Weighted Well-covered Graphs and Complexity Questions

A weighted graph G is called well-covered if all its maximal independent sets have the same weight. Let S be an independent set of G (possibly, S = ∅). The subgraph G − N [S] is called a co-stable subgraph of G. We denote by CSub(G) the set of all co-stable subgraphs of G considered up to isomorphism. A class of weighted graphs P is called co-hereditary if it is closed under taking co-stable subgraphs, i. e., G ∈ P implies CSub(G) ⊆ P. Note that the class WWELL of all weighted well-covered graphs is co-hereditary. We characterize WWELL in terms of forbidden co-stable subgraphs. Then we use a reduction from Satisfiability to show that the following decision problems are NP-complete. Decision Problem 1 (Co-Stable Subgraph). Instance: A graph G and a set U ⊆ V (G) that induces a subgraph H. Question: Is H a co-stable subgraph of G? Decision Problem 2 (Co-Stable Subgraph H). Instance: A graph G. Question: Is H a co-stable subgraph of G? Let ∆(G) be the maximum vertex degree of a graph G. We show that recognizing weighted well-covered graphs with bounded ∆(G) can be done in polynomial time. 2000 Math. Subj. Class. 05C85.