On Duality between Local Maximum Stable Sets of a Graph and its Line-Graph

TL;DR

This paper explores the relationship between local maximum stable sets in a graph and its line graph, establishing duality properties and conditions under which certain matchings are local maximum stable sets.

Contribution

It demonstrates that local maximum stable sets induce Koenig-Egervary subgraphs and that maximum matchings in these subgraphs correspond to local maximum stable sets in the line graph.

Findings

Induced subgraphs by local maximum stable sets are Koenig-Egervary graphs.

Maximum matchings in these subgraphs are local maximum stable sets in the line graph.

Establishes a duality relationship between stable sets and matchings in a graph and its line graph.

Abstract

G is a Koenig-Egervary graph provided alpha(G)+ mu(G)=|V(G)|, where mu(G) is the size of a maximum matching and alpha(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G if S is a maximum stable set of the closed neighborhood of S. Nemhauser and Trotter Jr. proved that any local maximum stable set is a subset of a maximum stable set of G. In this paper we demonstrate that if S is a local maximum stable set, the subgraph H induced by the closed neighborhood of S is a Koenig-Egervary graph, and M is a maximum matching in H, then M is a local maximum stable set in the line graph of G.