The family of all local maximum independent sets is an augmentoid

TL;DR

This paper proves that the family of all local maximum independent sets in any finite simple graph forms an augmentoid, providing explicit constructive methods and formulas for their structure, intersection, union, and enumeration.

Contribution

It establishes that the family of local maximum independent sets is an augmentoid for all graphs, with a constructive proof and explicit structural and counting formulas.

Findings

The family of local maximum independent sets forms an augmentoid in all graphs.

Provides explicit formulas for intersection, union, and enumeration of these sets.

Introduces a structural decomposition linking local maximum independent sets to graph substructures.

Abstract

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