On $\alpha$-excellent graphs

TL;DR

This paper characterizes various classes of graphs that are $ ext{alpha}$-excellent, meaning each vertex belongs to some maximum independent set, including bipartite, unicyclic, simplicial, chordal, block graphs, and all generalized Petersen graphs.

Contribution

It provides complete characterizations of $ ext{alpha}$-excellent graphs across multiple graph classes and proves that all generalized Petersen graphs are $ ext{alpha}$-excellent.

Findings

Characterizations of $ ext{alpha}$-excellent bipartite, unicyclic, simplicial, chordal, and block graphs.

Proof that every generalized Petersen graph is $ ext{alpha}$-excellent.

Extension of $ ext{alpha}$-excellence concept to various graph classes.

Abstract

A graph $G$ is $\alpha$-excellent if every vertex of $G$ is contained in some maximum independent set of $G$. In this paper, we characterize $\alpha$-excellent bipartite graphs, $\alpha$-excellent unicyclic graphs, $\alpha$-excellent simplicial graphs, $\alpha$-excellent chordal graphs, $\alpha$-excellent block graphs, and we show that every generalized Petersen graph is $\alpha$-excellent.