Critical and Maximum Independent Sets Revisited

Abstract

Let G be a simple graph with vertex set $V\left( G\right)$.

A set $S\subseteq V\left( G\right)$ is independent if no two vertices from S are adjacent, and by $\mathrm {Ind}(G)$ we mean the family of all independent sets of G.

The number $d\left( X\right) =$ $\left| X\right| -\left| N(X)\right|$ is the difference of $X\subseteq V\left( G\right)$, and a set $A\in \mathrm {Ind}(G)$ is critical if $d(A)=\max \{d\left( I\right) :I\in \mathrm {Ind}(G)\}$ [34].

Let us recall the following definitions:

• $\mathrm {core}\left( G\right) = {\displaystyle \bigcap } \left\{ S:S\textit{ is a maximum independent set}\right\}$ [16],

• $\mathrm {corona}\left( G\right) = {\displaystyle \bigcup } \left\{ S:S\textit{ is a maximum independent set}\right\}$ [5],

• $\mathrm {\ker }(G)= {\displaystyle \bigcap } \left\{ S:S\textit{ is a critical independent set}\right\}$ [18],

• $\mathrm {nucleus}(G)= {\displaystyle \bigcap } \left\{ S:S\textit{ is a maximum critical independent set}\right\}$ [12]

• $\mathrm {diadem}(G)= {\displaystyle \bigcup } \left\{ S:S\textit{ is a (maximum) critical independent set}\right\}$ [24].

In this paper we focus on interconnections between $\ker$, core, corona, $\mathrm {nucleus}$, and diadem.