A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth

Sergio Cabello; Alexander Dobler; Gašper Fijavž; Thekla Hamm; Mirko H. Wagner

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arXiv · 2511.14975

A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth

Sergio Cabello·Alexander Dobler·Gašper Fijavž·Thekla Hamm·Mirko H. Wagner

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Abstract

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.

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Sergio Cabello, Alexander Dobler, Gašper Fijavž, Thekla Hamm, Mirko H. Wagner. 2025-11-18. A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth. https://arxiv.org/abs/2511.14975

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