maximal unimodular systems of vectors

ArticleNo. eujc.1999.0298Availableonlineat http://www.idealibrary.com on

Europ. J. Combinatorics (1999) 20, 507–526

Maximal Unimodular Systems of Vectors

VLADIMIR DANILOV AND VIATCHESLAV GRISHUKHIN

A subsetR of a vector space V (or Rn) is called unimodular (or U -system) if every vector r ∈ Rhas an integral representation in every basisB ⊆ R. A U -systemR is called maximal if one cannotadd anon-zero vector not colinear to vectors of R such that thenew system is unimodular and spansRR. In thiswork, werefineassertionsof Seymour [7] and give adescription of maximal U -systems.We show that a maximal U -system can be obtained as amalgams (as 1- and 2-sums) of simplestmaximal U -systemscalled components. A component isamaximal U -system having no 1- and 2-de-compositions. It is shown that there are three types of components: the root systems An, which aregraphic, cographic systemsrelated to non-planar 3-connected cubic graphswithout separating cutsofcardinality 3, and aspecial system E5 representing thematroid R10 from [7] which isneither graphicnor cographic. Wegiveconditionsthat arenecessary and sufficient for maximality of an amalgamatedU -system. Wegive acompletedescription of all 11 maximal U -systems of dimension 6.

c© 1999 Academic Press

1. INTRODUCTION

Compositions, decompositionsandreductionsaremain toolsof mathematicsfor thestudy ofcomplicated objectsthrough moresimpleobjects. Someexamplesaredirect decomposition ofrepresentationsof groups(or modules), decomposition of root systemsin asum of irreducibleroot systems, etc. These tools can be used in the study of unimodular systems of vectors, orU -systems, which represent regular (or unimodular) matroids.

There are two simple operations defined on U -systems (and, in general, on matroids). Thefirst oneis adeletionof oneor morevectors. For thisoperation it isimportant tostudy maximalU -systemsthat cannot beobtainedby deletionfromany other U -system. Thesecondoperationis acontractionof oneor morevectors(or aprojectionalongoneor morevectors) of aU -system.Both theseoperations make, in asense, theoriginal U -system moresimple.

One of the operations that makes a U -system more complicated is the well-known directsum of U -systems (or matroids). Unfortunately, the direct sum of maximal U -systems is notmaximal. In [1], Brylawski considered amoregeneral ‘push-out’ construction or an amalgamof matroids. In some cases, this operation preserves unimodularity. Brylawski conjectured(seeResearch Problem 6.16 of [1]) that any U -system may beobtained using amalgams(anddeletions) from standard U -systems, namely, from graphic and cographic U -systems. Thiswas proved by Seymour [7], but the list of standard U -systems was enlarged by a specialU -system, theso called R10 (or E5, in our terms) system.

In this work, we refine assertions of Seymour [7] and apply them to maximal U -systems.Wedefinean amalgam of two U -systemsand 0-, 1- and 2-sumsasspecial casesof amalgams.We show that a U -system has a k-decomposition if and only if it is a k-sum of U -systems,k = 0,1,2. For k = 1,2, our k-sumsand k-decompositionsdiffer slightly from (k+ 1)-sumsand (k+ 1)-decompositionsof Seymour. Weshow that each connected U -system is the limitof a diagram of 1-sumsof U -systemshaving no 1-decompositions. Similarly, each U -systemhaving no 1-decompositionsisthelimi t of a diagram of 2-sumsof U -systemshaving no 1- and2-decompositions. Such diagrams of 1- and 2-sums are trees and are determined uniquely bytheoriginal U -system. Wesay that amaximal U -systemisablock if it hasno1-decompositions.A block having no 2-decomposition is called a component. We give the following completedescription of components: the root systems An which are equivalent to the set of columns

0195–6698/99/060407 + 20 $30.00/0 c© 1999 Academic Press

408 V. Danilov and V. Grishukhin

of the incidence matrix of the complete oriented graphKn+1, cographic systems related tonon-planar 3-connected cubic graphs without separating cuts of cardinality 3, and a specialsystemE5 representing the matroidR10 from [7] which is of neither graphic nor cographictype. Then we provide necessary and sufficient conditions for maximality of the amalgamatedU -systems in terms of trees of blocks and components.

As an example, we present all maximalU -systems of dimension less than or equal to 6(there are exactly 11 such systems in dimension 6; maximalU -systems in dimensions 5 andless were known earlier, see [3]).

There are two problems where the knowledge of maximal unimodular systems is impor-tant. Danilov and Koshevoy [2] consider a classP(R) of integral polytopes closed under theMinkowski sum and such that each edge of a polytopeP ∈ P(R) is parallel to a ray of agiven system of raysR. Let P(Z) be the set of integer vectors in a polytopeP. Obviously,P1(Z)+ P2(Z) ⊆ (P1+ P2)(Z). It is proved in [2] that the equality holds for allP1, P2 ∈ Pif and only ifR is spanned by a unimodular system of vectors.

Let R be a set of vectors spanningRn. The setR defines a familyH(R) of parallelhyperplanesH(r, z) = {x ∈ Rn : xr = z}, z ∈ Z. Let B ⊆ R be a basis forR. Then the setof intersection points of hyperplanes of the familyH(B) is a latticeL. Erdahl and Ryshkov [3]prove that the set of intersection points of hyperplanes of the whole familyH(R) is a lattice(which then coincides withL) if and only ifR is a unimodular system. In this case the familyH(R) is called alattice dicing. To classify alln-dimensional lattice dicings, we have to knowall maximaln-dimensional unimodular systems. The problem of classification of maximalunimodular systems was stated first by Erdahl and Ryshkov in [3].

2. U -SYSTEMS

Let V be a finite-dimensional vector space (over the real fieldR), andR ⊂ V be a set ofvectors. LetRR andZR denote sets of all linear combinations of vectors fromR with realand integer coefficients, respectively. The dimension of the spaceRR ⊂ V is said to be thedimensionofR, dimR. Note that dimR is a submodular function on the set of all subsets ofR, i.e., for anyR1,R2 ⊆ R, the following submodular inequality holds:

dimR1+ dimR2 ≥ dim(R1 ∩R2)+ dim(R1 ∪R2). (1)

This inequality follows from the fact thatRR1∩RR2 ⊇ R(R1∩R2). Note thatR(R1∪R2) =RR1+ RR2.

A subsetF ⊆ R is calledflat (or, equivalently, isclosedinR) if it is the set of all vectors ofR that lie in the spaceRF generated byF , i.e.,F = R ∩RF . A k-flat is a flat of dimensionk. A flatR1 is calledmodularif the inequality (1) holds as equality for any other flatR2. IfR′ ⊃ R andRR′ = RR, thenR′ is called anextensionof R. An extension istrivial if eachvector ofR′ −R is either zero vector, or colinear to a vector ofR.

A subsystemB ⊆ R is calledgeneratingif RB = RR.

DEFINITION. A system of vectorsR is calledunimodular (or a U -system) if, for anygenerating subsystemB ⊆ R, the latticeZB does not depend onB (in other words,RB = RRimpliesZB = ZR).

An equivalent definition is thatR ⊂ ZB for any generating subsystemB ⊆ R. Without lossof generality, we can assume thatV = RR. We denote byB a basis ofR and write down all thevectors ofR as columns vectors in this basis. We obtain a matrixU (R) = (In, A), wheren =dimR, andA is a totally unimodular matrix, i.e., each minor ofA is equal to 0 or±1. Hence a

Maximal unimodular systems of vectors 409

study of unimodular systems is a study of totally unimodular matrices in an invariant way (abouttotally unimodular matrices see, e.g., [6]). Once more, the invariant way is a study of regular(= unimodular) matroids, which are represented byU -systems. Matrix representation ofU -systems shows that ann-dimensionalU -system has at most 3n vectors (it has, in fact, much less).

A U -systemR is calledmaximalif one cannot add a non-zero vector not colinear to vectorsofR such that the new system is unimodular and spans the same space asR. In other words,R is maximal if all its non-trivial unimodular extensions coincide withR. A k-flat is calledmaximalif it is a maximalU -system of dimensionk.

Since aU -system represents a regular matroid, the notion of a maximal regular matroid isclear. Namely, a regular matroidM is maximalif for any regular matroidM ′, of the samerank asM that containsM as a submatroid, any element ofM ′ which is not an element ofMis either a loop or parallel to an element ofM . The notion of a maximal matroid is useful inthe theory of matroids. For example, ifM and its dualM∗ are maximal in a class of matroidsF , thenM is a splitter forF (see [7], Section 7). Proposition 7.1 of [7] can be reformlatedas follows. If N ∈ F , N∗ is maximal inF and N is a minor ofM ∈ F , then there is a setZ ⊆ E(M) such that the restriction ofM onto Z is a subdivision ofN.

A minimal by inclusion subsystem of linearly dependent vectors of a systemR is called acircuit. So, any subset ofR that includes a basis and at least one additional vector contains acircuit.

LetR be aU -system ofm vectors represented by ann×m matrixU (R) = (In, A). Thenthe matrixU (R∗) ≡ (−AT , Im−n), whereAT is the transpose ofA, is totally unimodular andtherefore represents aU -system ofm vectors. ThisU -system is called thedual of R and isdenoted byR∗. It is easy to verify that each row of the matrixU (R∗) is orthogonal to eachrow of the matrixU (R). The notions of the dual system are dualized by the prefix ‘co’, forexample, a circuit, a basis ofR∗ is a cocircuit, a cobasis ofR, respectively.

The following two operations can be applied to anyU -system (and, more generally, to anymatroid): deletion and contraction. IfR′ is a subset of aU -systemR (obtained fromRby deletion of some vectors), thenR′ is obviously aU -system as well. For matrices, thismeans that we delete corresponding columns. This property emphasizes the necessity to studythe maximalU -systems of any given dimension. Thus our work is concerned with maximalsystems. Note that a maximal system contains the zero vector 0 and is symmetric (i.e., itcontains−r with eachr ∈ R). However, sometimes it is convenient to considersimple U-systems, i.e.,U -systems without the zero vector and with exactly one vector from each pair ofopposite vectors.

Contraction, or projection, is the second operation. LetB be a subsystem ofR, V ′ = V/RBbe the factor-space andπ : V → V ′ be the canonical projection. Call such a projectionπalongRB feasible. It can be verified that the imageπ(R) = R′ of a feasible projectionπ is aU -system inV ′. In fact, an arbitrary contraction may be obtained as a sequence of consecutiveone-element contractions. IfB ⊆ R is a basis, then, for matrices, the contraction of an elementof B means that we delete the corresponding (unit) column and row (and leave one columnfrom each set of equals columns). Obviously, contraction preserves total unimodularity of thematrix. Deletions and contractions of vectors ofR correspond to contractions and deletionsof vectors ofR∗, respectively.

The nature of the operations suggests that it may be useful to considerU -systems as acategory. Namely, ifR andR′ areU -systems inV andV ′, respectively, then the linear mapf : V → V ′ such thatf (R) ⊆ R′ can be naturally considered as amorphismofR toR′. Inparticular, we obtain the notion ofisomorphismof U -systems.

For example, it is not difficult to see that there is a unique (up to an isomorphism) maximalone-dimensionalU -system represented by the matrix(1,0,−1). Similarly, it is easy to see


what the two-dimensional maximalU -systems are: they contain seven vectors, columns of thefollowing matrix: (

1 0 1 0 −1 0 −10 1 −1 0 1 −1 0

).

(A simple two-dimensional maximalU -system contains three vectors:e1, e2, ande1− e2.)In the next three subsections, we present examples ofU -systems to be used in the sequel.

2.1. Graphic U-systems.The following important systemAn is a generalization of theabove two-dimensionalU -system.An consists ofn2+n+1 vectors±ei , i = 1, . . . ,n (where{ei } is a basis ofRn), andei −ej , i, j = 1, . . . ,n. It is not difficult to verify thatAn is maximal.A simpleU -systemAn containsn(n+1)

2 vectors. Moreover, anyU -system inRn contains nomore thann2+n+1 vectors. Usually this fact is attributed to Heller [4], who proved it in 1957.However, Erdahl and Ryshkov [3] recall the work [5] by Korkine and Zolotarev in 1877, wherethis result was proved 80 years before.

Consider a simple systemAn. In the basis{ei }, An is represented by the matrix(In, A),where the totally unimodular matrixA containsn(n−1)

2 columns, and each column beside zeroscontains exactly one +1 and one−1. If we add to(In, A) the(n+ 1)th row containing−1s inthe firstn entries and zeros in other entries, we obtain the incidence matrixC of an orientedcomplete graphKn+1 onn+1 vertices. Any oriented graphG onn+1 vertices may be obtainedfrom a complete oriented graphKn+1 by deletion of some edges. The incidence matrixC(G)of G is then obtained by deleting corresponding columns fromC. Denote the correspondingU -systemR(G). Such aU -system is calledgraphic. So, any symmetric graphicU -systemR(G) of a graphG onn+ 1 vertices is a symmetric subsystem ofAn.

2.2. Cographic U-systems.The dual system of a graphicU -system is calledcographic.There is a one-to-one correspondence between circuits of a simple graphicU -systemR(G)

and circuits of the corresponding graphG. There is anotherU -systemR∗(G) related to anygraphG such that there is a one-to-one correspondence between circuits of simpleR∗(G)and minimal cuts ofG. A cut of a connected graphG is a subsetK ⊆ E(G) of edges ofGhaving end-vertices in distinct parts of a partition(V1,V2) of the setV(G) of vertices ofG.The cutK is calledminimal if both the graphsG(V1) andG(V2) induced onV1 andV2 areconnected. The cutK is calledseparatingif both the graphsG(V1) andG(V2) have edges. Inparticular, vertices of valency 1 and 2 determine cuts of cardinality 1 and 2, respectively. Cutsof cardinality 1 and 2 are represented in a cographic system by a zero vector and two colinearvectors, respectively. Also, ifG is not connected or has acut-vertex, i.e., a vertex, deletionof which disconnectsG, thenR∗(G) is not connected, i.e., it is direct sum ofU -systems. Weshall see that direct sum ofU -systems is never maximal. Hence, we will restrict our attentionto connected graphsG having no cut-vertex and all cuts of which have cardinality at least 3.Such graphs are called3-connected.

The deletion and the contraction of a vector ofR∗(G) correspond, respectively, to thecontraction and the deletion of the related edge ofG. The contraction of an edgee is thedeletion ofe with identifying of end-vertices ofe.

LetR(G) be represented in a basis by the matrixU (G) = (In, A(G)). If G hasm edges,then the matrixA(G) hasm−n columns andn rows. The matrixU∗(G) = (−AT (G), Im−n),whereAT (G) is transpose ofA(G), represents the dual, cographicU -systemR∗(G).

Note that dimension ofR∗(G) is equal to number of rows ofU∗(G), i.e.,

dimR∗(G) = m− n,


wherem is the number of edges andn + 1 is the number of vertices ofG. So, to increasethe numberm of vectors ofR∗(G), we have simultaneously to increase the numbern+ 1 ofvertices ofG. In other words, we have to split a vertex ofG.

2.3. The U-system E5. This U -system inR5 consists of 21 vectors. The correspondingsimple system of 10 vectors represents the matroidR10 of [7] and is represented by thefollowing totally unimodular matrix:

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

.Another, more symmetric representation is given by the 20 six-dimensional(±1)-vectors eachwith three+1 entries and three−1 entries plus the zero vector. This representation shows thatthe automorphism group of the corresponding matroidR10 is doubly transitive.

We denote the system byE5 since it can be obtained from the root systemE6. One can takea vectorr0 ∈ E6 and project allr ∈ E6 such thatrr 0 = 1 alongr0. The obtained system ofvectors span the unique maximal system of 10 equiangular lines inR5.

It is worth noting the following two properties ofE5. The first is thatE5 contains nosubsystem isomorphic toA2. The second is thatE5 is a one-element extension of the graphicsystemR(K3,3). Since the automorphism group ofE5 is transitive, all proper subsystems ofE5 are graphic.

3. AMALGAMS

The category point of view suggests the case of such constructions as the direct product,the direct sum, etc. Unfortunately, direct products do not exist in the category ofU -systems.However, direct sums do exist. IfR1 andR2 areU -systems inV1 andV2, we can considerthe union ofR1 andR2 in V1⊕ V2. It is easy to show that this system (denoted byR1⊕R2,orR1⊕0R2) is aU -system, and moreover, it is the direct sum in the category ofU -systems.

However, this simple and important operation has very little interest for us, because thedirect sum ofU -systems is never maximal. In fact, letr ∈ R1, r ′ ∈ R2 be non-zero vectors.Then we can add the vectorr − r ′ to the systemR1 ⊕ R2 such that the enlarged system isunimodular. (Justly this wayA2 is obtained fromA1⊕A1.) The fact that the enlarged systemis unimodular can be verified directly or by using a general construction considered below.

It turns out that a generalization of direct sum notion (the so-called amalgamated sum, orsimply amalgam) is more useful. Recall the example, where we considerR1 ⊕R2 with theadded vectorr − r ′. If we project alongr − r ′, we obtain a newU -systemR in the spaceV = V1 ⊕ V2/R(r − r ′). The spacesV1 and V2 are subspaces ofV (andR1 andR2 aresubsystems ofR), but now they intersect along the lineRr = Rr ′. In other words, theU -systemR is represented as union oramalgamof R1 andR2 with an identification ofr andr ′ (and, of course, of−r and−r ′). In the above example, we take subsystems ofR1 andR2isomorphic toA1, and identify them.

There is a generalization of this construction (for details see [1]). LetR1 andR2 be systemsof vectors spanningV1 andV2. For i = 1,2, letϕi : A→ Ri be an inclusion. LetW12 be thesubspace of the direct sumV1⊕V2 generated by vectorsϕ1(w)−ϕ2(w),w ∈ A. We form theamalgamV1 ⊕W V2 = V1 ⊕ V2/W12, whereW is a space isomorphic to either of the spaces


Rϕ1(A) ⊂ V1 andRϕ2(A) ⊂ V2. In other words,V1 ⊕W V2 is a sum of the spacesV1 andV2, whereRϕ1(A) andRϕ2(A) are identified.

DEFINITION. TheamalgamR1 ⊕A R2 alongA is the unionR1 ∪R2 considered in theamalgamated spaceV1⊕W V2.

The amalgam of unimodular systems always exists, but it is not necessarily unimodular. Thefollowing Proposition 1 proved by Brylawski [1] gives a sufficient condition for unimodularityof an amalgam.

PROPOSITION 1. The amalgamR1 ⊕A R2 of U-systemsR1 andR2 is unimodular ifAis a modular flat inR1 orR2.

We sharpen this proposition for maximalU -systems as follows.

PROPOSITION 2. LetR = R1⊕A R2 be an amalgam of two U-systems such thatA is amodular flat inR1. If R is maximal, thenR2 is also maximal.

PROOF. If R2 is not maximal, letR′2 be a unimodular extension ofR2. ThenR′ =R1⊕A R′2 is aU -system that is an extension ofR, a contradiction. 2

Modular flats ofU -systems are described by the following lemma.

LEMMA 1. LetA ⊆ R be a subsystem of a U-systemR.

(a) If A is maximal, then it is a modular flat.(b) If R is maximal andA is a modular flat, thenA is maximal.

PROOF. (a) Obviously,A is a flat. LetA′ be another flat, and letW = RA ∩ RA′. Weprove that (1) holds as equality forR1 = A andR2 = A′. SinceR(A ∪A′) = RA+RA′, itis sufficient to prove thatR(A ∩ A′) = W. Let p : R → RA be a feasible projection suchthat p(A′) ⊆ RA′. Thenp(A′) generatesW. Sincep(R) containsA, andA is maximal, wehavep(R) = A and p(A′) ⊂ p(R) = A ⊂ R. SinceA′ is a flat,A′ = RA′ ∩R ⊃ p(A′).Hencep(A′) ⊆ A ∩A′ andA ∩A′ generatesW, i.e.,R(A ∩A′) = W.

(b) This assertion is implied by Proposition 2, sinceR = R⊕A A. 2

Fortunately, we do not need the general construction. The following special case when thesubsystemA is isomorphic toAk is sufficient for our purposes. SinceAk is maximal, byLemma 1(a) it is a modular flat in anyU -system that contains it.

DEFINITION. If A is isomorphic toAk, then the amalgamR1 ⊕A R2 is calledk-sumanddenoted asR1⊕k R2.

Note that forp,q ≥ k, A p ⊕k Aq ⊆ A p+q−k.Proposition 2 implies the following corollary.

COROLLARY 1. If at least one of summands of a k-sum is not maximal, then the k-sum isnot maximal.

REMARK. For k = 1,2, our definition ofk-sums corresponds to but slightly differs fromthe definition of(k+1)-sums of [7]. Seymour’s sums are defined on the symmetric differenceR11R2 ofR1 andR2, i.e., the identified set is deleted. Seymour’sk-sum,k = 1,2,3, of twomaximal (and simple)U -systems isnevermaximal (in particular, it can be extended by theidentified set).


The matrix representations of amalgams are as follows (see Theorem 6.15 of [1]). LetU -systemsR1 of dimensionp andR2 of dimensionq be represented by matrices(I p, A) and(Iq, B), respectively. ThenR1⊕0R2 is represented by the matrix

(I p, A)⊕0 (Iq, B) =(

I p 0 A 00 Iq 0 B

).

Let A1 = {e}. We can take matrix representations ofR1 andR2 with bases containinge.Then the definition of 1-sum implies(

I p−1 0 A0 1 a

)⊕1

(1 0 b0 Iq−1 B

)=( I p−1 0 0 A 0

0 1 0 a b0 0 Iq−1 0 B

).

Similarly, let A2 = {e1,e2,e3 = e1 + e2}. We can take a matrix representations ofR1 andR2 with bases both containinge1 ande2. Then the definition of 2-sum implies( I p−2 0 0 0 A

0 1 0 1 a10 0 1 1 a2

)⊕2

(1 0 0 1 b10 1 0 1 b20 0 Iq−2 0 B

)

=

I p−2 0 0 0 0 A 00 1 0 0 1 a1 b10 0 1 0 1 a2 b20 0 0 Iq−2 0 0 B

.Definition of ak-sum implies that for a dimension of thek-sum the following equality holds:

dim(R1⊕k R2) = dimR1+ dimR2− k. (2)

4. DECOMPOSITIONS

The representation of aU -system as an amalgam shows that there is a decomposition of aU -system into more simple parts. For example, ifR = R1⊕k R2, then the systemR lies inthe union of two subspacesRR1 andRR2 intersecting along ak-dimensional subspace. It isintuitively clear that ifR lies in the union of two proper subspacesV1 andV2, thenRmay berepresented as an amalgam.

DEFINITION. We say thatR = R1∪R2 is ak-decompositionof a systemR if the subspacesV1 = RR1 andV2 = RR2 of the spaceV =RR are proper, the subspaceW = V1 ∩ V2 hasdimension dimW = k andRi = R ∩ Vi .

SinceV1 andV2 are proper subspaces, then dimV1, dimV2 ≥ k+ 1. By the definition of anamalgam,R is an amalgam ofR1 andR2 alongA = R1 ∩R2 if A generatesW.

REMARK. The notion of ak-decomposition of aU -system is closely related to the notionof anexact(k + 1)-separationof corresponding matroid, defined in [7]. Namely, a partitionR1 ∪R2 = R,R1 ∩R2 = ∅, is an exact(k+ 1)-separation ofR if |R1|, |R2| ≥ k+ 1 and

dimR1+ dimR2 = dimR+ k.

Recall that dimR is the dimension of the space spanned byR. Let Vi be the space spannedbyRi , i = 1,2, andW = V1 ∩ V2. Then the above equality shows that dimW = k. Hence ak-decomposition of aU -system is equivalent to an exact(k+ 1)-separation(R′1,R′2), whereR′i = Ri −R′′i , and(R′′1,R′′2) is a partition ofR ∩W such that dimR′i =dimRi .


A U -system isconnectedif it has no 0-decomposition. This notion of connectivity isequivalent to another useful definition of connectivity.R is calledconnectedif for any twovectors ofR there is a circuit containing them (see any book on Matroid Theory).

In Propositions 3+ k, k = 0,1,2, belowR = R1 ∪R2 andW = RR1 ∩ RR2.

PROPOSITION 3. If dimW = 0, thenR is the direct sum ofR1 andR2.

The proof is strightforward.

PROPOSITION 4. LetR be a maximal U-system, anddimW = 1, then

(a) A = R ∩W is isomorphic toA1;(b) R is an amalgam ofR1 andR2 with identifiedA;(c) Ri is maximal in Vi , i = 1,2.

PROOF. Let p1 : V → V2 be a feasible projection, identical onV2. SinceR ⊂ V1 ∪ V2,Kerp1 ⊂ V1, and p1 can be considered as a feasible projection ofV1 onto W. Similarassertions are true for a feasible projectionp2 : V → V1. SinceRi generatesVi , the imagepi (Ri ) generatesW, i.e., pi (Ri ) 6= 0. Since dimW=1, pi (Ri ) ∼= A1. Consider theU -systemsR′2 = p1(R) = R2 ∪ p1(R1) andR′1 = p2(R) = R1 ∪ p2(R2). Now p2(R′2) =p2(R2) ∪ p2(p1(R1)) = p2(R2) ∪ p1(R1) ⊇ p1(R1) ⊆ p1(R′1). Since p1(R1) ∼= A1is maximal, we have here equalities, i.e.,p2(R′2) = p1(R′1) = p1(R1) = p2(R2). Sincepi (Ri ) is maximal, by Lemma 1, it is a modular flat both inR′1 and inR′2. Hence the amalgamR′1⊕1R′2 alongp1(R1) = p2(R2) is unimodular. Since it containsR andR is maximal, wehaveR = R′1⊕1R′2,R′i = Ri , i = 1,2, andR1 ∩R2 = A is isomorphic toA1. ThereforeRi are maximal by Corollary 1. 2

PROPOSITION 5. LetR be a maximal U-system, anddimW = 2. Suppose thatR has no1-decomposition. Then

(a) A = R ∩W is isomorphic toA2;(b) R is an amalgam ofR1 andR2 with identifiedA;(c) Ri is maximal in Vi , i = 1,2.

PROOF. R is connected, since it is maximal. If at least one ofRi , sayR1, has a feasibleprojection onW of cardinality≤ 1, thenR1 spans a subspace ofV1 intersecting withV2 by aspace of dimension≤ 1. This contradicts the assumption thatR has no 1-decomposition and isconnected. So, each feasible projection ofV1 (andV2) onW contains at least two vectors. Weclaim that there is a feasible projectionp1 : V1 → W containing exactly three non-colinearvectors.

Note thatR1 is connected. Otherwise,R1 = R′1 ⊕0 R′′1 and V1 = V ′1 ⊕0 V ′′1 . Weobtain the following 1-decomposition ofV onto V ′1 and V ′′1 ⊕1 V2, a contradiction (R hasno 1-decomposition). Suppose that a feasible projectionp(R1) contains only two vectors.Consider preimagese1 ande2 of these vectors. SinceR1 is connected, there is a circuitCcontaining them. Lete3 ∈ C ande3 6= e1,e2. ThenC − {e3} is a linearly independent set.We complement the setC − {e3} up to a basisB of R1. Consider the projectionp1 of V1on W along the space spanned byB − {e1,e2}. Then p1(C) = p({e1,e2,e3}) and p1(ei ),i = 1,2,3, are three distinct vectors, sincee3 has non-zero coordinates ine1 ande2. So, thereis a feasible projectionp1 : V1 → W such thatp1(R1) ∼= A2. Similarly, there is a feasibleprojectionp2 : V2→ W such thatp2(R2) ∼= A2. Reasoning as in the proof of Proposition 4gives thatR1 ∩R2 ∼= A2. Ri are maximal by Corollary 1. 2


We conclude this section with a basic result onU -systems that was proved by Seymour [7].We formulate it in the following convenient form.

SEYMOUR THEOREM. If a U-systemR is neither graphic, nor cographic, and is not isomor-phic toE5, thenR has a 1- or 2-decomposition.

5. COMPONENTS

LetR be a maximalU -system. IfR has a 1-decomposition, then by using Proposition 4,one can representR as a 1-sum of more simple maximalU -systems. For theseU -systems,one can verify whether or not they have a 1-decomposition. If there is a 1-decomposition,it can be represented as a 1-sum. Finally, we obtain a decomposition of the original systeminto blocks that are maximal and have no 1-decomposition. Now, using Proposition 5, wedecompose these blocks into 2-sums. This implies our main result (see below).

DEFINITION.

(1) A maximalU -system having no 1-decomposition is called ablock.(2) A maximalU -system having no 1- and 2-decompositions is called acomponent.

REMARK. We shall see that a cographic component can have ak-decomposition fork ≥ 3.But, in the definition of a component we do not need to assume that a component has nok-decomposition fork ≥ 3. The main reason is the lack of an analogue of Propositions 3+ kfor k ≥ 3. Even if there is ak-decomposition fork ≥ 3 of a maximalU -system, the parts ofthe decomposition can be non-maximal. Also, the intersection of the parts is not isomorphicto Ak, and moreover it is not even a modular flat.

According to the Seymour Theorem, components are of the following three types.

5.1. The componentE5. It is proved in [7] thatE5 is a maximalU -system which has no 1-and 2-decompositions.

5.2. Graphic components. In Section 2 we noted thatAn is maximal. It has nok-decomposition for allk ≥ 0. In fact, suppose thatAn has ak-decomposition. Then thebasis{ei } of An is partitioned into two parts. Lete1 ande2 belong to different parts. Then thevectore1− e2 lies in none of the subspacesV1 andV2, a contradiction.

5.3. Cographic components.Obviously, a cographic component should be maximal in theclass of cographicU -systems. Recall that to increase the number of vectors ofR∗(G)we haveto split a vertex of the graphG.

Now we define the operation ofsplittingof a vertexv of G. This operation is a converse tothe contraction of an edge ofG. We partition the setEv of edges incident tov into two disjointsubsetsE1

v andE2v . We changev by two new verticesv1 andv2 connected by a new edge such

that the edges ofEiv are incident tovi , i = 1,2. The new edge defines an extension ofR∗(G).

Recall that for a cographicU -systemR∗(G), one can consider 3-connected graphs, i.e.,connected graphs with no cut-vertex and cut of cardinality 2. Each vertex of a 3-connectedgraph has a valency of at least 3. A graph is calledcubic if the valency of each of its vertexis 3.


LEMMA 2. A U-system of cographic typeR∗(G) is maximal in the class of cographicU-systems if and only if G is a 3-connected cubic graph.

PROOF. First, we show that the cographicU -systemR∗(G) is not maximal if the graphG is not 3-connected, using the operation of splitting. We consider splittings determiningnon-trivial extensions ofR∗(G).

Let v be a cut-vertex ofG. Then there is a splitting ofv into v1 andv2 such that the newedge(v1v2) form a cut of cardinality one. The edge(v1v2) provides a trivial extension ofR∗(G). Deleting the edge(v1v2), we partitionG into disjoint subgraphsG1 andG2. Thispartition defines a partition of the setEv of edges incident tov in G into partsE1 and E2such thatEi ⊆ E(Gi ). We take another partitionEv = E1

v ∪ E2v such that the intersections

Eiv ∩ E(G j ) 6= ∅ for all four pairsi j , 1≤ i, j ≤ 2. Then the new edge of this splitting defines

a non-trivial extension ofR∗(G).Similarly, let e1 ande2 be two edges of a cut ofG of cardinality 2. We contracte1 into a

vertexv and then splitv so that the new edge of this splitting does not form a cut of cardinality 2with e2. Then the new edge defines a non-trivial extension ofR∗(G).

Now, the valency of each vertex should be at least 3. If we split a vertex of valency 3, thenwe obtain a vertex of valency 2 giving a cut of cardinality 2, i.e., we obtain a trivial extension.Hence a vertex may be splitted if its valency is at least 4. If there is a vertex of valency atleast 4, then we can split it into vertices each of valency 3. ThereforeR∗(G) is maximal ifand only ifG is a cubic 3-connected graph. 2

Note that ifG is planar, then it defines thedualplanar graphG∗ as follows. WhenG is placedon a plane, it partitions the plane into connected domains. The vertices ofG∗ correspond to thedomains, two vertices being adjacent if and only if the corresponding domains have a commonboundary edge. It is easy to see that there is a bijection between (intersecting) edges ofGandG∗ such that each cut ofG corresponds to a circuit ofG∗. In other words, theU -systemsR∗(G) andR(G∗) are isomorphic. IfG∗ has at least five vertices, thenG∗ is not complete,since it is planar and 4≤ dimR(G∗) = dimR∗(G). Hence we have

LEMMA 3. A cographic n-dimensional U-systemR∗(G) is not maximal if G is planar andn ≥ 4.

Let X be the set of vertices of a graphG and letK = (X1, X2) be a separating cut ofG. Fori = 1,2, let E(Xi ) be the (non-empty) set of edges ofG with both ends inXi . The deletion ofelements ofR∗(G) related to edges ofE(X1) corresponds to the contraction of these edges.Denote byG(X1) andG(X2) the graphs obtained fromG by contracting edges ofE(X2) andE(X1), respectively.

If the separating cutK has cardinality 3, then the partition ofG into two graphsG(X1) andG(X2) corresponds to a 2-decomposition of theU -systemR∗(G) into subsystemsR∗(G(X1))

andR∗(G(X2)).Conversely, consider two cubic graphsG1 andG2. We choose a vertexvi in the graphGi ,

i = 1,2, and form a graphG by identifying edges of the one-vertex cuts ofv1 andv2. (Ofcourse,G depends on the identification of edges of the one-vertex cuts.) Obviously,G is acubic graph with a separating cut of cardinality 3. Since a cut of cardinality 3 in a graph isisomorphic toA2, theU -systemR∗(G) is an amalgam (or 2-sum) ofR∗(G1) andR∗(G2).

Let us see what is ak-decomposition of a cographicU -systemR∗(G) in terms ofG. IfR∗(G) has a 0-decomposition, i.e., it is disconnected, thenG is either disconnected or has acut-vertex. We suppose thatG is connected. Recall that there is a one-to-one correspondencebetween projections ofR∗(G) along vectors and deletions of edges ofG related to these


vectors. Hence ifR∗(G) is connected but has a 1-decomposition, thenG has an edge afterdeletion of whichG has a cut-vertex. For a cubic graph, this is equivalent to the assumptionthat G has a cut of cardinality 2. Since such a cut is a circuit ofR∗(G) of cardinality 2,the vectors related to edges of the cut are colinear and span a one-dimensional space. So ifR∗(G) has no 1-decomposition, thenG has no cut-vertex and cuts of cardinality 2, i.e.,G is3-connected.

LetR∗(G) has no 1-decomposition. ThenR∗(G) has a 2-decomposition if and only ifGhas a separating cut of cardinality 3. The vectors representing edges of such a cut form a circuitof R∗(G) isomorphic toA2. Hence the cut is a modular flat ofR∗(G). Call a 3-connectedgraphstrongly3-connected if it has no separating cuts of cardinality 3. Note thatK4 is aunique strongly 3-connected cubic graph on four vertices. Also,R∗(K4) is isomorphic toR(K4) and maximal in the classes of graphic, cographic and allU -systems. In other words,R∗(K4) ∼= R(K4)=A3, similarly asR∗(K ∗3) = R(K3) ∼=A2, are components, which are bothgraphic and cographic.

LEMMA 4. A cographic systemR∗(G) is a component if and only if either G= K ∗3 , K4,or G is a strongly 3-connected non-planar cubic graph.

PROOF. LetR∗(G) be a component,G 6= K ∗3 , K4. Then by definition of a component andby Lemmas 2 and 3,G is a strongly 3-connected non-planar cubic graph.

Conversely, letG be a strongly 3-connected non-planar cubic graph. ThenR∗(G) has no 1-and 2-decompositions. We show that the corresponding cographic systemR∗(G) is maximal.If not, letR′ be an extension ofR∗(G). SinceR∗(G) is maximal in the class of cographicsystems andG is non-planar,R′ is neither graphic, nor cographic. Obviously,R′ is notisomorphic toE5, since all subsystems ofE5 are graphic. Then by the Seymour Theorem,R′has a 1- or 2-decomposition, and this decomposition generates a decomposition ofR∗(G), acontradiction. 2

EXAMPLE. We give the following infinite family of cographic components. For an integerk ≥ 1, we construct a cubic graphQk as follows. Letvi , 1≤ i ≤ 2k, be consecutive vertices ofa circuit of length 2k. We obtain the graphQk if, for 1 ≤ i ≤ k, we connect the verticesvi andvi+k by an edge.Qk has 2k vertices and 3k edges. Hence dimR∗(Qk) = 3k−2k+1= k+1.For example,Q1 = K ∗3, Q2 = K4, Q3 = K3,3, and, fork ≥ 3, Qk is not planar. It is notdifficult to see that the only cuts of cardinality 3 are the one-vertex cuts, i.e.,Qk is strongly3-connected for allk ≥ 1. Hence, by Lemma 4, fork ≥ 1, R∗(Qk) is a maximalU -systems.

REMARK. A 2-sum ofR∗(Gi ), 1 ≤ i ≤ k, for k ≥ 3 summands along the same setK ,which is a one-vertex cut of eachGi , is not cographic. Letk = 3. First, consider a subsumof two cographic summandsR∗(G1) andR∗(G2) along a one-vertex cutK . Then the setKis a separating cut inG of R∗(G) = R∗(G1)⊕2R∗(G2). Note thatR∗(K4)⊕2R∗(K4) =R∗(G0), whereG0 is a planar cubic graph on six vertices such thatG∗0 = K5 − e. Theseparating cut ofG0 corresponds to the triangle ofG∗0 not adjacent toe. Obviously, theabove graphsG1 and G2 contain K4 as a minor. HenceR∗(G) = R∗(G1) ⊕2 R∗(G2)

containsR(K5 − e) as a minor such that the triangle (not adjacent toe) corresponds to theseparating cutK of G. A cubic non-planar graph containsK3,3 as a minor. Hence the 2-sumR∗(G)⊕2R∗(G3) contains the minorR(K5− e)⊕2R∗(K3,3) which is neither graphic, norcographic. Note that the 2-sumR∗(G)⊕2R∗(G3) is taken along the setK which is a cut ofdifferent types inG andG3. So, if a 2-sum of two cographicU -systems is taken along a setK that is a separating 3-cut of one graph and an arbitrary cut in another graph, we obtain, ingeneral, neither graphic nor cographicU -system.


Also, note that ifG is a cubic non-planar graph andn ≥ 4, the 2-sumR∗(G)⊕2 An containsR∗(K3,3)⊕2 A4 as a minor. Hence the 2-sum is neither graphic, nor cographic.

5.4. We know all components.Recall that the Seymour Theorem implies that the aboveexamples of components give the complete list of all components.

THEOREM 1. Any component is isomorphic to eitherAn for some n, or toE5, or to acographic U-systemR∗(G) for a strongly 3-connected non-planar cubic graph G.

It is convenient to classify components with respect to theirA-rank, which is equal todimension of a maximal subsystem of typeAr . According toA-rank, all components aresubdivided into three classes:

(1) A1 andE5, with A-rank 1;(2) cographic componentsR∗(G) for G 6= K4, with A-rank 2;(3) graphic componentsAn, n ≥ 3, with A-rankn ≥ 3.

Note that the components of the classes (2) and (3) have the following important property:any element of such a component is contained in a subsystem of typeA2. Moreover, for acographic component, any non-zero element is contained in exactly two subsystems of typeA2. This implies that the related graphG is uniquely determined by the cographic componentR∗(G). To see that, take subystems of typesA2 andA1 as vertices and edges ofG, respectively.

In fact, Theorem 1 is the only proposition, where we use (and very heavily) the SeymourTheorem. We hope that one can obtain an independent description of components. If it isindeed possible, such a description would give another proof of the Seymour Theorem.

6. TREES OF U -SYSTEMS AND THEIR DECOMPOSITIONS

In the beginning of Section 5, we explained that every maximalU -systemR is partitionedinto blocks (and, according to Proposition 4, it may be represented as a sequence of 1-sums ofblocks). Similarly, every block may be represented as a sequence of 2-sums of components.Now, we have to determine which 2-sums of components give blocks, and which 1-sums ofblocks give maximalU -systems.

For us to study these problems, it is inconvenient to work with sequences of amalgams,or 1- and 2-sums. Hence, we introduce a notion that describes the whole plan of gluing ofconstituent parts, namely the notion of a tree (or a forest) ofU -systems.

DEFINITION. The following data are called aforest of U-systems(T,R(·)):(a) an oriented forest (a graph without circuits)T with a set of verticesSwhich is partitioned

into disjoint parts:J (‘joining’ vertices) andC (‘capital’ vertices), and a set of arrows( j, c); the arrow( j, c) goes fromj ∈ J into c ∈ C;

(b) aU -systemR(s), for eachs ∈ S;(c) a morphism ofU -systemsR( j, c) : R( j )→ R(c), for each arrow( j, c).

Therefore, a forest ofU -systems(T,R(·)) is a functor fromT to the category ofU -systems. For a forest ofU -systems, it is natural to define its (direct or inductive) limitR(T) = lim−→(T,R(·)) in the category ofU -systems. In these terms, an amalgam is thelimit of the diagram

R1◦ ←−A◦−→R2◦ .


To assure the existence of the limit (see Section 3), we suppose that theU -systemR( j ) ismaximal for each joining vertexj ∈ J, and each gluing morphismR( j, c) : R( j ) → R(c)is an embedding. If these conditions hold, the limitR(T) = lim−→(T,R(·)) exists and can beexplicitly defined by two ways. In one way, we construct an amalgam of the vector spacesRR(c), c ∈ C, with identified subspacesRR( j ), j ∈ J. Then the limit is simply the unionof all R(c) in the amalgamated space. In another way, we construct the limit by induction.Let c0 ∈ C be a pendant vertex ofT . Let c0 be adjacent toj0 ∈ J, and letT ′ = T − c0.ThenR(T) is the amalgamR(c0)⊕R( j0) R(T ′) along the subsystemR( j0) according to theembeddingR( j0, c0).

Both of these methods are useful: the first way shows that the result of the second does notdepend on the choice of a pendent vertex. The second way makes it obvious that the limitsystemR(T) is unimodular.

Since we are not interested in direct sums, we suppose that the forestT is connected (i.e.,T is a tree) and thatR( j ) 6= ∅ for all j ∈ J. It is clear that we shall takeA1 andA2 asR( j ),and components and blocks asR(c).

Below we provide conditions for the limit of a tree ofU -systems to be maximal. To clarifythe nature of decompositions of the limit in terms of the tree, we assume that trees satisfy somenatural and not very restrictive conditions.

DEFINITION. A tree ofU -systems is calledfeasibleif the following conditions hold:

(I) for each joining vertexj there are at least two arrows( j, c);(II) for each arrow( j, c), the corresponding embeddingR( j, c) : R( j )→ R(c) is not an

isomorphism;(III) if two distinct arrows( j1, c) and( j2, c) come into the same vertexc, then the images of

embeddingsR( j1, c) andR( j2, c) are distinct.

It is clear that the joining vertices of a feasible treeT relate to decompositions of the limitU -systemR(T).

Consider a feasible tree ofU -systems. Take a joining vertexj ∈ J, and letCj be the subsetof capital vertices adjacent toj . Call by splitting of j the following operation. PartitionCj

into two disjoint non-empty partsC1j andC2

j of cardinalityq1 andq2.If q1 = q2 = 1, then delete the vertexj with incident arrows.If q1 = 1, q2 > 1 (or q1 > 1, q2 = 1), then delete the arrow( j, c) such that{c} = C1

j (or{c} = C2

j , respectively).If q1,q2 > 1, then change the vertexj by two non-adjacent verticesj1 and j2 adjacent to

all c ∈ C1j andc ∈ C2

j , respectively.It is clear that the splitting determines both a decomposition of the treeT into two parts

T1 andT2 and a 1- or 2-decomposition of the limitU -systemR(T) = lim−→(T,R(·)) into two

subsystemsR1 andR2 intersecting byR( j ). The condition (I) implies that the treesT1 andT2 are feasible. Proposition 6 below shows that the converse holds.

PROPOSITION 6.

(a) Let (T,R(·)) be a feasible tree of U-systems such thatR( j ) ∼= A1 for all j ∈ J andR(c) have no 1-decomposition for all c∈ C. Then any 1-decomposition of the limitU-systemR(T) is given by the splitting of a unique joining vertex j of the feasible tree.

(b) Let (T,R(·)) be a feasible tree of U-systems such thatR( j ) ∼= A2 for all j ∈ J andR(c) have no 2-decomposition for all c∈ C. Then any 2-decomposition of the limitU-systemR(T) is given by the splitting of a unique joining vertex j of the feasible tree.


PROOF. Since the proofs of (a) and (b) are similar, we give a proof of (a). LetR(T) =R1 ∪R2 be a 1-decomposition. Note that the intersectionR12 = R1 ∩R2 has dimension 1.The condition (II) implies that the dimension of each capitalU -systemR(c) is at least 2.SinceR(c) has no 1-decomposition, it belongs only to one of partsR1 or R2. Hence thesetC is partioned into two disjoint parts. But a joining systemR( j ) can lie inR12. Theequality dimR12 = 1 implies thatR12 contains the joining systemR( j ) only for one j . Thecondition (III) determines uniquely the joining systemR( j ). Now, the partition of arrows( j, c), i.e., the splitting of the vertexj , is determined uniquely. 2

7. STRUCTURE OF BLOCKS

Obviously, any component is a block. But there are also decomposable blocks, namely2-decomposable blocks. It is clear that a decomposable block is represented as the limit ofa feasible tree ofU -systems, where joining systemsR( j ) are isomorphic toA2 and capitalsystemsR(c) are components of typeAk with k ≥ 3 and cographic systems (withA-rank 2).(If a block is non-decomposable, then it is represented as a trivial tree consisting of one vertex.)

In fact, letR = R1 ∪R2 be a 2-decomposition of a blockR. According to Proposition 5,R = R1⊕AR2 such thatR1 andR2 are blocks andA = R1 ∩R2 ∼= A2. By induction, theblocksR1 andR2 are represented by feasible trees(T1,R1(·)) and(T2,R2(·)).

Obviously,A, as a subsystem ofR1, lies in some (maybe in several) componentsR(c1),c1 ∈ C1. If A lies in several components, then it lies in their intersection, i.e., in a joiningU -systemR( j1), j1 ∈ J1, where j1 is determined uniquely by condition (III). Similar assertionsare true forR2.

Now it is clear how the treesT1 andT2 are connected in a feasible tree. We have to definean operation that is a converse of the splitting of a joining vertex. Namely, we add new joiningvertices and arrows to the union ofT1 andT2 as follows.

If A lies only inR(c1) andR(c2), ci ∈ Ci , i = 1,2, then we add a new vertexj and arrows( j, c1) and( j, c2).

If A lies inR(c1) andR( j2) (or inR( j1) andR(c2)), then we add the arrow( j2, c1) (or( j1, c2), respectively).

If A lies inR( j1) andR( j2), then we identify the joining verticesj1 and j2 in a new vertexj .It is easy to see that the obtained tree is feasible.So, we obtain a (rather simple) assertion that any block is represented by a feasible tree of

components. By construction, it is clear that the tree is uniquely determined by the block.It should be noted that the obtained feasible tree has a special property. To formulate thisimportant property, we note that if a block is decomposable (i.e., it is not a component), thenthe components contained in it are distinct fromE5, i.e., all these components haveA-rank≥ 2. The above property may be formalized as follows:

(*) In any feasible tree of a block, the components ofA-rank≥ 3 cannot be neighbouring.We call verticesc1 andc2 neighbouringif there are arrows( j, c1) and( j, c2) for the same

joining vertex j .In fact, let two graphic componentsAn andAk be neighbouring, wheren, k ≥ 3. Then the

2-sumAn ⊕2 Ak is not a maximalU -system, since it hasAn+k−2 as an extension. Then, byCorollary 1, the limitU -system is not maximal.

The condition (*) is necessary for the limitU -system to be maximal. The theorem belowshows that this condition is also sufficient.

THEOREM 2. Let (T,R(·)) be a feasible tree of U-systems such that the capital verticescorrespond to components ofA-rank≥ 2, andR( j ) ∼= A2 for all j ∈ J . Let the components of


A-rank≥ 3 not neighbour. Then the limit U-systemR = lim−→(T,R(·)) is a block. Conversely,

every block is uniquely represented by such a tree.

PROOF. It suffices to prove thatR is maximal. We proceed by induction on the number ofvertices ofT . If the treeT consists of one vertex, thenR is a component, and the proof isobvious.

Now, we consider the case whenT contains more that one vertex (or, equivalently, whenJ 6= ∅, or whenR is decomposable). Suppose thatR is not maximal, andR′ ⊃ R is amaximal extension ofR. First, we prove thatR′ is not a component, the more so as we havethe complete list of components.(α) LetR′ be a component.R′ is not graphic and not isomorphic toE5. Otherwise,R is

graphic, and therefore all its componentsR(c) are graphic, too, and haveA-rank≥ 3. SinceJ 6= ∅, there are neighbouring components ofA-rank≥ 3. This is in contradiction to (*).(β) R′ is not cographic. In fact, letR′ = R∗(G′), where, according to Lemma 4,G′

is a strongly 3-connected non-planar cubic graph. ThenR is also cographic and has theform R∗(G), whereG is obtained fromG′ by contracting some edges. SinceR has a 2-decomposition, the graphG has a separating minimal cutK of cardinality 3 (we noted thisfact in Section 5.3). But thenK is a separating minimal cut ofG′, too. This contradicts thefact thatG′ is strongly 3-connected.(γ ) Thus,R′ is not a component. SinceR′ is maximal, it has a 1- or 2-decompositionR′ = R′1∪R′2. This decomposition generates a 1- or 2-decomposition ofR intoR1 = R∩R′1andR2 = R∩R′2. But, by construction,R has no 1-decomposition. HenceR = R1∪R2 isa 2-decomposition. At least one of theU -systems,R1 orR2, is not maximal. We saw abovethat any 2-decomposition ofR is obtained by the splitting of a joining vertexj ∈ J of the treeT representingR. Hence, fori = 1,2, theU -systemRi is also represented by a feasible treeTi , where the components ofA-rank≥ 3 are not neigboring. By induction,Ri is maximal.This contradiction proves the theorem. 2

As an example, consider the sumAn⊕2R⊕2 Ak, whereR is a cographic component. Thesum is a block if and only ifAn andAk are connected with the cographic componentR bydistinct subsystemsA2 ofR.

8. STRUCTURE OF MAXIMAL U -SYSTEMS

As an arbitrary block was constructed from components by 2-sums, an arbitrary maximalU -system may be constructed from blocks by 1-sums. The first part of the previous section isapplicable to this case without significant changes (of course, we have to change componentsby blocks and 2-sums by 1-sums). And similarly, as for blocks, the following analogue of theproperty (*) is necessary (and, as we shall show, sufficient) for the limit system to be maximal:

(**) In any feasible tree of U-systems, where blocks are related to capital vertices, the blocksof A-rank≥ 2 cannot be neighbouring.

In other words, if verticesc andc′ of our feasible tree are neighboring, then one of blocksR(c) or/andR(c′) should be isomorphic toE5. In fact, the following lemma holds.

LEMMA 5. LetR1 andR2 be blocks distinct fromE5. Then the 1-sumR1 ⊕1 R2 is anon-maximal U-system.

PROOF. We saw in the previous section that blocks distinct fromE5 are obtained as 2-sumsof graphic and cographic components. Hence they have the following property: any elementbelongs to a subsystem isomorphic toA2. Let the amalgam ofR1 andR2 be taken along


vectors±r1 ∈ R1 and±r2 ∈ R2, and let, fori = 1,2,Ai ⊆ Ri be a subsystem containing±ri and isomorphic toA2. Using the equalityRi = Ri ⊕2Ai we can represent the amalgamas the sumR1⊕2 (A1⊕1A2)⊕2R2. The middle sumA1⊕1A2 ∼= A2⊕1 A2 is not maximal,since it has an extension isomorphic toA3. Hence the 1-sumR1⊕1R2 hasR1⊕2 A3⊕2R2as an extension. 2

Note that if a block is distinct fromE5, then it does not containE5 as a subsystem. This factand the above lemma prove the theorem below.

THEOREM 3. Let (T,R(·)) be a feasible tree of U-systems such that the capital verticescorrespond to blocks andR( j ) ∼= A1 for all j ∈ J . Let the blocks ofA-rank ≥ 2 arenot neighbouring. Then the limit U-systemR(T) = lim−→(T,R(·)) is a maximal U-system.

Conversely, every maximal U-system is uniquely represented by such a tree.

PROOF. The proof of this theorem is parallel to that of Theorem 2. As in the proof ofTheorem 2, we can assume thatT contains more than one vertex. So, by (**)R containsE5 asa subsystem. IfR is not maximal, then letR′ be its maximal extension. SinceR′ 6= E5, butcontainsE5 as a subsystem,R′ is not a block, and therefore it has a 1-decomposition. Now,as in the proof of Theorem 2, we obtain a contradiction, using induction. 2

As an example, consider the sumA2⊕1 E5⊕1 A2. The sum is a maximalU -system if andonly if bothA2 are connected with the blockE5 by distinct (non-colinear) vectors.

Proposition 6 and Theorems 2 and 3 give a complete description of maximal unimodularsystems. Namely, a maximal unimodular system is a ‘tree connection’ of components. Adescription of the ‘tree connection’ is given in Proposition 6. The neighbouring conditionsmean that intersecting components (and intersecting blocks) should be maximally inserted ineach other.

9. MAXIMAL U -SYSTEMS OF SMALL DIMENSIONS

Now we describe maximalU -systems of dimensionn ≤ 6. We give some matrix represen-tations forn = 6. Matrix representations forn ≤ 5 can be found elsewhere.

Recall the familyQk, k ≥ 1, (introduced in Section 5.3) of graphs determining cographiccomponents. For simplicity sake, we denoteR∗(Qn−1) =Wn. Recall thatR(Kn+1) = An.

DIMENSION 1. There is only one systemA1.

DIMENSION 2. There is only one systemA2 = R(K3) = R∗(Q1) =W2, whereQ1 = K ∗3is planar and consists of three mutually parallel edges, andQ∗1 = K3 is a triangle.

DIMENSION 3. There is only one systemA3 = R(K4) = R∗(Q2) =W3, whereQ2 = K4is planar andQ∗2 is isomorphicQ2.

DIMENSION 4. There are two maximal systems:A4 = R(K5), containing 10 vectors, andW4 = R∗(Q3), containing nine vectors, whereQ3 = K3,3 is non-planar.

DIMENSION 5. There are four maximalU -systems:A5 = R(K6), containing 15 vectors,E5, containing 10 vectors, and two cographic systems each containing 12 vectors:W5 =R∗(Q4), whereQ4 is a unique strongly 3-connected cubic non-planar graph on eight vertices,andA3⊕2 W4 = R∗(G), whereG is a non-planar cubic 3-connected graph with a separatingcut of cardinality 3.


DIMENSION 6. There are 11 maximalU -systems. Let the matrix(In, A(R)) represents aunimodular systemR of dimensionn. Below we give 11 matricesA(R) for the 11 maximalunimodular systems of dimension 6.

The first one is, of course, the graphic systemA6 = R(K7), having 21 vectors.

A(A5) =

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

.There are eight cographic maximalU -systemsR∗(Gi ), 1≤ i ≤ 8, each containing 15 vec-

tors. EachGi is a connected non-planar cubic graph on 10 vertices. The first four graphsGi , 1 ≤ i ≤ 4, are strongly 3-connected, andQ5 = G1 and the Petersen graphPe = G2are among them. HenceR∗(Gi ), 1 ≤ i ≤ 4, are components. The seven graphsGi (exceptG2 = Pe) have a Hamiltonian circuit, but the Petersen graph has a Hamiltonian path. Takingthe complement of a Hamiltonian path as a basis ofR∗(Gi ), we obtain a matrix representationconsisting only of zeros and ones. We abbreviateA(R∗(Gi )) to A(Gi ).

It is easy to reconstruct a graphG by the matrixA(G). In fact, take a path of nine edges.Each rowri , 1 ≤ i ≤ 6, of A(G) contains a set of consecutive ones, which corresponds to asetEi of consecutive edges of the path. Add to the path an edgeei such thatei andEi form acircuit. The nine edges of the path and the six edgesei form the graphG.

A(W6) = A(Q5) = A(G1) =

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

A(Pe) = A(G2) =

1 1 1 1 0 0 0 0 01 1 1 1 1 1 1 0 00 1 1 1 1 1 1 1 10 0 1 1 1 1 0 0 00 0 0 1 1 1 1 1 00 0 0 0 0 1 1 1 1

A(G3) =

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

A(G4) =

1 1 1 1 1 1 1 1 11 1 1 1 1 0 0 0 00 1 1 1 1 1 1 0 00 0 1 1 1 1 0 0 00 0 0 1 1 1 1 1 10 0 0 0 1 1 1 1 0

.


The other four maximal cographic systems correspond to graphsGi , 5 ≤ i ≤ 8, withseparating cuts of cardinality 3:R∗(G5) = A3⊕2 W4⊕2 A3 =W3⊕2 W4⊕2 W3,R∗(G6) = A3⊕2 W4⊕2 A3 =W3⊕2 W4⊕2 W3,R∗(G7) =W4⊕2 W4,R∗(G8) = A3⊕2 W5 =W3⊕2 W5.The 2-sums ofW4 = R∗(K3,3)with twoA3 = R∗(K4) are taken by one-vertex cuts ofK3,3

that correspond to adjacent and non-adjacent vertices ofK3,3 in G5 andG6, respectively. Inother words, the joining systems intersect inR∗(G5) and do not intersect inR∗(G6). Since thegraphic components are not neighbouring, theU -systemsR∗(G5) andR∗(G6) are maximal.For these two cographic systems we give additionally matrices giving explicit representationsas sums.

There is once more connected cubic non-planar graphG′ on 10 vertices such thatR∗(G′) =(A3⊕2 A3)⊕2 W4. But thisU -system is not maximal, since the two graphic componentsA3are neighbouring.

A(G5) =

1 1 1 1 1 1 1 1 11 1 0 0 0 0 0 0 00 1 1 1 1 1 0 0 00 0 0 1 1 1 1 1 00 0 0 0 1 1 1 1 10 0 0 0 0 1 1 0 0

∼=

1 1 0 0 0 0 0 0 01 1 1 1 0 1 0 0 00 1 1 1 0 0 1 1 00 0 0 1 1 1 0 0 00 0 0 1 1 0 1 1 10 0 0 0 0 0 0 1 1

A(G6) =

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

∼=

1 1 0 0 0 0 0 0 01 1 1 1 1 0 0 0 00 1 1 1 0 1 0 0 00 0 0 1 1 0 1 1 00 0 0 1 0 1 1 1 10 0 0 0 0 0 0 1 1

A(G7) =

1 1 1 0 0 0 0 0 00 1 1 1 0 0 0 0 00 0 1 1 1 1 1 0 01 1 1 1 1 1 1 1 10 0 0 0 0 1 1 1 00 0 0 0 0 0 1 1 1

A(G8) =

1 1 0 0 0 0 0 0 00 1 1 1 1 1 0 0 01 1 1 1 1 1 1 1 10 0 0 1 1 1 1 0 00 0 0 0 1 1 1 1 00 0 0 0 0 1 1 1 1

.

There are two more maximalU -systems of dimension 6:A2⊕1 E5, having 12 vectors, andR16 =A4⊕2W4, having 16 vectors, which are neither graphic nor cographic.R16 is mentionedfor the first time in [1, pp. 36–37]. Note thatR16 is a unique maximal extension of Seymour’smatroidR12 from [7]. This extension is taken by an element ofA4 and the common 3-circuitA2of A4 andW4 along which the 2-sum is taken. SinceA3⊕2 A3 = R∗((K5−e)∗) = R(K5−e),theU -systemR16 is a unique one-element extension ofR∗(G′) = A3 ⊕2 W4⊕2 A3, wherethe two sums are taken along the same systemA2 which is a one-vertex cut in each graph, i.e.,


bothA3 are neighbouring.

A(A2⊕1 E5) =

1 0 0 0 0 01 1 0 0 1 −10 −1 1 0 0 10 1 −1 1 0 00 0 1 −1 1 00 0 0 0 −1 1

A(R16) = A(A4⊕2 W4) =

0 1 0 −1 1 0 0 0 0 01 0 −1 0 −1 0 0 0 0 01 1 0 0 0 1 1 1 1 10 0 1 1 0 1 1 1 0 00 0 0 0 0 0 1 1 1 00 0 0 0 0 0 0 1 1 1

.In total, we have 11 non-isomorphic maximalU -systems of dimension 6. Among them,

the graphic system has 21 vectors, and the others have≤16 vectors. This implies that if asix-dimensionalU -system has more than 16 vectors, then it is graphic.

So we come to a curious corollary: if aU -system contains sufficiently many (with respectto its dimension) vectors, then it is graphic. The exact assertion is as follows:

(*) If a unimodular vector systemR of dimension n≥ 4has m> m(n) ≡ 7+ n2−3n2 vectors,

thenR is graphic.In fact, we know that the number of vectorsm= n(n+1)

2 of the maximal graphic systemAn is aquadratic function onn. Similarly, the number of vectorsm= 3(n−1) of a maximal cographicsystemR∗(G)of dimensionn is a linear function onn. The cardinality ofE5 equals 10. Hence,for us to prove (*), we have to show that any maximal non-graphic system of dimensionn hasnot more than 7+ n2−3n

2 vectors. It is sufficient to consider non-graphical systems minimallydifferent from graphic systems. Such systems of dimensionn areE5 ⊕1 An−4 for n ≥ 6,W5 ⊕2 An−3 for n ≥ 6 andW4 ⊕2 An−2 for n ≥ 5 . The numbers of their vectors arem1 = 15+ n2−7n

2 , m2 = 12+ n2−5n2 andm3 = m(n) = 7+ n2−3n

2 , respectively. Forn ≥ 6,m(n) = m3 > m2 > m1. Forn = 5 andn = 4, m(5) = m3 = m2 = 12, andm(4) = 9 arethe numbers of elements of maximal five- and four-dimensional cographic systems. Thereforewe obtain (*).

For example, as we saw, a five-dimensional system with more than 12 vectors, a six-dimensional system with more than 16 vectors, a seven-dimensional system with more than21 vectors, eight-dimensional system with more than 27 vectors, and so on are graphic. Thiscan be considered as a generalization of the result of Korkin and Zolotarev (more known asa result of Heller) that ann-dimensional system withn(n+1)

2 vectors is graphic (since it isisomorphic toAn).

We note that the same method can be used for construction of maximal seven-dimensionalU -systems. But it is a huge work to enumerate all cubic graphs on 12 vertices for us to obtainmaximal cographicU -systems of dimension 7. Besides, for dimensionsn ≥ 7, there aredistinct amalgamsR1⊕Ai R2, i = 1,2 for the sameR1,R2 andAi depending on embeddingsA i intoR1 andR2.

ACKNOWLEDGEMENT

The authors would like to thank K. Sonin for corrections of their English. Additionally, wenote that it was a desire of one of referees to have explicit matrix representations of maximalsix-dimensionalU -systems.


REFERENCES

1. }}T. Brylawski, Modular constructions for combinatorial geometries,Trans. AMS, 203(1975), 1–44.2. }}V. Danilov and G. Koshevoy, Discrete convexity and unimodularity I, submitted toAdv. Math.3. }}R. H. Erdahl and S. S. Ryshkov, On lattice dicing,Europ. J. Combinatorics, 15 (1994), 459–481.4. }}I. Heller, On unimodular sets of vectors, in:Recent Advances in Math. Progr., R. R. Graves and

P. Wolfe (eds.), McGraw-Hill, New York, 1963, pp. 39–53.5. }}A. Korkine and G. Zolotarev, Sur les formes quadratiques positives,Math. Ann., 11(1877), 242–292.6. }}A. Schrijver,Theory of Linear and Integer Programming, John Wiley and Sons, Chichester, 1986.7. }}P. D. Seymour, Decomposition of regular matroids,J. Comb. Theory, Ser. B, 28 (1980), 305–359.

Received 13 August 1998 and accepted 20 March 1999

V. DANILOV AND V. GRISHUKHIN

CEMI,Russian Academy of Sciences,

Nakhimovskii Prospect 47, 117418Moscow

E-mail: [email protected]

maximal unimodular systems of vectors

Documents