Minimal Obstructions for Polarity, Monopolarity, Unipolarity and (s, 1)-Polarity in Generalizations of Cographs

It is known that every hereditary property can be characterized by finitely many minimal obstructions when restricted to either the class of cographs or the class of P₄ P4 -reducible graphs. In this work, we prove that the same is true when restricted to some other superclasses of cographs, including P₄ P4 -sparse and P₄ P4 -extendible graphs (both of which extend P₄ P4 -reducible graphs). We also present complete lists of P₄ P4 -sparse and P₄ P4 -extendible minimal obstructions for polarity, monopolarity, unipolarity, and ( s , 1)-polarity, where s is a positive integer. In parallel to the case of P₄ P4 -reducible graphs, all the P₄ P4 -sparse minimal obstructions for these hereditary properties are cographs.