The Structure of Well-Covered Graphs and the Complexity of Their Recognition Problems

A graph is well-covered if all its maximal independent sets are of the same cardinality. Deciding whether a given graph is well-covered is known to beNP-hard in general, and solvable in polynomial time, if the input is restricted to certain families of graphs. We present here a simple structural characterization of well-covered graphs and then apply it to the recognition problem. Apparently, polynomial algorithms become easier to design. In particular we present a new polynomial time algorithm for the case where the input graph contains no induced subgraph isomorphic toK_{1, 3}. Considering the line-graph of an input graph, this result provides a short and simple alternative to a proof by Lesk, Plummer and Pulleyblank, who showed that equimatchable graphs can be recognized in polynomial time.