A CHARACTERIZATION OF WELL-COVERED GRAPHS THAT CONTAIN NEITHER 4-CYCLES NOR 5-CYCLES

A graph is well covered if every maximal independent set has the same cardinality. A vertex x, in a well-covered graph G, is called extendable if G - {x} is well covered and beta(G) = beta(G - {x}). If G f is a connected, well-covered graph containing no 4- nor 5-cycles as subgraphs and G contains an extendable vertex, then G is the disjoint union of edges and triangles together with a restricted set of edges joining extendable vertices. There are only 3 other connected, well-covered graphs of this type that do not contain an extendable vertex. Moreover, all these graphs can be recognized in polynomial time. (C) 1994 John Wiley & Sons, Inc.