Graphs with core(G) = nucleus(G)

TL;DR

This paper characterizes graphs where the core equals the nucleus, using Larson's decomposition, and establishes conditions involving the boundary between components and the diadem and corona sets.

Contribution

It provides a complete characterization of graphs satisfying core(G)=nucleus(G) using Larson's independence decomposition and boundary conditions.

Findings

core(G)=nucleus(G) iff core(L_G^c)=∅ and no vertex of corona(G) lies on the boundary

The boundary condition is equivalent to diadem(G)=corona(G)∩L(G)

Several structural properties related to these conditions are derived.

Abstract

Let $G$ be a finite simple graph. 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 $G$. A critical independent set is maximum if it has maximum cardinality. The $core$ and the $nucleus$ of $G$ are defined as the intersection of all maximum independent sets and the intersection of all maximum critical independent sets, respectively. In 2019, Jarden, Levit, and Mandrescu posed the problem of characterizing the graphs satisfying $core(G)=nucleus(G)$. In this paper, we provide a complete solution to this problem. Using Larson's independence decomposition, which partitions any graph into a K\"onig--Egerv\'ary component $L_G$ an a $2$-bicritical component $L_G^c$, we establish that $core(G)=nucleus(G)$ holds if and only if $core ({L_G^c})=\emptyset$ and no vertex of $corona(G)$ lies in the boundary between $L_G$ and $L_G^c$. We also show that the same boundary condition is equivalent to the identity $diadem(G)=corona(G) \cap L(G)$. Several consequences and related structural properties are also derived.