Note on Min-k-Planar Drawings of Graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866913524252409856 |
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| author | Hliněný, Petr Ködmön, Lili |
| author_facet | Hliněný, Petr Ködmön, Lili |
| contents | The k-planar graphs, which are (usually with small values of k such as 1, 2, 3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently introduced [Binucci et al., GD 2023] min-k-planar drawings of graphs, edges may possibly carry more than k crossings, but in any two crossing edges, at least one of the two must have at most k crossings. In both concepts, one may consider general drawings or a popular restricted concept of drawings called simple. In a simple drawing, every two edges are allowed to cross at most once, and any two edges which share a vertex are forbidden to cross. While, regarding the former concept, it is for k<=3 known (but perhaps not widely known) that every general k-planar graph admits a simple k-planar drawing and this ceases to be true for any k>=4, the difference between general and simple drawings in the latter concept is more striking. We prove that there exist graphs with a min-2-planar drawing, or with a min-3-planar drawing avoiding crossings of adjacent edges, which have no simple min-k-planar drawings for arbitrarily large fixed k. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_11610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Note on Min-k-Planar Drawings of Graphs Hliněný, Petr Ködmön, Lili Combinatorics Discrete Mathematics 68R10 The k-planar graphs, which are (usually with small values of k such as 1, 2, 3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently introduced [Binucci et al., GD 2023] min-k-planar drawings of graphs, edges may possibly carry more than k crossings, but in any two crossing edges, at least one of the two must have at most k crossings. In both concepts, one may consider general drawings or a popular restricted concept of drawings called simple. In a simple drawing, every two edges are allowed to cross at most once, and any two edges which share a vertex are forbidden to cross. While, regarding the former concept, it is for k<=3 known (but perhaps not widely known) that every general k-planar graph admits a simple k-planar drawing and this ceases to be true for any k>=4, the difference between general and simple drawings in the latter concept is more striking. We prove that there exist graphs with a min-2-planar drawing, or with a min-3-planar drawing avoiding crossings of adjacent edges, which have no simple min-k-planar drawings for arbitrarily large fixed k. |
| title | Note on Min-k-Planar Drawings of Graphs |
| topic | Combinatorics Discrete Mathematics 68R10 |
| url | https://arxiv.org/abs/2401.11610 |