On almost bipartite non-König–Egerváry graphs

Abstract

A set $S\subseteq V$ is independent in a graph $G=\left(V,E\right)$ if no two vertices from $S$ are adjacent. The independence number $\alpha \left(G\right)$ is the cardinality of a maximum independent set, while $\mu \left(G\right)$ is the size of a maximum matching in $G$. If $\alpha \left(G\right)+\mu \left(G\right)$ equals the order of $G$, then $G$ is a König–Egerváry graph (Deming, 1979; Gavril, 1977; Sterboul, 1979). The number $d\left(G\right)=max\{\left(A\right)-\left(N\left(A\right)\right):A\subseteq V\}$ is the critical difference of $G$ (Zhang, 1990) (where $N\left(A\right)=\left(v:v\in V,N\left(v\right)\cap A\ne 0̸\right)$). It is known that the inequality $\alpha \left(G\right)-\mu \left(G\right)\le d\left(G\right)$ 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 $\ker \left(G\right)=\bigcap \{S:S$ is a critical independent set of $G\}$, core$\left(G\right)$ be the intersection of all maximum independent sets, and corona$\left(G\right)$ be the union of all maximum independent sets of $G$. It is known that $\ker \left(G\right)\subseteq \mathrm{core}\left(G\right)$ 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$\left(G\right)=$ core$\left(G\right)$, corona$\left(G\right)\cup N$(core($G$)) = $V\left(G\right)$, and $\left(\mathrm{corona}\left(G\right)\right)+\left(\mathrm{core}\left(G\right)\right)=2\alpha \left(G\right)+1$.