Well-covered and uniformly well-covered graphs

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 $\mathcal{G}$ 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 $\mathcal{G}$ 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.