Extending simple monotone drawings
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911186160713728 |
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| author | Kynčl, Jan Soukup, Jan |
| author_facet | Kynčl, Jan Soukup, Jan |
| contents | We prove the following variant of Levi's Enlargement Lemma: for an arbitrary arrangement $\mathcal{A}$ of $x$-monotone pseudosegments in the plane and a pair of points $a,b$ with distinct $x$-coordinates and not on the same pseudosegment, there exists a simple $x$-monotone curve with endpoints $a,b$ that intersects every curve of $\mathcal{A}$ at most once. As a consequence, every simple monotone drawing of a graph can be extended to a simple monotone drawing of a complete graph. We also show that extending an arrangement of cylindrically monotone pseudosegments is not always possible; in fact, the corresponding decision problem is NP-hard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17675 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extending simple monotone drawings Kynčl, Jan Soukup, Jan Combinatorics Computational Geometry 05C10 (Primary) 68R10 (Secondary) We prove the following variant of Levi's Enlargement Lemma: for an arbitrary arrangement $\mathcal{A}$ of $x$-monotone pseudosegments in the plane and a pair of points $a,b$ with distinct $x$-coordinates and not on the same pseudosegment, there exists a simple $x$-monotone curve with endpoints $a,b$ that intersects every curve of $\mathcal{A}$ at most once. As a consequence, every simple monotone drawing of a graph can be extended to a simple monotone drawing of a complete graph. We also show that extending an arrangement of cylindrically monotone pseudosegments is not always possible; in fact, the corresponding decision problem is NP-hard. |
| title | Extending simple monotone drawings |
| topic | Combinatorics Computational Geometry 05C10 (Primary) 68R10 (Secondary) |
| url | https://arxiv.org/abs/2312.17675 |