Edge-apexing in hereditary classes of graphs
<|reference_start|>Edge-apexing in hereditary classes of graphs: 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.<|reference_end|>
@article{singh2024edge-apexing,
    title={Edge-apexing in hereditary classes of graphs},
    author={Jagdeep Singh and Vaidy Sivaraman},
    journal={arXiv preprint arXiv:2403.09456},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.09456},
    primaryClass={math.CO cs.DM}
}