On graphs with a simple structure of maximal cliques
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908337159798784 |
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| author | Gollin, J. Pascal Hatzel, Meike Wiederrecht, Sebastian |
| author_facet | Gollin, J. Pascal Hatzel, Meike Wiederrecht, Sebastian |
| contents | We say that a hereditary graph class $\mathcal{G}$ is \emph{clique-sparse} if there is a constant $k=k(\mathcal{G})$ such that for every graph $G\in\mathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal clique of $G$ can be intersected in at most $k$ different ways by other maximal cliques.
We provide various characterisations of clique-sparse graph classes, including a list of five parametric forbidden induced subgraphs. We show that recent techniques for proving induced analogues of Menger's Theorem and the Grid Theorem of Robertson and Seymour can be lifted to prove induced variants in clique-sparse graph classes when replacing ``treewidth'' by ''tree-independence number''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16863 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On graphs with a simple structure of maximal cliques Gollin, J. Pascal Hatzel, Meike Wiederrecht, Sebastian Combinatorics Discrete Mathematics We say that a hereditary graph class $\mathcal{G}$ is \emph{clique-sparse} if there is a constant $k=k(\mathcal{G})$ such that for every graph $G\in\mathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal clique of $G$ can be intersected in at most $k$ different ways by other maximal cliques. We provide various characterisations of clique-sparse graph classes, including a list of five parametric forbidden induced subgraphs. We show that recent techniques for proving induced analogues of Menger's Theorem and the Grid Theorem of Robertson and Seymour can be lifted to prove induced variants in clique-sparse graph classes when replacing ``treewidth'' by ''tree-independence number''. |
| title | On graphs with a simple structure of maximal cliques |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2504.16863 |