Critical independent sets and Konig-Egervary graphs

Let Ind(G) be the family of all independent sets of graph G and (G) be the cardinality of an independent set of maximum size, while the set core(G) is the intersection of all maximum independent sets. An independent set A is called critical if jAj jN(A)j = maxfjSj jN(S)j : S 2 Ind(G)g. Let (G) be the size of a maximum matching, and def(G) = jV (G)j 2 (G) be the de…ciency of G. Our main …nding is the following series of equalities def (G) = (G) (G) = jcore(G)j jN(core(G))j = maxfjSj jN(S)j : S 2 Ind(G)g holding for every König–Egerváry graph G. 1 Introduction The neighborhood of a vertex v 2 V is the set N(v) = fw : w 2 V and vw 2 Eg, and N(A) = [fN(v) : v 2 Ag, N [A] = A [ N(A) for A V . A set S V (G) is independent if no two vertices from S are adjacent, and by Ind(G) we mean the set of all the independent sets of G. The independence number of G is (G) = maxfjSj : S 2 Ind(G)g. Let us denote the set fS : S is a maximum independent set of Gg by (G), and let core(G) = \fS : S 2 (G)g [5]. The matching number (G) is the cardinality of a maximum matching of G. If (G) + (G) = jV j, then G is called a König-Egerváry graph [3, 8]. The number def(G) = jV (G)j 2 (G) is known as the de…ciency of G. The number d(G) = maxfjSj jN(S)j : S 2 Ind(G)g is called the critical di¤erence of G. An independent set A is critical if jAj jN(A)j = d(G). The critical independence number c(G) is the cardinality of a maximum critical independent set [9]. Clearly, c(G) (G). In [4] it was shown that G is a König-Egerváry graph if and only if c(G) = (G), thus giving a positive answer to the Gra¢ ti.pc 329 conjecture [2]. In this paper we give a new characterization of König-Egerváry graphs claiming that G is a König-Egerváry graph if and only if each of its maximum independent sets is critical. On the one hand, it is similar in form to Sterboul’s theorem [8]. On the other hand it extends Larson’s …nding [4].