A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917091775348736 |
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| author | Cabello, Sergio Dobler, Alexander Fijavž, Gašper Hamm, Thekla Wagner, Mirko H. |
| author_facet | Cabello, Sergio Dobler, Alexander Fijavž, Gašper Hamm, Thekla Wagner, Mirko H. |
| contents | A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets $\mathcal{S}$ of crossing types gives a recognition problem: does a given graph admit an $\mathcal{S}$-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in $\mathcal{S}$?
We show that there is a set $\mathcal{S}_{\rm bad}$ with three crossing types and the following properties: If $\mathcal{S}$ contains no crossing type from $\mathcal{S}_{\rm bad}$, then the recognition of graphs that admit an $\mathcal{S}$-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. If $\mathcal{S}$ contains any crossing type from $\mathcal{S}_{\rm bad}$, then it is NP-hard to decide whether a graph has an $\mathcal{S}$-restricted drawing, even when considering graphs of constant pathwidth.
We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth Cabello, Sergio Dobler, Alexander Fijavž, Gašper Hamm, Thekla Wagner, Mirko H. Data Structures and Algorithms Computational Geometry Combinatorics A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets $\mathcal{S}$ of crossing types gives a recognition problem: does a given graph admit an $\mathcal{S}$-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in $\mathcal{S}$? We show that there is a set $\mathcal{S}_{\rm bad}$ with three crossing types and the following properties: If $\mathcal{S}$ contains no crossing type from $\mathcal{S}_{\rm bad}$, then the recognition of graphs that admit an $\mathcal{S}$-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. If $\mathcal{S}$ contains any crossing type from $\mathcal{S}_{\rm bad}$, then it is NP-hard to decide whether a graph has an $\mathcal{S}$-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth. |
| title | A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth |
| topic | Data Structures and Algorithms Computational Geometry Combinatorics |
| url | https://arxiv.org/abs/2511.14975 |