Abstract
A graph G is called well covered if every two maximal independent sets of G have the same number of vertices. In this paper we shall use the modular and primeval decomposition techniques to decide well coveredness of graphs such that, either all their P 4-connected components (in short, P 4-components) are separable or they belong to well known classes of graphs that, in some local sense, contain few P 4's. In particular, we shall consider the class of cographs, P 4-reducible, P 4-sparse, extended P 4-reducible, extended P 4-sparse graphs, P 4-extendible graphs, P 4-lite graphs, and P 4-tidy graphs.
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