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A graph $G$ is said to be an $(s, k)$-polar graph if its vertex set admits apartition $(A, B)$ such that $A$ and $B$ induce, respectively, a complete$s$-partite graph and the disjoint union of at most $k$ complete graphs. Polargraphs and monopolar graphs are defined as $(\infty, \infty)$- and $(1,\infty)$-polar graphs, respectively, and unipolar graphs are those graphs witha polar partition $(A, B)$ such that $A$ is a clique.  The problems of deciding whether an arbitrary graph is a polar graph or amonopolar graph are known to be NP-complete. In contrast, deciding whether agraph is a unipolar graph can be done in polynomial time. In this work we provethat the three previous problems can be solved in linear time on the classes of$P_4$-sparse and $P_4$-extendible graphs, generalizing analogous resultspreviously known for cographs.  Additionally, we provide finite forbidden subgraph characterizations for$(2,2)$-polar graphs on $P_4$-sparse and $P_4$-extendible graphs, alsogeneralizing analogous results recently obtained for the class of cographs.

$2$-polarity and algorithmic aspects of polarity variants on cograph superclasses