Recognizing well covered graphs of families with special $P_4$-components

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 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.

2009