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