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Information Processing Letters 88 (2003) 143–147 www.elsevier.com/locate/ipl A notion of cross-perfect bipartite graphs Milind Dawande The University of Texas at Dallas, Richardson, TX 75083-0688, USA Received 24 February 2003; received in revised form 27 July 2003 Communicated by M. Yamashita Abstract In this note, we consider four quantities defined on a bipartite graph B: the cross-chromatic index χ (B), the biclique number w (B), the cross-free matching number m (B) and the biclique edge covering number β (B). We mention several applications of these numbers and define a notion of cross-perfect bipartite graphs. A duality between these numbers for the class of cross- perfect graphs is examined. 2003 Published by Elsevier B.V. Keywords: Combinatorial problems; Bipartite graph; Perfect graph; Integral polytope 1. Definitions Given a bipartite graph B = (U V,E), two non- adjacent edges e,e E with e = (u 1 ,v 1 ) and e = (u 2 ,v 2 ) are said to form a cross if (u 1 ,v 2 ) E and (u 2 ,v 1 ) E. Two edges are said to be cross-adjacent if either they are adjacent (i.e., share a common node) or they form a cross. A cross-free matching in B is a set of edges E E with the property that no two edges e,e E are cross-adjacent. A cross-free coloring of B is a coloring of the edge set E subject to the restriction that no pair of cross-adjacent edges has the same color. A set of edges E E is said to form a biclique if the induced subgraph corresponding to E is a complete bipartite subgraph of B .A biclique edge cover for B is a covering of the edge set E by bicliques. E-mail address: [email protected] (M. Dawande). We now consider the following four quantities on B : (1) The cross-chromatic index, χ (B), of B which is the minimum number of colors required to get a cross-free coloring of B . (2) The biclique number, w (B), of B defined as the cardinality of the maximum edge biclique in B . (3) The cross-free matching number of B , m (B), defined as the edge cardinality of the maximum cross-free matching in B . (4) The biclique edge covering number of B , β (B), defined as the minimum number of bicliques required to cover the edges of B . This note is concerned with a duality between the four numbers defined above. The duality, as stated here, resembles the classical duality between the four quantities, namely, stability number α(G), clique partition number θ(G), chromatic number γ (G) and 0020-0190/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.ipl.2003.08.006

A notion of cross-perfect bipartite graphs

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Information Processing Letters 88 (2003) 143–147

www.elsevier.com/locate/ip

A notion of cross-perfect bipartite graphs

Milind Dawande

The University of Texas at Dallas, Richardson, TX 75083-0688, USA

Received 24 February 2003; received in revised form 27 July 2003

Communicated by M. Yamashita

Abstract

In this note, we consider four quantities defined on a bipartite graphB: the cross-chromatic indexχ∗(B), the biclique numberw∗(B), the cross-free matching numberm∗(B) and the biclique edge covering numberβ∗(B). We mention several applicationof these numbers and define a notion of cross-perfect bipartite graphs. A duality between these numbers for the clasperfect graphs is examined. 2003 Published by Elsevier B.V.

Keywords:Combinatorial problems; Bipartite graph; Perfect graph; Integral polytope

1. Definitions We now consider the following four quantitie

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Given a bipartite graphB = (U ∪ V,E), two non-adjacent edgese, e′ ∈ E with e = (u1, v1) and e′ =(u2, v2) are said to form across if (u1, v2) ∈ E and(u2, v1) ∈ E. Two edges are said to becross-adjacenif either they are adjacent (i.e., share a common noor they form a cross. Across-free matchingin Bis a set of edgesE′ ⊆ E with the property that notwo edgese, e′ ∈ E′ are cross-adjacent. Across-freecoloring of B is a coloring of the edge setE subjectto the restriction that no pair of cross-adjacent edhas the same color. A set of edgesE′ ⊆ E is said toform abicliqueif the induced subgraph corresponditoE′ is a complete bipartite subgraph ofB. A bicliqueedge coverfor B is a covering of the edge setE bybicliques.

E-mail address:[email protected] (M. Dawande).

0020-0190/$ – see front matter 2003 Published by Elsevier B.Vdoi:10.1016/j.ipl.2003.08.006

(1) Thecross-chromatic index, χ∗(B), of B which isthe minimum number of colors required to gecross-free coloring ofB.

(2) Thebiclique number, w∗(B), of B defined as thecardinality of the maximum edge biclique inB.

(3) The cross-free matching numberof B, m∗(B),defined as the edge cardinality of the maximcross-free matching inB.

(4) Thebiclique edge covering numberof B, β∗(B),defined as the minimum number of bicliqurequired to cover the edges ofB.

This note is concerned with a duality betweenfour numbers defined above. The duality, as stahere, resembles the classical duality betweenfour quantities, namely, stability numberα(G), cliquepartition numberθ(G), chromatic numberγ (G) and

144 M. Dawande / Information Processing Letters 88 (2003) 143–147

the maximum clique numberω(G) for a perfect graphG [1,2,9]. In fact, we show that the duality between

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w∗(B), χ∗(B), m∗(B) andβ∗(B) can be studied byinvestigating when a certain class of graphs is perfThese numbers arise in many theoretical as welpractical settings. We review some applications innext section.

2. Applications and complexity results

The problem of finding a maximum cross-frmatching in a bipartite graph is discussed in [19]. Tproblem was studied as a special case of a moreeral problem of findingalternate cycle-free matchingin bipartite graphs. Alternate cycle-free matchinwere first investigated in [12] in the context of depedence graphs of matroids: IfM is a matroid with ba-sisB, finding a maximum cardinality alternating cycfree matching is related to finding a basis ofM forwhich the intersection withB is of minimum cardinal-ity. Another application studied in [3] and mentionin [19] is as follows: LetG be an acyclic directedgraph. Thesetup numberor step numberν(G) is de-fined to be the minimum number of arcs that mustadded toG so that the resulting graph remains acycbut contains a directed Hamiltonian path. It is shown[3] that computingν(G) is equivalent to determininthe maximum cardinality of an alternating cycle-frmatching in the bipartite graph obtained by replaceach arc with an undirected edge.

The problem of finding a maximum edge cardinity biclique has applications in manufacturing in tcomputer industry. This application is discussed intail in [5]. In short, the relationship between a setproducts and a set of components which constituteproducts can be represented by a bipartite graph.way to reduce manufacturing lead times is to redthe final assembly times for the products by creatsub-assemblies. Finding good sub-assemblies incontext can be done by finding large edge-cardinabicliques in the bipartite graph. Another interesting aplication, also described in [5], occurs in the areaformal concept analysis [7].

Covering the edges of a bipartite graph by bicliquhas an interesting application [4] in combinatorchemistry:Split Synthesis[6,13] is a method of choiceto build libraries for combinatorial chemistry. Th

the other end is extended one residue at a timparallel across a set of beads. Library construcproceeds through an interleaved sequence of divgrow and combine operations. A set of beads candividedor partitioned into different chambers, andthe molecules on the beads in a particular chamgrown or extended by the same set of residues. Athis reaction, the sets of beads can becombinedsothat they can be grown within the same chambOnce combined, two sets of beads cannot be sepaagain, although the bead mixture can be dividedwill. Split synthesis can be modeled by a direcacyclic graph in which each node represents a gstep on a particular subset of beads. In [4], itshown that optimizing split synthesis constructionintimately related to covering the edges of a bipargraph by bicliques.

Findingβ∗(B) is NP-hard [8]. In [19], it is shownthat findingm∗(B) is NP-hard. Recently, Peeters [1showed that findingw∗(B) is also NP-hard. To the beof our knowledge, the complexity of findingχ∗(B) isopen.

The problem of finding complete bipartite sugraphs in bipartite graphs has recently received sattention. We mention the following:

(i) In [17], a semidefinite relaxation for the probleof findingw∗(B) is considered.

(ii) In [10], bounds on the size of the maximucomplete bipartite subgraph of a general graphGare derived using the eigenvalues of the marepresentation ofG.

(iii) In [11], approximation algorithms are presentfor the problem (related to the maximum weigbiclique) of minimizing the total weight of nodeand edges deleted in a bipartite graph so thatremaining subgraph is a biclique.

3. Formulations and results

Clearly, a biclique cannot have more than one efrom a cross-free matching. This immediately givthe following integer programming formulation fow∗(B)w∗(B)= max

{1x: K∗x � 1, x ∈ {0,1}n},

M. Dawande / Information Processing Letters 88 (2003) 143–147 145

wheren = |E| andK∗ is a (0,1) incidence matrixwhose rows correspond to maximal cross-free match-

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B iff V (E′) is a independent set inL′(B). The cliquepolytope forL′(B) isQ= {x: Sx � 1, x ∈Rn+} where

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ings ofB and whose columns correspond to the edof B. Let m be the number of maximal cross-frematchings inB. It is also easy to see that

χ∗(B)= min{1y: yK∗ � 1, y ∈ {0,1}m}

.

The fractional biclique polytopeof B is P = {x ∈Rn+: K∗x � 1}. We denote the convex hull of feasibinteger solutions ofP by PI .

We generalize the definitions ofw∗ and χ∗ toconsider binary objective functions. Forc ∈ {0,1}n

w∗(Bc)= max{cx: K∗x � 1, x ∈ {0,1}n},

χ∗(Bc)= min{1y: yK∗ � c, y ∈ {0,1}m}

.

The definition ofm∗(Bc) (respectivelyβ∗(Bc)) issimilar to that ofw∗(Bc) (respectivelyχ∗(Bc)) where,instead ofK∗, we use the(0,1) incidence matrixwhose rows correspond to maximal bicliques inBand columns correspond to edges ofB. Note that thedefinitions ofw∗(B) andw∗(Bc) differ only in theobjective function. The constraint matrixK∗ is thesame for both. Also, ifc �= 1, thenw∗(Bc) doesnotcorrespond to the optimal edge cardinality bicliquethe subgraph (either edge-induced or node-induccorresponding to the vectorc. This is an importandistinction between the definition ofw∗(Bc) and thecorresponding definition for the maximum cliquea graph: For a graphG, the formulation for themaximum clique number,ω(G), generalized to binarobjective functions, can be written as

ω(Gc)= max{cx: Kx � 1, x ∈ {0,1}p},

wherep = |V (G)| andK is the clique matrix ofG[16]. Here,ω(Gc) corresponds to the maximum cliquon the node-induced subgraph ofG correspondingto c. A similar observation can be made aboutχ∗(Bc).

Definition. B is cross-perfect ifw∗(Bc)= χ∗(Bc) forall vectorsc ∈ {0,1}n.

Theorem 3.1. B is cross-perfect iffP is integral.

Proof. Consider the line graph,L(B), ofB. For every4-hole inL(B), add the two chords. Let the new grabe calledL′(B). We refer toL′(B) as themodifiedlinegraph ofB. Then,E′ ⊆ E is a cross-free matching i

the rows ofS correspond to maximal independent sand the columns correspond to the nodes ofL′(B).But Q = P . Thus, B is cross-perfect implies thaω(H) = γ (H) for every node induced subgraphHof L′(B) whereω(H) denotes the cardinality of thmaximum clique inH and γ (H) is the chromaticnumber ofH . Then,L′(B) is perfect by the weakperfect graph theorem [14,15]. It follows thatQ andhenceP is integral. Conversely, ifP is integral, so isQ and henceω(H) = γ (H) for every node inducedsubgraphH of L′(B). Thus,w∗(Bc)= χ∗(Bc) for allvectorsc ∈ {0,1}n andB is cross-perfect. ✷Note 1. An alternative proof of Theorem 3.1 wasuggested by Prof. R. Chandrasekaran: The coluintersection graph of the matrixK∗ is the complemengraph of the modified line graphL′(B) defined aboveThe definition of the cross-perfectness and the pergraph theorem [14] then imply thatP is integral.

Note 2. Even though the proof of Theorem 3.1 usthe correspondence between the biclique numberw∗and the clique numberω, the structure of these twproblems is fundamentally different. In fact, even tcomplexity question of computingw∗ was open untilthe recent result of Peeters [18]. As noted above,duality in Theorems 3.1 and 3.4 (given below) cbe derived from the classical duality in the perfegraph literature. However, the quantities involvedthis duality are fundamentally different from thoseperfect graphs.

In general, the biclique polytope,P , is not integral.This follows because ifP were integral, so wouldthe clique polytope,Q, of the graphL′(B) which inturn would mean thatL′(B) is perfect. However, iis easy to demonstrate a bipartite graphB such thatL′(B) contains a 5-hole as a node-induced subgr(see Fig. 1).

Corollary 3.2. B is cross-perfect iff the modified lingraphL′(B) is perfect.

This observation suggests that the propertycross-perfectness in bipartite graphs can be stuby studying perfectness in modified line graphs. IfB

146 M. Dawande / Information Processing Letters 88 (2003) 143–147

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Theorem 3.5 and the definition of cross-perfectnessimply the following result.

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Fig. 1. A bipartite graph whose biclique polytope is not integra

is cross-perfect then, by Theorem 3.1, the constraK∗x � 1 are sufficient to describePI . Note that everyclique inL′(B) does not correspond to a biclique inB.However, everymaximalclique inL′(B) does corre-spond to a biclique. Hence, a maximum clique, whis also a maximal clique, corresponds to a maximbiclique inB. This observation immediately gives uthe following result.

Theorem 3.3. If B is cross-perfect thenw∗(B) andχ∗(B) are polynomially solvable.

Every independent set inL′(B) corresponds to across-free matching inB. Consider a covering,C,of the nodes ofL′(B) by cliques. A cliquec∗ in Ceither corresponds to a biclique inB or if not, c∗is contained in a maximal clique which corresponto a biclique inB. Thus, α(L′(B)) = m∗(B) andθ(L′(B))= β∗(B). LetH be a node induced subgrapof L′(B) and c ∈ {0,1}n be the incidence vector othe nodes corresponding toH . Then, it is easy tosee thatα(H) = m∗(Bc) and θ(H) = β∗(Bc) (seeSection 3 for a definition ofm∗(Bc) andβ∗(Bc)). Thiscorrespondence along with the weak perfect grtheorem [14,15] gives us the following result.

Theorem 3.4. The following are equivalent:

(1) w∗(Bc)= χ∗(Bc) for all vectorsc ∈ {0,1}n.(2) m∗(Bc)= β∗(Bc) for all vectorsc ∈ {0,1}n.

Theorem 3.5. Node-induced subgraphs of cross-pfect graphs are cross-perfect.

Proof. The fractional biclique polytope,PH , corre-sponding to a node induced subgraphH of B is a faceof P . Hence ifP is integral, so isPH . ✷

Corollary 3.6. If B is cross-perfect thenw∗(H) =χ∗(H) for all node induced subgraphsH ofB.

It is not known whether the class of modified lingraphs of cross-perfect bipartite graphs is a new cof perfect graphs.

We end this note by summarizing its contents:have examined the relationship between four qutities defined on a bipartite graphB viz., the cross-chromatic indexχ∗(B), the biclique numberw∗(B),the cross-free matching numberm∗(B) and the bi-clique edge covering numberβ∗(B). We have mentioned several applications of these numbers. A duabetween these numbers for the class of cross-pebipartite graphs can be studied using classical pergraph literature. However, the structure of these qutities is fundamentally different from that of the corrsponding quantities arising in perfect graphs.

Acknowledgements

I thank Prof. R. Chandrasekaran for helpful discsions and suggestions.

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