisionsum� see �� k�� Y�sum

sum �special�� with T nodes� ��

Graphic� see also Characteriza�tion� Planar� Regular

matrix� �matroid� ��

Graphicness testbinary matroid� ��matroid� �subroutine� ��

Grid graph� ��Groundsetbinary matroid� �graphic matroid� �matroid� �

H

Hadwiger�s conjecture �graph col�oring��

Hamiltonian cycle problem� ��Hasa G minor� terminology� �an M minor� terminology� �

Homeomorphism� graph� �Hyperplane� ��

I

Identi�cation� nodes� ��Incidence� edge� �Independent setbinary matroid� �graphic matroid� �matroid� �

Independent vertex set problem��

Induced k�separation� �� see

also Separation algorithmgraph� ��su cient conditions� ��under L�isomorphism� ��

Induced subgraph� by node sub�set� ��

Integerhull of polyhedron� ��program� ��

Internal vertex� �Internally node�disjoint paths� �Intersectionalgorithm� �� circuits and cocircuits� ��problem� applications� ��

problem� generalization� ��problem of at least three ma�troids� ��

problem of two matroids� ��

sets� �theorem� ��

Isolated node� �Isomorphismgraph� �matrix� ��matroid� �

Isthmus� �

K

K graph� ��


��splitter� ��

K� graph� �� see also

Signed graph� W� graphK�� graph� ��Km�n graph� ��Kn graph� ��K graph� ��K�� graph� ��

�� Ki

�� graph� �� K�� graph� �� k�connected� see k�connectivityk�connectivityabstract matrix� ��binary matrix� �binary matroid� �graph� ��graphic matrix� �graphic matroid� �matroid� ��

k�separable� see k�separationk�separation� see also Polynomial

algorithmabstract matrix� ��binary matrix� �binary matroid� �with given subsets� �� graph� ��graphic matrix� �graphic matroid� �link between graph and ma�troid separations� ��

matroid� ��minimal k� �rank function� ��

k�star� �k�sum� see also �� Y�

sumbinary matroid� ��components� ��composition� binary matroid��

connecting matroid� ��

connecting minor� ��decomposition� binary ma�troid� ��

regular matroid �k � ��

L

Label sum� graph� ��Lattice viewpoint� ��Leafedge� �node� �

Lengthcircuit� �cycle� �matrix� ��path� �

Line of graph� ��Linear programduality theorem version� �intersection problem� ��max �ow problem� �min cut problem� �partitioning problem� ��polynomial algorithm� �with totally unimodular coe �cient matrix� ��

Link between graph and matroidseparations� ��

l�isomorphism� ��L�isomorphism� ��Loopbinary matroid� ��graph� �matroid� ��

M

M�C�

�� matroid� ��

M�G�� matroid� ��M�K� matroid� ��

�splitter� ��

M�K�� matroid� ��


�� graph with T nodes� ��

M�K�� matroid� ��M�K�� matroid� ��

�� M�K�� matroid� ��

�� graph with T nodes� ��

M�W�� matroid� �� M�Wn� matroid� ��M�W�� matroid� ��

��M�W�� matroid� ��Matchinggraph� �� matroid� �

Matrix� �� see also Construc�tion� Polynomial algorithm

decomposition� see �� k�� Y�sum

determinant� see Determinantdisplay convention� see Dis�play

over GF�� see Fieldover GF�� see Fieldisomorphism� see Isomorphismlength� see Lengthpivot� see Pivotover IR� see Fieldrank� see Rankover the real numbers� see

Fieldrepresentation� see Represen�tation

support� see Supportterminology� ��viewpoint� ��

Matroid� see also Characteriza�tion� Construction� Polyno�mial algorithm

addition� see Additionaxioms� see Axiomscomposition� see �� k�� Y�sum

connectivity� see Connectivitycontraction� see Contractiondecomposition� see �� k�� Y�sum

de�nition� see Axiomsdeletion� see Deletiondelta�sum� see ��sumdual� see Dual��sum� see ��sumexpansion� see Expansionextension� see Extensionintersection algorithm� see In�tersection

intersection theorem� see In�tersection

isomorphism� see Isomorphismk�connectivity� see k�connec�tivity

k�separation� see k�separationk�sum� see k�summax �ow problem� see Max�ow problem

min cut problem� see Min cutproblem

minor� see Minor��sum see ��sumoracle� see Oraclerank function� see Rankreduction� see Reductionregularity� see Regular� Regu�larity test

representability� see Repre�sentability� Representation

representation� see Represent�ability� Representation

separation algorithm� see Sep�aration algorithm

split� see Splitsplitter� see Splitterstandard representation ma�trix� see Standard represen�tation matrix

submodularity� see Submodu�larity


sum� see �� k�� Y�sum

�sum� see �sum�sum� see �sumwye�sum� see Y�sumY�sum� see Y�sum

Max cut problem� �� Max �ow problemgraph� ��matroid� ��

Max��ow min�cut� see also Char�acterization� Construction�Polynomial algorithm

matroid� ��property� ��property test� �theorem� graph� ��

Maximalsubgraph� �submatrix� �subset� �

Maximum matching� see Match�ing

Menger�s Theorem� �� Min cut problemgraph� ��matroid� ��

Mindependence� �Minimalin context of induced separa�tions� ��

in context of K�� minor� ��cutset� �GF��dependence� �under isomorphism� withoutinduced k�separation� ��

under L�isomorphism� withoutinduced k�separation� ��

matroid without induced k�se�paration� ��

nonregular matrix� �� non�totally unimodular ma�trix� ��

subgraph� �submatrix� �subset� �support vector� binary space��

violation matrix of a property��

violation matrix of regularity��

violation matrix of total uni�modularity� �� see also Character�ization� Construction� Poly�nomial algorithm

Minimum node cover� ��Minorbinary matroid� ��with given elements� �� graph� �graph� terminology� �graphic matroid� �matroid� ��matroid� terminology� �

N

N� matroid� �� see also B�

matrixN� matroid� �� see also B�

matrixn�cycle� ��Nested �connected minorsalgorithm� ��binary matroid� ��graph� ��

Network �owproblem� ��property� ��property test� ��

Network matrix� ��Node� �cover� ��

Node�arc incidence matrix� ��


Node�edge incidence matrix� ��

Nonsingular matrix �abstract��

Nonstandard representation ma�trix� ��

NP� ��complete� ��hard� �

n�path� ��n�subgraph� ��Nullspace� ��

O

Odd� see also Signed graphcycle� ��cycle �signed graph�� double triangle� ��edge� ��K� graph� ��

Odd�K� minor� �� seealso Signed graph

��sumbinary matroid� �� components� ��composition� binary matroid��


decomposition� graph� ��graph� regular matroid� ��

Optimum branching� ��Oracle� ��Orderalgorithm� �matrix� ��

Orthogonal complement� binaryspace� ��

Outerplanar graph� �� see also

Characterization� Planar� SP

P

Parallel� see also SPclass� graph� �class� graphic matroid� edges� graph� �edges� graphic matroid� elements� binary matroid� ��elements� matroid� ��vectors�

Parity condition� almost regularmatroid� ��

Partially odd vertex� see Signedgraph

Partitioning problem� �� see also Intersection al�gorithm� Intersection prob�lem

generalization� ��graph into forests� ��

Path� �length� �

Path shortening technique� ��

applications� ��Perfectgraph� �� matrix� �

Petersen graph� �� PG�� projective geometry�

�� Pivot� see Abstract pivot� GF��

pivot� F�pivotPivot element� �� Planar� see also Characterizationgraph� �graph� �Y reducibility� ��matrix� �matroid� �matroid� ��element extension��

Plane graph� �� Point �� vertex�� Polyhedral combinatorics� ��


�� Polymatroid� ��Polynomial algorithm� �� see

also Characterization� Con�struction

graphicness testing subroutine�see Graphicness test

induced separation� see Sepa�ration algorithm

induced separation or exten�sion� ��

intersection problem� see In�tersection algorithm

k�separation with given sub�sets� see k�separation

linear program� �max �ow problem� max��owmin�cut matroid� ��

min cut problem� max��owmin�cut matroid� ��

�� or �sum decomposition��

partitioning problem� see In�tersection algorithm

shortest circuit problem� max��ow min�cut matroid� ��

signing of regular matrix� see

Signingtest of graphicness� see Graph�icness test

test of matrix total unimodu�larity� see Total unimodu�larity test

test of matroid regularity� seeRegularity test

test of max��ow min�cut prop�erty� see Max��ow min�cutproperty test

test of network �ow property�see Network �ow propertytest

test of representability over ev�ery �eld� ��

�separation or sequence ofnested �connected minors��

Polynomial reduction� �Power set� �Problem instance� �Projective geometry PG�� see

PG��Propersubgraph� �submatrix� ��subset� �

R

IR �real numbers�� see FieldR�� matroid� ��

�� see also B��B�� B�� matrix

alternate representation ma�trix� �

graph with T nodes� ��splitter� �

R�� matroid� �� see also B�� matrix

complement totally unimodu�lar representation matrix��

graph with T nodes� ��Rankabstract matrix� ��binary matroid� ��eld F � see F�rank�eld GF�� see F�rank�eld GF�� see F�rankfunction� binary matroid� ��

function� matroid� �� function viewpoint� ��gap� ��graph� �matrix� ��


matroid� �Real numbers �IR�� see FieldRecognition problem� generalized

network� �Reductionbinary matroid� ��graph� ��graphic matroid� �� matroid� ��

Regular� see also Characteriza�tion� Cographic� Construc�tion� Graphic� Planar� Poly�nomial algorithm� Regular�ity test� Representabilityover every �eld� Total uni�modularity test� Totally uni�modular matrix

matrix� �� matroid� �� matroid decomposition theo�rem� ��

Regularity test� �� seealso Regular

Removal� edge� �Representability� see also Char�

acterization� Construction�Polynomial algorithm� Reg�ular� Regularity test

over every �eld �� regularity��

general discussion� �� over GF�� over GF�� over GF�� and GF�q�� over GF��

Representation� see also Repre�sentability

abstract matrix� �� matrix� binary matroid� �matrix� graphic matroid� �of matroid by abstract matrix��

Restricted� see also Almost regu�lar matroid

�Y exchange� �� Y extension sequence� ��

SP step� �� Return edge �SP graph�� see SPRim� wheel graph� ��Rowcomplement of matrix� ��Eulerian matrix� �submatrix� ��vector� terminology� ��

S

Scalinggraph� ��signed graph� see Signed graph

Separable� see k�separationSeparation� see k�separationSeparation algorithm� ��

�� Sequence of nested �connected

minors� ��Series� see also SPclass� graph� ��class� graphic matroid� � �edges� graph� ��edges� graphic matroid� � �elements� binary matroid� ��elements� matroid� ��

Series�parallel� see SPSet� �Shortest circuit problem� �� see

also Polynomial algorithmSigned graph �with odd and even

edges�� see also Max��ow min�cut

��sum� see ��sumequivalence� ��even edge� ��odd cycle� ��odd double triangle� ��odd edge� ��odd�K� minor� ��


without odd�K� minors� ��

partially odd vertex� �scaling� ��two odd faces� ��sum of type �� see �sum�sum of type � see �sum

Signinggraph �for ��balancedness��

regular matrix �for total uni�modularity��

Simplegraph� ��rounding of vector� ��

Singular matrix �abstract�� Solidstaircase matrix� �triangular matrix� �

SP� see also Characterization�Construction� Parallel� Se�ries

extension� graph� ��extension� matroid� ��graph� ��graph� return edge� ��matroid� ��reduction� graph� ��reduction� matroid� ��step� almost regular matroid��

step� restricted� �� Special edge� max �ow problem�

��Split �� separation�� Splitter� �� see also

F�� F ��

matroid� G�� K

graph�M�K�� R�� matroid�W� graph

theorem� binary matroid� ��

theorem� graph� �� Spoke� wheel graph� ��Standard representation matrix�

�� Star� �Steinitz�s Theorem� ��Straddling element� �separation�

� ��Strong perfect graph conjecture�

�Subdivisionedge� ��graph� ��

Subgraph� �Submatrix� ��Submodular� see SubmodularitySubmodularityabstract matrix rank function��

binary matroid rank function��

F�rank function� function on sets� ��matrix rank function� � ��

matroid rank function� �Subset� �Subvector� ��Sum� see also �� k��

Y�sumof circuits property� ��of several graphs �special��

Support� matrix� ��Suspended tree� see TreeSwitching� �� graph with T nodes� ��

Symmetric di�erencecircuits� binary matroid� ��disjoint unions of circuits� bi�nary matroid� ��

sets� �

T

T node� ��Ternary� see also Characteriza�

tion


�eld �� GF�� see Fieldmatroid� ��matroid without M�K�� mi�nors� ��

Test� see Polynomial algorithm�connected extensionbinary matroid� �� graph� �

�connectednessbinary matroid� ��matroid� �

�sumbinary matroid� ��components� ��composition� binary matroid��

composition� graph� �� decomposition� binary ma�troid� ��

decomposition� graph� ��

graph� �� regular matroid� ��

Tip node� �Total dual integrality� �Total unimodularity test� ��

�� see also Signing� Totallyunimodular matrix

Totally unimodular matrix� �� see also B��B�� matrix� B�� matrix�totally unimodular version�Complement totally uni�modular matrix� Construc�tion� Linear program� Mini�mal violation matrix of totalunimodularity� Polynomialalgorithm�Regular� Regular�ity test� Signing� U i matrix

Transpose� abstract matrix� ��Tree� �� decomposition� �graphic matroid� �suspended� �

Triad� binary matroid� �� Triangle� binary matroid� �� Triples in circuits� ��trivialmatrix� ��matroid� �

Tutteconnectivity� graphs� ��k�connected� see k�connectiv�ity

k�connectivity� see k�connec�tivity

k�separable� see k�separationk�separation� see k�separation

Two odd faces� see Signed graph�connected components� graph�

�� isomorphism� ��sumbinary matroid� ��components� ��composition� binary matroid��


decomposition� graph� ��graph� �� max��ow min�cut matroid��

regular matroid� �� of type � �signed graph�� of type �signed graph��

U

Ui matrix� ��U�

matroid� ��

U�

matroid� ��

U�

�matroid� ��

minor with given element� ��Umn matroid� ��

Union� sets� �Uniqueness of reductions


binary matroid� ��graph� ��

V

Valid inequality� ��Vector� terminology� ��Vertex� �connectivity� ��k�connected� ��k�separable� ��k�separation� ��isomorphism� ��

Visible minor� binary matroid��

W

W� graph� �� Wn graph� �� W� graph� �� see

also K� graphsplitter� ��

Wi matroid� �� see also U�

�

matroidWheelgraph� ��

see also K�� W�� Wn� W�

graphmatroid� �� see also

M�W�� M�Wn�� M�W��M�W�� matroid

theorem� �� Wheels and whirls theorem� ��Whirl� �� Wye�sum� see Y�sum� ��

Y

Y�sumbinary matroid� ��components� ��composition� binary matroid��


decomposition� graph� ��graph� ��regular matroid� ��

Y�to�� graph� ��

matroid decomposition

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