Well-covered and uniformly well-covered graphs

TL;DR

This paper characterizes well-covered and uniformly well-covered graphs within a specific class using algebraic properties of their edge rings, providing necessary and sufficient conditions for recognition.

Contribution

It introduces new criteria for identifying well-covered and uniformly well-covered graphs in class G, linking graph theory with algebraic properties of edge rings.

Findings

Characterization of well-covered graphs in class G.

Conditions for uniformly well-covered graphs.

Algebraic interpretation via zero-divisors in edge rings.

Abstract

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A well-covered graph $G$ is called uniformly well-covered if there is a partition of the set of vertices of $G$ such that each maximal independent set of vertices has exactly one vertex in common with each part in the partition. The problem of determining which graphs is well-covered, was proposed in 1970 by M.D. Plummer. Let $\cal G$ be the class of graphs with some disjoint maximal cliques covering all vertices. In this paper, some necessary and sufficient conditions are presented to recognize which graphs in the class $\cal G$ are well-covered or uniformly well-covered. This characterization has a nice algebraic interpretation according to zero-divisor elements of edge ring of graphs which is illustrated in this paper.