On almost bipartite non-König–Egerváry graphs

A set S⊆V is independent in a graph G=V,E if no two vertices from S are adjacent. The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. If α(G)+μ(G) equals the order of G, then G is a König–Egerváry graph (Deming, 1979; Gavril, 1977; Sterboul, 1979). The number dG=max{A−NA:A⊆V} is the critical difference of G (Zhang, 1990) (where NA=v:v∈V,Nv∩A≠0̸). It is known that the inequality α(G)−μ(G)≤dG holds for every graph (Levit and Mandrescu, 2012; Lorentzen, 1966; Schrijver, 2003). A graph G is (i) unicyclic if it has a unique cycle, (ii) almost bipartite if it has only one odd cycle. Let ker(G)=⋂{S:S is a critical independent set of G}, coreG be the intersection of all maximum independent sets, and coronaG be the union of all maximum independent sets of G. It is known that ker(G)⊆core(G) for every graph (Levit and Mandrescu, 2012), while the equality holds for bipartite graphs (Levit and Mandrescu, 2013), and for unicyclic non-König–Egerváry graphs (Levit and Mandrescu, 2014). In this paper, we prove that if G is an almost bipartite non-König–Egerváry graph, then ker(G)= core(G), corona(G)∪N(core(G)) = V(G), and corona(G)+core(G)=2α(G)+1.