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

TL;DR

This paper classifies the computational complexity of recognizing restricted 1-planar graph drawings based on crossing types, showing fixed-parameter tractability or NP-hardness depending on the crossing type set and graph treewidth.

Contribution

It introduces a dichotomy for recognizing restricted 1-planar graphs based on crossing types, with complexity results parameterized by treewidth.

Findings

Recognition is fixed-parameter tractable when crossing types avoid a specific bad set.

Recognition becomes NP-hard when crossing types include any from the bad set.

Results extend to straight-line 1-planar drawings with similar complexity behavior.

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.