A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number. We answer a question of Topp and Volkmann [5] and prove that if the Cartesian product of two graphs is well-covered, then at least one of them must be well-covered.

‎in this paper we give a characterization of unmixed tripartite‎ ‎graphs under certain conditions which is a generalization of a‎ ‎result of villarreal on bipartite graphs‎. ‎for bipartite graphs two‎ ‎different characterizations were given by ravindra and villarreal‎. ‎we show that these two characterizations imply each other‎.

A graph G is well-covered if every maximal independent set of maximum. (k,ℓ)-partition a partition its vertex into k sets and ℓ cliques. (k,ℓ)-well-covered it admits (k,ℓ)-partition. The recognition graphs polynomial-time solvable for the cases (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), hard, otherwise. In sandwich problem property Π, we are given pair G1=(V,E1) G2=(V,E2) with E1⊆E2, asked wheth...

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97–108] that there are no 5-connected planar w...

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x)= ∑ ik xk . In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤β (this is...

We study graphs in which the maximum and the minimum sizes of a maximal independent set differ by exactly one. We call these graphs almost well-covered, in analogy with the class of well-covered graphs, in which all maximal independent sets have the same size. A characterization of graphs of girth at least 8 having exactly two different sizes of maximal independent sets due to Finbow, Hartnell,...

A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all ...