2010/09/27 by Rashid Zaare-Nahandi, Zaare-Nahandi, Rashid
Mathematics · #05C25 #05E45 #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1009.5242
openalex publication_date 2010/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.