balanced graphs and network flows

Balanced Graphs and Network Flows

Stephen G. Penrice*

S.U.N.Y. College at Cortland, Cortland, New York 13045

A graph G is balanced if the maximum ratio of edges to vertices, taken over all subgraphs of G, occurs atG itself. This note uses the max-flow/min-cut theorem to prove a good characterization of balanced graphs.This characterization is then applied to some results on how balanced graphs may be combined to form alarger balanced graph. In particular, we show that edge-transitive graphs and complete m-partite graphs arebalanced, that a product or lexicographic product of balanced graphs is balanced, and that the normalproduct of a balanced graph and a regular graph is balanced. q 1997 John Wiley & Sons, Inc.

1. INTRODUCTION give a unified presentation of several facts about balancedgraphs. We use the characterization to prove a result onhow balanced graphs can be joined together to form aThroughout this paper, graphs are assumed to be finitelarger balanced graph. As corollaries, we will show thatwithout loops or multiple edges. Let G be a graph withall edge-transitive graphs are balanced and that if twovertex set V (G) and edge set E(G) . Let d(G) Å e(G) /balanced graphs are joined by placing all possible edgesn(G) , where n(G) and e(G) denote ÉV (G)É andbetween them, the resulting graph is balanced. We closeÉE(G)É, respectively. We say that G is balanced if forthe paper by showing that a product or lexicographicall subgraphs H of Gproduct of balanced graphs is balanced and that a normalproduct of a balanced graph and a regular graph is bal-d(H) ° d(G) , (1)anced. These latter results are also consequences of ourcharacterization.where V ( H) is assumed to be nonempty. Clearly, to

Balanced graphs first arose in the study of randomcheck whether a given graph G is balanced, it suffices tographs. See [1] . More recently, there have been severalverify (1) for all nonempty induced subgraphs H . Thepapers written on balanced extensions of graphs. Amongmost commonly cited examples of balanced graphs in-these is [4] , which gives references to several more pa-clude trees, complete bipartite graphs, and regular graphs.pers on this topic. In [5] , a result is proved which impliesIn Section 2, we prove a good characterization of bal-that maximal planar graphs and maximal outerplanaranced graphs. The proof shows that deciding whether agraphs are balanced.given graph G is balanced is equivalent to finding a mini-

mum cut in a certain network related to G . The main ideaof this proof first appeared in [3] , which gives an efficientalgorithm for the more general problem of finding a sub- 2. A CHARACTERIZATION OF BALANCEDgraph H of a given graph G such that d(H) is maximum. GRAPHSHowever, our characterization of balanced graphs appearsto be new. In Section 3, this characterization is used to

Let G be a graph. We define the incidence graph I(G)to be the bipartite graph with independent sets A Å E(G)and B Å V (G) and with e √ A adjacent to £ √ B if and*Current address: Mail Stop 40P, Educational Testing Service,

Princeton, NJ 08541. only if £ is an endpoint of e in G . A normalized edge

NETWORKS, Vol. 29 (1997) 77–80q 1997 John Wiley & Sons, Inc. CCC 0028-3045/97/020077-04

77

742/ 8u0d$$0004 01-20-97 18:17:23 netwa W: Networks

78 PENRICE

weight assignment is an assignment of nonnegative real Since no cut has capacity less than n(G)e(G) , this im-plies that d(H) ° d(G) . jweights to the edges of I(G) such that

Using the proof of Theorem 1 and any efficient mini-1. For every e √ A , the sum of the weights of the edgesmum cut algorithm gives a polynomial algorithm for de-of I(G) incident to e is n(G) ;termining whether a given graph G is balanced. In the

2. For every £ √ B , the sum of the weights of the edgescase where G is not balanced, we may use the minimum

of I(G) incident to £ is e(G) .cut to find an induced subgraph H such that d (G )õ d(H) ; the vertices of H will be the vertices correspond-

Theorem 1. Let G be a graph. Then, G is balanced if and ing to nodes in S > B . However, there are examples ofonly if I(G) has a normalized edge weight assignment. graphs G where the algorithm produces a subgraph H

with d(H) ú d(G) , but d(H) is not maximum. For aProof. We adopt the network terminology used in [2] . version of this algorithm which does find a subgraph of

In particular, if D is a network with source s and sink t , maximum density in polynomial time, see [3] .an s–t separator is a set S such that s √ S and t √/ S ;thus, each s–t separator determines a cut in D . Considerthe network D formed by

3. APPLICATIONS

• Orienting all edges of I(G) from A to B and giving In this section, we apply Theorem 1 to prove some resultsthese arcs capacity n(G)e(G) . on how balanced graphs may be combined to form a

• Creating a source s with an arc of capacity n(G) going larger balanced graph. The advantage in using Theoremfrom s to each node in A . 1 is that it gives a structure, the normalized edge weight

assignment, which, under the right conditions, can be• Creating a sink t with an arc of capacity e(G) enteringextended from the smaller graphs to the larger one.t from each node in B .

If G is a graph and U ⊆ V (G) , let GU denote thesubgraph induced by U . If x √ V (G) , let N(x) denote

By the definition of D , I(G) has a normalized edgethe set of neighbors of x . A regular partition of V (G)

weight assignment if and only if D admits a flow of valueis a partition of V (G) into sets V1 , V2 , . . . , Vm such that

n(G)e(G) . Since D has a cut of capacity n(G)e(G) , bythere exist integers k1 , k2 , . . . , km where ÉN(x) > ViÉthe max-flow/min-cut theorem, D has a flow of value Å ki for all i √ {1, . . . , m} and x √Q Vi .

n(G)e(G) if and only if it has no cut of capacity strictlyless than n(G)e(G) . Thus, it suffices to show that G is

Lemma 1. Let G be a graph. Suppose that V1 and V2balanced if and only if D has no cut of capacity strictlyform a regular partition of V (G) with d(GV1

) ¢ d(GV2) ,less than n(G)e(G) .

and suppose that GV1and GV2

are balanced. Then, theSuppose that G is balanced. Let S be an s–t separatorfollowing are equivalent:in D which determines a cut of minimum capacity. Note

that since the cut determined by S has minimum capacity,for every x √ A , x √ S if and only if both of x’s neighbors 1. G is balanced;in B are in S . Thus, if H is the subgraph of G induced 2. d(GV1

) ° d(G);by S > B , E(H) Å S > A . Therefore, the capacity of the 3. d(GV1

) ° d(GV2) / k1 .

cut determined by S isProof. By the definition of a balanced graph, Condi-

tion 1 implies Condition 2. Condition 2 can be writtenn(G)e(G) 0 n(G)e(H) / n(H)e(G) ,

e(GV1)

n(GV1)°

e(GV1) / e(GV2

) / k1n(GV2)

n(GV1) / n(GV2

).which is less than n(G)e(G) only if d(G)õ d(H) . Since

G is balanced, the capacity of the cut determined by S isat least n(G)e(G) .

The equivalence of Condition 2 and Condition 3 followsConversely, suppose that no cut in D has capacity lessfrom straightforward algebra. Thus, we need only showthan n(G)e(G) . Let H be an induced subgraph of G .that Condition 2 implies Condition 1.Define an s– t separator S in D by setting S Å {s}

Suppose that Condition 2 holds. Then, d(GV2) ° d(G).< E(H) < V ( H) . The capacity of the cut determined

Since GV1and GV2

are balanced, we may assume we haveby S isfixed normalized edge weight assignments for I(GV1

) andI(GV2

) . Our goal is to produce a normalized edge weightn(G)e(G) 0 n(G)e(H) / n(H)e(G) .


BALANCED GRAPHS AND NETWORK FLOWS 79

assignment for I(G) . Recall that a typical edge of I(G) d(GV1<rrr< Vm02) ° d(GVm01

) / k1 /rrr/ km02 ,is of the form (e , £) where e √ E(G) , £ √ V (G) , and £

is an endpoint of e . There are two cases to consider: and by the hypothesis of the theorem, we have

CASE 1. e √ E(GVi) . Give (e , £) the weight

d(GVm01) ° d(GVm

) / km01 .

n(G)n(GVi

)w , It follows that

d(GV1<rrr< Vm02) ° d(GVm

) / k1 /rrr/ km01 ,where w is the weight of (e , £) in the normalized edgeweight assignment for I(GVi

) .and

CASE 2. £ √ Vi and e joins £ to a vertex in Vj , i x j .d(GVm01

) / k1 /rrr/ km02 ° d(GVm) / k1 /rrrGive (e , £) the edge weight

/ km01 .1kjSe(G) 0 e(GVi

)n(G)n(GVi

) D .Applying Lemma 2 gives us

d(GV1<rrr>Vm01) ° d(GVm

) / k1 /rrr/ km01 .Clearly, the edge weights in Case 1 are nonnegative,because each is the product of a positive rational and

By Lemma 1, this implies that G is balanced. jan edge weight from a normalized edge weight assign-ment. Condition 2 ensures that

Corollary 1. Let G be a graph. Suppose that V1 , . . . ,Vm form a regular partition of V (G ) with d (GV1

)e(G) 0 e(GVi

)n(G)n(GVi

)¢ 0, Å d(GV2

) ÅrrrÅ d(GVm) and that GVi

is balanced for1 ° i ° m. Then, G is balanced. j

and thus the weights are nonnegative in Case 2.An automorphism of a graph G is a bijection fromIt is straightforward to verify that the remaining condi-

V (G) to itself which preserves edges and nonedges. Ations of a normalized edge weight assignment are satis-graph G is edge-transitive if for every pair of edges e1fied at nodes of D that correspond to vertices of G orand e2 there is an automorphism which maps e1 to e2 .to edges of G that do not join vertices of G1 to vertices

of G2 . The condition can be verified at the remainingCorollary 2. If G is an edge-transitive graph, then G isnodes of D by noting that the sum of the edge weightsbalanced.at any two of these nodes are equal and that the sum

of the weights of all of the edges of D is n(G)e(G) . Proof. If G is regular, then clearly it is balanced. If it[Note that these latter nodes are not present in D when is not regular, then there exist integers r and s , r x s ,d(G) Å d(GV1

) Å d(GV2) , because there are no edges such that every edge of G joins a vertex of degree r to a

between V1 and V2 .] vertex of degree s . The vertices of degree r and the verti-ces of degree s form a pair of independent sets, so we

Lemma 2. Let a, b, c, d, x be positive real numbers such may apply Corollary 1 by letting V1 be the set of verticesthat a/b ° x and c/d ° x. Then, [(a / c) /(b / d)] of degree r and letting V2 be the set of vertices of de-° x. j gree s . j

Corollary 3. Let G be a graph. Suppose that V1 , . . . ,Theorem 2. Let G be a graph. Suppose that V1 , V2 , . . . ,Vm form a partition of V (G) such that GVi

is balancedVm, m ¢ 2 , form a regular partition of V (G) withd(GV1

) ¢ d(GV2) ¢rrr¢ d(GVm

) , and suppose that for 1 ° i ° m, and x Ç y whenever x √ Vi , y √ Vj andi x j. Then, G is balanced.GV1

, . . . , GVmare balanced. If d(GVi

) ° d(GVi/1) / ki

for 1 ° i ° m 0 1 , then G is balanced. Proof. Without loss of generality, d(GV1) ¢ d(GV2

)¢rrr° d(GVm

) . Apply Theorem 2, with ki Å ÉViÉ,Proof. We induct on m . The case m Å 2 follows fromthe lemma. Suppose that mú 2. By the induction hypoth- noting that d(H) ° n(H) for all graphs H . jesis, GV1<rrr< Vm02

and GV1<rrr< Vm01are balanced.

Applying Lemma 1, it follows that A complete m-partite graph is a graph G such that


80 PENRICE

V (G) can be partitioned into m independent sets V1 , . . . , of ({ i , j}, i) in the normalized edge weight assignmentfor I(G) .Vm such that x Ç y whenever x √ Vi , y √ Vj , and i x j .

Note that Corollary 1 and Corollary 3 each imply theWe leave it to the reader to verify that we have definedfollowing:

a normalized edge weight assignment for I(P) . j

Corollary 4. If G is a complete m-partite graph, then GCorollary 5. If G and H are balanced graphs, then Gis balanced. j1 H is balanced.

Proof. Apply Theorem 3 with k Å 1. jFinally, we prove a result which tells us how the bal-anced property behaves under three kinds of products of

Corollary 6. If G and H are balanced graphs, the G[H]graphs. Let G and H be graphs. The product G 1 H isis balanced.the graph on the vertex set V (G) 1 V ( H) with (x , y)

Ç (x *, y*) if and only if x Å x * and y Ç y *, or x Ç x * Proof. Apply Theorem 3 with k Å n(H) . jand y Å y *. The lexicographic product G[H] is the graphon V (G) 1 V ( H) with (x , y) Ç (x *, y*) if and only if Corollary 7. If G is a balanced graph and H is a regularx Ç x *, or x Å x * and y Ç y *. The normal product GrH graph of degree d, then GrH is balanced.is the graph on the vertex set V (G) 1 V ( H) with (x , y)

Proof. Apply Theorem 3 with k Å d / 1. jÇ (x *, y*) if and only if x Å x * and y Ç y *, or x Ç x *and y Å y *, or x Ç x * and y Ç y *.

This author received support as a DIMACS Research/Edu-cation Postdoctoral Fellow from August 1, 1994 to July 31,

Theorem 3. Let G and H be balanced graphs with V (G) 1995. This position was funded through NSF Contract STC 91-Å {1, 2, . . . , m}. Suppose that P is a graph and k is a 19999.positive integer such that V ( P) can be partitioned intosets V1 , V2 , . . . , Vm with PVi

isomorphic to H for 1 ° iREFERENCES° m, ÉN(x) > VjÉ Å k whenever x √ Vi and i Ç j in G,

and ÉN(x) > VjÉ Å 0 whenever x √ Vi and i Ç/ j in G.[1] P. Erdos and A. Renyi, On the evolution of randomThen, P is balanced.

graphs, Pub. Math. Hung. Acad. Sci. 5 (1960) 17–61.Proof. By Theorem 1, we may assume we have fixed [ 2] L. Lovasz and M. Plummer, Matching Theory, North-

normalized edge weight assignments for I(G) and I(H) , Holland, Amsterdam (1986).and thus for I(PVi

) , 1 ° i ° m . We must produce a [ 3] J. C. Picard and M. Queyranne, A network flow solutionto some non-linear 0-1 programming problems, with ap-normalized edge weight assignment for I(P) . A typicalplications to graph theory. Networks 12 (1982) 141–159.edge of I(P) is of the form (e , £) , where e √ E(P) , £

[ 4] A. Rucinski and A. Vince, The solution to an extremal√ V ( P) , and £ is an endpoint of e . There are two casesproblem on balanced extensions of graphs. J. Graph The-to consider:ory 17 (1993) 417–431.

CASE 1. e √ E(PVi) for some i . Assign (e , £) the weight [ 5] N. Veerapandiyan and S. Arumugam, On balanced

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CASE 2. £ √ Vi and e joins £ to a vertex in Vj , j x i . Received March 24, 1995Accepted June 5, 1996Assign (e , £) the weight n(H)w , where w is the weight


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