Subdivisions,parity and well-covered graphs

A graph is well-covered if every maximal independent set is maximum. This concept, introduced by Plummer in 1970 (J. Combin. Theory 8 (1970)), is the focal point of much interest and current research. We consider well-covered 2-degenerate graphs and supply a structural (and polynomial time algorithm) characterization of the class called 3-separable graphs. Also we consider parity graphs studied by Finbow and Hartnell and answer the question posed by them (Ars. Combin. 40 (1995)) by proving, among other results, that the decision problem: “given a graph G which is a parity graph, is G also well-covered graph?” is in the class CO-NPC. In addition we supply some complexity results that answer some problems due to Plummer (Quaestiones Math. 16 (1993)) and Finbow, Hartnell, and Whitehead (Discrete Math. 125 (1994)). © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 85–94, 1997