The family of all local maximum independent sets is an augmentoid

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 $Ψ(G)$ 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.

Figures (3)

• Figure 1: A roadmap of the framework developed in LM2022 and of the present result. The theorem proved below upgrades the two earlier augmentation mechanisms for $\Psi(G)$ to a full augmentoid statement valid for all graphs.
• Figure 2: Two small examples separating the families $\mathrm{CritIndep}(G)$, $\mathrm{Crown}(G)$, and $\Psi(G)$. The star from Figure \ref{['fig:star']} witnesses the failure of the equality $\mathrm{CritIndep}(G)=\mathrm{Crown}(G)$, while the graph from Figure \ref{['fig:triangle']} witnesses the failure of the equality $\mathrm{Crown}(G)=\Psi(G)$.
• Figure 3: A concrete augmentation for Theorem \ref{['thm:main']}. In panel (a), gray vertices belong to $S - T$, double-circled vertices belong to $T - S$, and the black vertex belongs to $S\cap T$. Panels (b) and (c) show the two augmented local maximum independent sets produced by the theorem.

Theorems & Definitions (20)

• Theorem 1.2
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3