The family of all local maximum independent sets is an augmentoid

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 $\mathrm{\Psi }(G)$ of local maximum independent sets, and asked in Problem~5.5 to characterize the graphs whose family $\mathrm{\Psi }(G)$ is an augmentoid. We prove that the answer is positive in full generality: for every finite simple graph $G$, the set system $(V(G),\mathrm{\Psi }(G))$ is an augmentoid. The proof is constructive. If $S,T\in \mathrm{\Psi }(G)$, then the explicit choice $$A=S\setminus N[T],B=T\setminus N[S]$$ satisfies $$T\cup A\in \mathrm{\Psi }(G),S\cup B\in \mathrm{\Psi }(G),\mathrm{\mid }T\cup A\mathrm{\mid }=\mathrm{\mid }S\cup B\mathrm{\mid }\mathrm{.}$$ As a structural consequence, for every fixed $S\in \mathrm{\Psi }(G)$ the map $T\mapsto S\cup T$ induces a canonical bijection from $\mathrm{\Psi }(G-N[S])$ onto the members of $\mathrm{\Psi }(G)$ containing $S$, and $$\alpha (G)=\mathrm{\mid }S\mathrm{\mid }+\alpha (G-N[S])\mathrm{.}$$ 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.