A Characterization of Konig–Egervary GraphsUsing a Common Property of All Maximum

Matchings

Vadim E. Levit 1

Department of Computer Science and MathematicsAriel University Center of Samaria

Ariel, Israel

Eugen Mandrescu 2

Department of Computer ScienceHolon Institute of Technology

Holon, Israel

Abstract

The independence number of a graph G, denoted by α(G), is the cardinality ofan independent set of maximum size in G, while μ(G) is the size of a maximummatching in G, i.e., its matching number. G is a Konig–Egervary graph if its orderequals α(G)+μ(G). In this paper we give a new characterization of Konig–Egervarygraphs. We also deduce some properties of vertices belonging to all maximumindependent sets of a Konig–Egervary graph.

Keywords: maximum independent set, maximum matching, core.

1 Email: levitv@ariel.ac.il2 Email: eugen m@hit.ac.il

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 38 (2011) 565–570

1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2011.09.092

1 Introduction

Throughout this paper G = (V,E) is a simple (i.e., a finite, undirected, loop-less and without multiple edges) graph with vertex set V = V (G), edge setE = E(G), and order n = |V |. If X ⊂ V , then G[X] is the subgraph of Gspanned by X. By G −W we mean the subgraph G[V −W ], if W ⊂ V (G).For F ⊂ E(G), by G − F we denote the partial subgraph of G obtained bydeleting the edges of F , and we use G − e, if W = {e}. If A,B ⊂ V andA ∩ B = ∅, then (A,B) stands for the set {e = ab : a ∈ A, b ∈ B, e ∈ E}.

A set S of vertices is independent, i.e., S ∈ Ind (G), if no two verticesfrom S are adjacent. The independence number of G, denoted by α(G), is thecardinality of a maximum independent set of G. Let Ω(G) denote the set ofall maximum independent sets of G and core(G) = ∩{S : S ∈ Ω(G)} [12]. Amatching (i.e., a set of non-incident edges of G) of maximum cardinality μ(G)is a maximum matching, and a perfect matching is one covering all vertices ofG. If α(G)+μ(G) = n, then G is called a Konig-Egervary graph. We attributethis definition to Deming [4], and Sterboul [20]. These graphs were studiedin [2,8,18], and generalized in [1,19]. Several properties of Konig-Egervarygraphs are presented in [11,13,14,15].

Theorem 1.1 [6,7] Every bipartite graph is a Konig-Egervary graph.

Let M be a maximum matching of a graph G. To adopt Edmonds’sterminology [5], we recall the following terms for G relative to M . An al-ternating path from a vertex x to a vertex y is an x, y-path whose edgesare in and out of M alternatively. A vertex x is exposed relative to M ifx is not the endpoint of an edge belonging to M . An odd cycle C withV (C) = {x0, x1, ..., x2k} and E(C) = {xixi+1 : 0 ≤ i ≤ 2k − 1} ∪ {x2k, x0},such that x1x2, x3x4, ..., x2k−1x2k ∈M is a blossom relative to M . The vertexx0 is the base of the blossom. The stem is an even length alternating pathjoining the base of a blossom and an exposed vertex for M . The base is theonly common vertex to the blossom and the stem. A flower is a blossom to-gether with its stem. A posy consists of two (not necessarily disjoint) blossomsjoined by an odd length alternating path whose first and last edges belong toM . The endpoints of the path are exactly the bases of the two blossoms.

Theorem 1.2 (Sterboul) [20] For a graph G, the following properties areequivalent:

(i) G is a Konig-Egervary graph;

(ii) there are no flower and no posy relative to some maximum matching;

(iii) there are no flower and no posy relative to every maximum matching.

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 38 (2011) 565–570566

In [9], Korach et al. described Konig-Egervary graphs in terms of excludedconfigurations. In [17], Lovasz gave a characterization of Konig-Egervarygraphs having a perfect matching, in terms of certain forbidden subgraphswith respect to a specific perfect matching of the graph.

The number d(G) = max{|S| − |N(S)| : S ∈ Ind(G)} is called the criticaldifference of G. An independent set A is critical if |A| − |N(A)| = d(G),and the critical independence number αc(G) is the cardinality of a maximumcritical independent set [21]. In [10] it was shown that G is a Konig-Egervarygraph if and only if αc(G) = α(G), thus giving a positive answer to theGraffiti.pc 329 conjecture [3]. Extending these results, we have proved thefollowing Sterboul style characterization of Konig-Egervary graphs.

Theorem 1.3 [16] For a graph G, the following assertions are equivalent:

(i) G is a Konig-Egervary graph;

(ii) there is S ∈ Ω(G), such that S is critical, i.e., αc(G) = α(G);

(iii) every S ∈ Ω(G) is critical.

In this paper we give a new characterization of Konig-Egervary graphsbased on a common property of its maximum matchings.

2 Main results

Notice that all the maximum matchings of the graphs G1 and G2 from Figure1 are included in (S, V (Gi)− S), i = 1, 2, for each S ∈ Ω(Gi), i = 1, 2. On theother hand, M1 = {xu, yz} and M2 = {xu, vz} are maximum matchings of thegraph G3 from Figure 1, and S = {u, v} ∈ Ω(G3), but M1 � (S, V (G3)− S),while M2 ⊆ (S, V (G3)− S).

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G2 � � �� �

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u v

xz y

G3

Fig. 1. G1 and G2 are Konig–Egervary graphs, but only G2 has a perfect matching.G3 is not a Konig–Egervary graph.

Theorem 2.1 The following properties are equivalent:

(i) G = (V,E) is a Konig-Egervary graph;

(ii) each maximum matching of G is contained in (S, V − S) for someS ∈ Ω(G);

(iii) each maximum matching of G is contained in (S, V − S) for everyS ∈ Ω(G).

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 38 (2011) 565–570 567

Proof. [The principle of the proof]

The implications (i) =⇒ (iii) and (iii) =⇒ (ii) are simple.

The implication (ii) =⇒ (i) goes by reductio ad absurdum. The proofmakes a careful study of the vertices reachable by alternating paths originatingat a vertex exposed by a maximum matching. It reminds the techniquesneeded to justify both Hall’s Theorem and the Konig-Egervary Theorem 1.1.On the other hand, it works differently, since the graphs under investigationare not assumed to be bipartite.

Assume, to the contrary, that G is not a Konig-Egervary graph, and letM = {akbk : 1 ≤ k ≤ μ(G)} be a maximum matching. Since M ⊆ (S, V − S),we infer that μ(G) ≤ |S| = α(G), and one may suppose that:

A = {ak : 1 ≤ k ≤ μ(G)} ⊆ S, while B = {bk : 1 ≤ k ≤ μ(G)} ⊆ V − S.Let x ∈ V − S −B and Sx be the set of vertices v ∈ B such that there existsa path x = v1, v2, v3, ..., v2k+1 = v, where v2iv2i+1 ∈M, v2i ∈ A and v2i+1 ∈ B.Let us also define M(Sx) = {aj ∈ A : bj ∈ Sx}.

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V − S

G

S

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Sx

M(Sx)

Fig. 2. S ∈ Ω(G) and S1 = {x} ∪ Sx ∪ (S −M (Sx)).

Further we gradually approach the end of the proof justifying the followingclaims:

Claim 1. {x} ∪ (S −M(Sx)) is an independent set in G.

Claim 2. Sx is independent.

Claim 3. No edge joins x to some vertex of Sx.

Claim 4. No edge joins a vertex from S −M(Sx) to a vertex of Sx.

Finally, we conclude that S1 = {x} ∪ Sx ∪ (S −M(Sx)) is an independentset in G, but this leads to the inequality |S1| = |S|+ 1 > α(G), which clearlycontradicts the fact that S is a maximum independent set in G. �

Proposition 2.2 If G = (V,E) is a Konig-Egervary graph, then

(i) for every maximum matching, each exposed vertex belongs to core(G).

(ii) at least one of the endpoints of every edge of G is a μ-critical vertex.

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 38 (2011) 565–570568

Proposition 2.3 Let G be a Konig-Egervary graph and v ∈ V (G) be suchthat G − v is still a Konig-Egervary graph. Then v ∈ core(G) if and only ifthere exists a maximum matching that does not saturate v.

Remark 2.4 The conclusion of Proposition 2.3 fails if the hypothesis thatG− v is a Konig-Egervary graph is omitted; e.g., each maximum matching ofthe graph G from Figure 3 saturates c ∈ core(G) = {a, b, c}.

� � � � �� �

���

a

b

cu v

x

y

G

Fig. 3. G is a Konig-Egervary graph with core(G) = {a, b, c}.

Corollary 2.5 For every bipartite graph G, the vertex v ∈ core(G) if andonly if there exists a maximum matching that does not saturate v.

3 Concluding remarks

In this paper we give a new characterization of Konig-Egervary graphs similarin form to Sterboul’s Theorem 1.2. It seems to be interesting to characterizeKonig-Egervary graphs with unique maximum independent sets.

References

[1] J. M. Bourjolly, P. L. Hammer, B. Simeone, Node weighted graphs having Konig-Egervary property, Math. Programming Study 22 (1984) 44-63.

[2] J. M. Bourjolly, W. R. Pulleyblank, Konig-Egervary graphs, 2-bicritical graphsand fractional matchings, Discrete Applied Mathematics 24 (1989) 63–82.

[3] E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc,http://cms.dt.uh.edu/faculty/delavinae/research/wowII/

[4] R. W. Deming, Independence numbers of graphs - an extension of the Konig-Egervary theorem, Discrete Mathematics 27 (1979) 23–33.

[5] J. Edmonds, Paths, trees and flowers, Canadian Journal of Mathematics 17(1965) 449-467.

[6] E. Egervary, On combinatorial properties of matrices, Matematikai Lapok 38(1931) 16–28.

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 38 (2011) 565–570 569

[7] D. Konig, Graphen und Matrizen, Matematikai Lapok 38 (1931) 116–119.

[8] E. Korach, On dual integrality, min-max equalities and algorithmsin combinatorial programming, University of Waterloo, Department ofCombinatorics and Optimization, Ph.D. Thesis, 1982.

[9] E. Korach, T. Nguyen, B. Peis, Subgraph characterization of red/blue-splitgraphs and Konig-Egervary graphs, Proceedings of the Seventeenth AnnualACM-SIAM Symposium on Discrete Algorithms, ACM Press (2006) 842-850.

[10] C. E. Larson, The critical independence number and an independencedecomposition, European Journal of Combinatorics 32 (2011) 294-300.

[11] V. E. Levit, E. Mandrescu, Well-covered and Konig-Egervary graphs,Congressus Numerantium 130 (1998) 209–218.

[12] V. E. Levit, E. Mandrescu, Combinatorial properties of the family of maximumstable sets of a graph, Discrete Applied Mathematics 117 (2002) 149-161.

[13] V. E. Levit, E. Mandrescu, On α+-stable Konig-Egervary graphs, DiscreteMathematics 263 (2003) 179–190.

[14] V. E. Levit, E. Mandrescu, On α-critical edges in Konig-Egervary graphs,Discrete Mathematics 306 (2006) 1684-1693.

[15] V. E. Levit, E. Mandrescu, Partial unimodality for independence polynomialsof Konig-Egervary graphs, Congressus Numerantium 179 (2006) 109–119.

[16] V. E. Levit, E. Mandrescu, Critical independent sets and Konig–Egervarygraphs, Graphs and Combinatorics (2011) (accepted), arXiv:0906.4609[math.CO], 8pp.

[17] L. Lovasz, Ear decomposition of matching covered graphs, Combinatorica 3(1983) 105-117.

[18] L. Lovasz, M. D. Plummer, Matching Theory, Annals of Discrete Mathematics29 (1986) North-Holland.

[19] V. T. Paschos, M. Demange, A generalization of Konig-Egervary graphsand heuristics for the maximum independent set problem with improvedapproximation ratios, European Journal of Operational Research 97 (1997)580–592.

[20] F. Sterboul, A characterization of the graphs in which the transversal numberequals the matching number, Journal of Combinatorial Theory Series B 27(1979) 228–229.

[21] C. Q. Zhang, Finding critical independent sets and critical vertex subsets arepolynomial problems, SIAM J. Discrete Mathematics 3 (1990) 431-438.

V.E. Levit, E. Mandrescu / Electronic Notes in Discrete Mathematics 38 (2011) 565–570570