Recursively decomposable well-covered graphs

We give an alternative characterization for well-covered graphs and restrict this to a characterization for very well covered graphs. We state the conditions under which the intersection of a pair of maximal independent sets of a well-covered graph is maximal and use this result to deene and characterize two recursively decomposable sub-classes of well-covered graphs, one properly containing the other. We show that the smaller sub-class, in turn, properly contains the family of very well covered graphs without isolated vertices.