The sandwich problem for odd-hole-free and even-hole-free graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2024
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| author | Cameron, Kathie Chaniotis, Aristotelis de Figueiredo, Celina M. H. Spirkl, Sophie |
| author_facet | Cameron, Kathie Chaniotis, Aristotelis de Figueiredo, Celina M. H. Spirkl, Sophie |
| contents | For a property $\mathcal{P}$ of graphs, the $\mathcal{P}$-\textsc{Sandwich-Problem}, introduced by Golumbic and Shamir (1993), is the following: Given a pair of graphs $(G_1, G_2)$ on the same vertex set $V$, does there exist a graph $G$ such that $V(G)=V$, $E(G_{1})\subseteq E(G) \subseteq E(G_{2})$, and $G$ satisfies $\mathcal{P}$? A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. An odd (respectively even) hole is a hole of odd (respectively even) length. Given a class of graphs $\mathcal{C}$ and a graph $G$ we say that $G$ is {\em $\mathcal{C}$-free} if it contains no induced subgraph isomorphic to a member of $\mathcal{C}$. In this paper we prove that if $\mathcal{P}$ is the property of being odd-hole-free or the property of being even-hole-free, then the $\mathcal{P}$-\textsc{Sandwich-Problem} is NP-hard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10888 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The sandwich problem for odd-hole-free and even-hole-free graphs Cameron, Kathie Chaniotis, Aristotelis de Figueiredo, Celina M. H. Spirkl, Sophie Combinatorics Discrete Mathematics For a property $\mathcal{P}$ of graphs, the $\mathcal{P}$-\textsc{Sandwich-Problem}, introduced by Golumbic and Shamir (1993), is the following: Given a pair of graphs $(G_1, G_2)$ on the same vertex set $V$, does there exist a graph $G$ such that $V(G)=V$, $E(G_{1})\subseteq E(G) \subseteq E(G_{2})$, and $G$ satisfies $\mathcal{P}$? A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. An odd (respectively even) hole is a hole of odd (respectively even) length. Given a class of graphs $\mathcal{C}$ and a graph $G$ we say that $G$ is {\em $\mathcal{C}$-free} if it contains no induced subgraph isomorphic to a member of $\mathcal{C}$. In this paper we prove that if $\mathcal{P}$ is the property of being odd-hole-free or the property of being even-hole-free, then the $\mathcal{P}$-\textsc{Sandwich-Problem} is NP-hard. |
| title | The sandwich problem for odd-hole-free and even-hole-free graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2404.10888 |