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

03/09/2022

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by   Fernando Esteban Contreras-Mendoza, et al.

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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.