Complexity results for well-covered graphs

A graph with n vertices is well-covered if every maximal independent set is a maximum independent set, and very well-covered if every maximal independent set has size n ⁄2. In this paper, we study these graphs from an algorithmic complexity point of view. We show that wellcovered graph recognition is co-NP-complete and that several other problems are NP-complete for well-covered graphs. A number of these problems remain NP-complete on very well-covered graphs, while some admit polynomial time solutions for the smaller class. For both families, the isomorphism problem is as hard as general graph isomorphism. 1. Definitions and notation A graph is a pair G =(V ,E ) where V is a finite set of vertices , and E is a set of unordered pairs (u ,v ) of distinct vertices of V . Each such pair is called an edge . In what follows, G will denote a simple, undirected, finite graph of order n =  V  . Two vertices u and v are ad jacent if (u ,v )∈E . The degree d (v ) of a vertex v is the number of vertices adjacent to v . The neighbourhood Γ(v ) of v is the set of vertices that are adjacent to v . Two edges are adjacent if they have a vertex in common. A vertex of degree one is called a lea f . An edge which is incident with a leaf is called a pendant edge. A set of vertices is independent if no two of them are adjacent. A graph is a clique if every two vertices are adjacent. A set of vertices in G forms a vertex cover for G if every edge in G is incident on at least one vertex in the set. A subset of E is a matching if no two edges of the set are adjacent. A perfect matching is one in