Edge-apexing in hereditary classes of graphs

Abstract

A class $G$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $G^{\mathrm{epex}}$ the class of graphs that are at most one edge away from being in $G$. We note that $G^{\mathrm{epex}}$ is hereditary and prove that if a hereditary class $G$ has finitely many forbidden induced subgraphs, then so does $G^{\mathrm{epex}}$.

The hereditary class of cographs consists of all graphs G that can be generated from $K_1$ using complementation and disjoint union. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. For the class of edge-apex cographs our main result bounds the order of such forbidden induced subgraphs by 8 and finds all of them by computer search.