Inequalities Involving Core, Corona, and Critical Sets in General Graphs

Abstract

Let $\alpha(G)$ denote the cardinality of a maximum independent set. An independent set $I$ of $G$ is critical if $\left|I\right|-\left|N(I)\right|\ge\left|J\right|-\left|N(J)\right|$ for every independent set $J$ of $G$. Let $\text{core}(G)$ and $\text{corona}(G)$ be the intersection/union of all maximum independent sets of $G$. Let $\text{ker}(G)$ and $\text{diadem}(G)$ be the intersection/union of all critical independent sets of $G$. In this paper we prove that \[ \left|\text{corona}(G)\right|+\left|\text{core}(G)\right|\le2\alpha(G)+k, \] \noindent where $k$ is the number of vertex-distinct odd cycles in $G$, thus confirming a recent conjecture in the area. Moreover, we prove that \[ \left|\text{nucleus}(G)\right|+\left|\text{diadem}(G)\right|\le2\alpha(G), \] \noindent thereby confirming another conjecture (Levit--Mandrescu 2014). As an application of these facts, we obtain a chain of inequalities \[ \left|\text{nucleus}(G)\right|+\left|\text{diadem}(G)\right|\le2\alpha(G)\le\left|\text{corona}(G)\right|+\left|\text{core}(G)\right|\le2\alpha(G)+k. \] \noindent The paper concludes with a collection of related open problems.