The complexity of recognizing $ABAB$-free hypergraphs
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912351820709888 |
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| author | Damásdi, Gábor Keszegh, Balázs Pálvölgyi, Dömötör Singh, Karamjeet |
| author_facet | Damásdi, Gábor Keszegh, Balázs Pálvölgyi, Dömötör Singh, Karamjeet |
| contents | The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices $v_1,v_2,v_3,v_4$ in this order satisfying $v_1,v_3\in A\setminus B$ and $v_2,v_4\in B\setminus A$. In this paper, we prove that it is NP-complete to decide if a hypergraph is $ABAB$-free. We show a number of analogous results for hypergraphs with similar forbidden patterns, such as $ABABA$-free hypergraphs. As an application, we show that deciding whether a hypergraph is realizable as the incidence hypergraph of points and pseudodisks is also NP-complete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01680 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The complexity of recognizing $ABAB$-free hypergraphs Damásdi, Gábor Keszegh, Balázs Pálvölgyi, Dömötör Singh, Karamjeet Combinatorics 05C65 The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices $v_1,v_2,v_3,v_4$ in this order satisfying $v_1,v_3\in A\setminus B$ and $v_2,v_4\in B\setminus A$. In this paper, we prove that it is NP-complete to decide if a hypergraph is $ABAB$-free. We show a number of analogous results for hypergraphs with similar forbidden patterns, such as $ABABA$-free hypergraphs. As an application, we show that deciding whether a hypergraph is realizable as the incidence hypergraph of points and pseudodisks is also NP-complete. |
| title | The complexity of recognizing $ABAB$-free hypergraphs |
| topic | Combinatorics 05C65 |
| url | https://arxiv.org/abs/2409.01680 |