On the number of vertices belonging to all maximum stable sets of a graph Original Research Article

Let us denote by α(G) the size of a maximum stable set, and by μ(G) the size of a maximum matching of a graph G, and let ξ(G) be the number of vertices which belong to all maximum stable sets. We shall show that ξ(G)⩾1+α(G)−μ(G) holds for any connected graph, whenever α(G)>μ(G). This inequality improves on related results by Hammer et al. (SIAM J. Algebraic Discrete Methods 3 (1982) 511) and by Levit and Mandrescu [(prE-print math. CO/9912047 (1999) 13pp.)]. We also prove that on one hand, ξ(G)>0 can be recognized in polynomial time whenever μ(G)<|V(G)|/3, and on the other hand determining whether ξ(G)>k is, in general, NP-complete for any fixed k⩾0.