On $\alpha $-Critical Edges in K\"{o}nig-Egerv\'{a}ry Graphs

TL;DR

This paper explores the properties of alpha-critical edges in K"{o}nig-Egerváry graphs, generalizing known results from bipartite graphs and characterizing specific graph classes with these properties.

Contribution

It extends the understanding of alpha-critical edges from bipartite graphs to K"{o}nig-Egerváry graphs and characterizes graphs where the stability number relates to critical vertices and edges.

Findings

Alpha-critical edges form a matching in bipartite graphs.

In K"{o}nig-Egerváry graphs, alpha-critical edges are also mu-critical.

Characterization of trees based on alpha-critical vertices and edges.

Abstract

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. If alpha(G-e) > alpha(G), then e is an alpha-critical edge, and if mu(G-e) < mu(G), then e is a mu-critical edge, where mu(G) is the cardinality of a maximum matching in G. G is a Koenig-Egervary graph if alpha(G) + mu(G) equals its order. Beineke, Harary and Plummer have shown that the set of alpha-critical edges of a bipartite graph is a matching. In this paper we generalize this statement to Koenig-Egervary graphs. We also prove that in a Koenig-Egervary graph alpha-critical edges are also mu-critical, and that they coincide in bipartite graphs. We obtain that for any tree its stability number equals the sum of the cardinality of the set of its alpha-critical vertices and the size of the set of its alpha-critical edges. Eventually, we characterize the Koenig-Egervary graphs enjoying this property.