Polynomial Time Recognition of Essential Graphs Having Stability Number Equal to Matching Number

Abstract

For a graph $G$ let $\alpha (G)$ and $\mu (G)$ denote respectively the cardinality of a maximum stable set and of a maximum matching of $G$. It is well-known that computing $\alpha (G)$ is NP-hard and that computing $\mu (G)$ can be done in polynomial time. In particular checking if $\alpha (G)=\mu (G)$ is NP-complete and relies on the fact that computing $\alpha (G)$ is NP-hard (Mosca, Graphs Combinat 18:367–379, 2002). A well known result of Hammer et al. (SIAM J Alg Disc Math 3(4):511–522, 1982). states that the vertex-set of a graph $G$ can be efficiently and uniquely partitioned in two subsets (possibly empty) $A$ and $B$, such that $G[A]$ has the König-Egerváry property while $G[B]$ can be covered by pairwise disjoint edges and odd cycles: furthermore, one has $\alpha (G) = \alpha (G[A]) + \alpha (G[B])$, where computing $\alpha (G[A])$ can be done in polynomial time. For that let us call $essential$ those graphs which can be covered by pairwise disjoint edges and odd cycles (in particular computing $\alpha (G)$ remains NP-hard for such graphs). This paper shows that: (i) for every essential graph $G$, checking if $\alpha (G)=\mu (G)$ can be done in polynomial time; (ii) essential graphs for which $\alpha (G)=\mu (G)$ can be recognized in polynomial time and for such graph a maximum stable set can be computed in polynomial time; (iii) a new characterization of graphs which have the König-Egerváry property can be derived in that context.