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A class 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 G. We note that G^epex is hereditary and prove that if a hereditary class 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 K₁ 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.
Singh et al. (Thu,) studied this question.