A characterization of graphs with $\a{\corona G}+\a{\core G}=2\alpha(G)+1$

TL;DR

This paper characterizes graphs where the sum of the sizes of the corona and core equals twice the independence number plus one, extending known results to graphs with multiple odd cycles.

Contribution

It provides a complete characterization of such graphs, solving an open problem and generalizing previous results to graphs with many odd cycles.

Findings

Characterization of graphs with $ extnormal{corona}(G)+ extnormal{core}(G)=2 ext{α}(G)+1$

Extension of known properties from graphs with a single odd cycle to those with many odd cycles

Solution to an open problem posed by Levit and Mandrescu.

Abstract

A K\H{o}nig--Egerv\'ary graph is a graph $G$ satisfying $\alpha(G)+\mu(G)=n(G)$, where $\alpha(G)$, $\mu(G)$, and $n(G)$ denote the independence number, the matching number, and the order of $G$, respectively. Let $\textnormal{core}(G)$ and $\textnormal{corona}(G)$ be the intersection and the union of all maximum independent sets of $G$. In this paper, we provide a complete characterization of graphs satisfying $\a{\corona G}+\a{\core G}=2\alpha(G)+1$, thus giving a solution to an open problem posed by Levit and Mandrescu. It is known that for a non-K\H{o}nig--Egerv\'ary graph with a unique odd cycle, the following hold: $\ker G=\textnormal{core}(G),\allowbreak\ \left|\textnormal{corona}(G)\right| +\left|\textnormal{core}(G)\right| =2\alpha(G)+1,\allowbreak\ \textnormal{corona}(G)\cup N(\textnormal{core}(G))=V(G)$. We extend these three results to a family of graphs containing an arbitrarily large number of odd cycles.