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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 β(G) = β(G − {x}). If G is a connected, well covered graph of girth ≥ 5 and G contains an extendable vertex then G is the disjoint union of edges and 5-cycles together with a restricted set of edges joining these subgraphs. There are only six connected, well covered graphs of girth ≥ 5 which do not contain an extendable vertex.

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ikdenote the number of independent sets of cardinality k in G. We investigate the roots...