Graphs with core(G) = nucleus(G)

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.