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.

翻译：设 $G$ 为有限简单图。$G$ 的独立集 $I$ 称为临界独立集，若对于 $G$ 的任意独立集 $J$ 均有 $\left|I\right|-\left|N(I)\right|\ge\left|J\right|-\left|N(J)\right|$；临界独立集若具有最大基数则称为最大临界独立集。$G$ 的核与核心分别定义为所有最大独立集的交集与所有最大临界独立集的交集。2019年，Jarden、Levit 与 Mandrescu 提出了刻画满足 $core(G)=nucleus(G)$ 的图的问题。本文给出此问题的完整解答。通过利用 Larson 的独立分解（将任意图划分为 König--Egerváry 分量 $L_G$ 与 $2$-双临界分量 $L_G^c$），我们证明 $core(G)=nucleus(G)$ 成立当且仅当 $core({L_G^c})=\emptyset$ 且 $corona(G)$ 中无顶点位于 $L_G$ 与 $L_G^c$ 的边界上。同时证明这一边界条件等价于恒等式 $diadem(G)=corona(G) \cap L(G)$。文中还推导了若干推论及相关的结构性质。