On Konig-Egervary Square-Stable Graphs

TL;DR

This paper characterizes Konig-Egervary square-stable graphs, showing they have perfect matchings of pendant edges, and identifies well-covered trees as exactly the square-stable trees.

Contribution

It provides a complete characterization of Konig-Egervary square-stable graphs and links square-stability to well-covered trees.

Findings

Konig-Egervary graphs are square-stable iff they have a perfect matching of pendant edges.

Well-covered trees are precisely the square-stable trees.

Square-stability relates to the structure of perfect matchings in these graphs.

Abstract

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G)+mu(G) equals its order, then G is a Konig-Egervary graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality alpha(G)=alpha(G^{2}), where G^{2} denotes the second power of G. In particular, we show that a Konig-Egervary graph is square-stable if and only if it has a perfect matching consisting of pendant edges, and in consequence, we deduce that well-covered trees are exactly the square-stable trees.