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 $P_4$-reducible graphs. In this work, we prove that also when restricted to the classes of $P_4$-sparse graphs and $P_4$-extendible graphs (both of which extend $P_4$-reducible graphs) every hereditary property can be characterized by finitely many minimal obstructions. We present complete lists of $P_4$-sparse and $P_4$-extendible minimal obstructions for polarity, monopolarity, unipolarity, and $(s,1)$-polarity, where $s$ is a positive integer. In parallel to the case of $P_4$-reducible graphs, all the $P_4$-sparse minimal obstructions for these hereditary properties are cographs.