We prove the existence of a polynomial time algorithm to tell whether a graph, with no induced subgraph isomorphic to K1.3 , is well covered. A graph is wellcovered if all its maximal independent sets are of the same cardinality. The problem is known to be polynomialy solvable where the input graph is a line graph and it is NP-hard for the larger family of all graphs which do not contain an ind...

G is a well-covered graph provided all its maximal stable sets are of the same size (Plummer, 1970). S is a local maximum stable set of G, and we denote by S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. In 2002 we have proved that Ψ(G) is a greedoid for every forest G. The bipartite graphs and the trianglefree graphs, whose famil...

We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of Kn × G is found, provided that G has a largest greedy independent decomposition of length c < n. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, an...

A graph G is well covered, if any two maximal independent sets of G have the same number of vertices. A graph is called a block-cactus graph if each block is complete or a cycle. In this paper we characterize the well covered block-cactus graphs. 1. TERMINOLOGY AND INTRODUCTION In this paper we consider finite, undirected, and simple graphs G with the vertex set V(G). The degree d(x, G) of a ve...

A graph is well covered if every maximal independent set has the same size and very well covered if every maximal independent set contains exactly half the number of vertices. In this paper, we present an alternative characterization of a certain sub-class of well-covered graphs and show that this generalizes a characterization of very well covered graphs given by Favaron 3].