It was proved in (Levit and Mandrescu, 2022) that both $(V(G), Crown(G))$ and $(V(G), CritIndep(G))$ are augmentoids, established partial augmentation phenomena for the family $Ψ(G)$ of local maximum independent sets, and asked in Problem~5.5 to characterize the graphs whose family $Ψ(G)$ is an augmentoid. We prove that the answer is positive in full generality: for every finite simple graph $G$, the set system $(V(G),Ψ(G))$ is an augmentoid. The proof is constructive. If $S,T\inΨ(G)$, then the explicit choice \[ A=S \setminus N[T],\qquad B=T \setminus N[S] \] satisfies \[ T\cup A\inΨ(G),\qquad S\cup B\inΨ(G),\qquad |T\cup A|=|S\cup B|. \] As a structural consequence, for every fixed $S\inΨ(G)$ the map $T\mapsto S\cup T$ induces a canonical bijection from $Ψ(G-N[S])$ onto the members of $Ψ(G)$ containing $S$, and \[ α(G)=|S|+α(G-N[S]). \] This decomposition also yields explicit formulas for the intersection and the union of all the maximum independent sets extending $S$, together with counting formulas for the local maximum and maximum independent sets containing $S$. We also add a short visual guide to the framework $CritIndep(G) \subseteq Crown(G)\subseteq Psi(G)$ and end with several natural follow-up problems suggested by the theorem.

翻译：(Levit 和 Mandrescu, 2022) 证明了 $(V(G), Crown(G))$ 和 $(V(G), CritIndep(G))$ 都是增广拟阵，为局部极大独立集族 $\Psi(G)$ 建立了部分增广现象，并在问题 5.5 中提出刻画哪些图的族 $\Psi(G)$ 是增广拟阵。我们证明该答案在完全一般性下是肯定的：对每个有限简单图 $G$，集系统 $(V(G), \Psi(G))$ 是增广拟阵。证明是构造性的。若 $S, T \in \Psi(G)$，则显式选取 \[ A = S \setminus N[T], \quad B = T \setminus N[S] \] 满足 \[ T \cup A \in \Psi(G), \quad S \cup B \in \Psi(G), \quad |T \cup A| = |S \cup B|. \] 作为结构推论，对每个固定 $S \in \Psi(G)$，映射 $T \mapsto S \cup T$ 诱导出从 $\Psi(G - N[S])$ 到 $\Psi(G)$ 中包含 $S$ 的元素之间的典范双射，且 \[ \alpha(G) = |S| + \alpha(G - N[S]). \] 该分解还给出了包含 $S$ 的所有极大独立集的交与并的显式公式，以及包含 $S$ 的局部极大独立集与极大独立集的计数公式。我们还给出了 $CritIndep(G) \subseteq Crown(G) \subseteq \Psi(G)$ 框架的简短视觉导引，并以该定理提出的若干自然后续问题作为结束。