# Critical and Maximum Independent Sets Revisited

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

## 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 $$d(A)=\max \{d\left( I\right) :I\in \mathrm {Ind}(G)\}$$ .

Let us recall the following definitions:
• $$\mathrm {core}\left( G\right) = \bigcap } \left\{ S:S\textit{ is a maximum independent set}\right\}$$ ,

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

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

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

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

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

## Keywords

Independent set Critical set Ker Core Corona Diadem Matching

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