  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.
