Graphs with core(G) = nucleus(G)

2026-03-29

AI Summary

This work characterizes the class of graphs for which the core equals the nucleus—that is, graphs where the intersection of all maximum independent sets coincides with that of all maximum critical independent sets. Building upon Larson’s independence decomposition, the graph is partitioned into a König–Egerváry part and a 2-bicritical part. By analyzing the boundary between these components and the structure of the corona, the study establishes, for the first time, a complete necessary and sufficient condition for core(G) = nucleus(G): the core of the 2-bicritical component must be empty, and no vertex of the corona may lie on the decomposition boundary. This condition is equivalent to diadem(G) = corona(G) ∩ L(G), offering deeper insight into the structural properties of independent sets and yielding several structural corollaries.

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 König--Egerváry 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.

Problem

• core
• nucleus
• independent set
• König–Egerváry graph
• critical independent set

Innovation

• critical independent set
• core
• nucleus
• Larson decomposition
• König–Egerváry graph