Abstract
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 define and characterize two recursively decomposable subclasses of well-covered graphs, one properly containing the other. We show that the smaller subclass, in turn, properly contains the family of very well covered graphs without isolated vertices.
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