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.

翻译：若图类$\mathcal{G}$在取诱导子图运算下封闭，则称其为遗传类。记$G^{epex}$为至多删除一条边即可属于$\mathcal{G}$的图类。我们指出$G^{epex}$具有遗传性，并证明若遗传类$\mathcal{G}$仅有有限个禁用诱导子图，则$G^{epex}$亦然。余图的遗传类由所有可通过补运算与不交并从$K_1$生成的图$G$构成，其充要条件为不含4顶点路作为诱导子图。针对边顶点化余图类，本文主要结果将该类禁用诱导子图的阶数限定为8，并通过计算机搜索完整枚举了所有此类子图。