The family of all local maximum independent sets is an augmentoid

2026-03-23

AI Summary

This work characterizes the structural properties of the family \( \Psi(G) \) of locally maximal independent sets in any finite simple graph \( G \), establishing that \( \Psi(G) \) always forms an augmentoid. Through a constructive approach, the paper provides the first complete proof of this property and introduces a structural decomposition of \( \Psi(G) \) relative to an arbitrary fixed vertex set \( S \). By integrating tools from graph theory, combinatorial construction, and set-system analysis, the authors define an explicit augmentation operation between any two sets in \( \Psi(G) \). This operation yields a decomposition formula for maximum independent sets and exact counting formulas for both local and global maximal independent sets containing a given set, thereby uncovering the deep combinatorial structure underlying \( \Psi(G) \).

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Local Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• augmentoid
• local maximum independent set
• graph theory
• independent set
• set system

Innovation

• augmentoid
• local maximum independent set
• constructive proof
• graph decomposition
• independent set counting