Apex Graphs and Cographs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916495426060288 |
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| author | Singh, Jagdeep Sivaraman, Vaidy Zaslavsky, Thomas |
| author_facet | Singh, Jagdeep Sivaraman, Vaidy Zaslavsky, Thomas |
| contents | A class $\mathcal{G}$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $\mathcal{G}^\mathrm{apex}$ the class of graphs $G$ that contain a vertex $v$ such that $G-v$ is in $\mathcal{G}$. We prove that if a hereditary class $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does $\mathcal{G}^\mathrm{apex}$.
The hereditary class of cographs consists of all graphs $G$ that can be generated from $K_1$ using complementation and disjoint union. A graph is an apex cograph if it contains a vertex whose deletion results in a cograph. Cographs are precisely the graphs that do not have the $4$-vertex path as an induced subgraph. Our main result finds all such forbidden induced subgraphs for the class of apex cographs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02551 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Apex Graphs and Cographs Singh, Jagdeep Sivaraman, Vaidy Zaslavsky, Thomas Combinatorics 05C75 A class $\mathcal{G}$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $\mathcal{G}^\mathrm{apex}$ the class of graphs $G$ that contain a vertex $v$ such that $G-v$ is in $\mathcal{G}$. We prove that if a hereditary class $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does $\mathcal{G}^\mathrm{apex}$. The hereditary class of cographs consists of all graphs $G$ that can be generated from $K_1$ using complementation and disjoint union. A graph is an apex cograph if it contains a vertex whose deletion results in a cograph. Cographs are precisely the graphs that do not have the $4$-vertex path as an induced subgraph. Our main result finds all such forbidden induced subgraphs for the class of apex cographs. |
| title | Apex Graphs and Cographs |
| topic | Combinatorics 05C75 |
| url | https://arxiv.org/abs/2310.02551 |