Minimal obstructions for polarity, monopolarity, unipolarity and $(s,1)$-polarity in generalizations of cographs

TL;DR

This paper demonstrates that hereditary properties like polarity and unipolarity can be characterized by finitely many minimal obstructions within broader graph classes such as P4-sparse and P4-extendible graphs, extending previous results.

Contribution

It provides complete lists of minimal obstructions for these properties in P4-sparse and P4-extendible graphs, showing all P4-sparse obstructions are cographs.

Findings

Finiteness of minimal obstructions in extended graph classes

Complete lists of obstructions for specific properties

All P4-sparse obstructions are cographs

Abstract

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.