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

2025-11-18

AI Summary

This paper investigates the problem of recognizing *S-restricted 1-planarity*: given a graph and a set *S* of admissible crossing types, decide whether the graph admits a 1-planar embedding where all crossings belong to *S*. By systematically analyzing six fundamental crossing configurations, we identify a minimal “bad” set *S_bad* of three crossing types. We prove that the problem is fixed-parameter tractable (FPT) parameterized by treewidth if and only if *S ∩ S_bad = ∅*; otherwise, it is NP-hard—establishing the first precise complexity dichotomy for 1-planarity under crossing constraints. Leveraging tree decompositions, bespoke parameterized algorithms, and intricate reductions, we fully classify the computational complexity of all 63 nonempty subsets *S*. The results extend to straight-line 1-planar drawings. Our work uncovers a fundamental structural boundary linking crossing-type restrictions with graph-theoretic properties.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Planning, Routing, and Scheduling: Activity and Plan Recognition
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web applications in cross-disciplinary domains and verticals such as mixed reality, smart cities, and digital health
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $mathcal{S}_{ m bad}$ with three crossing types and the following properties: If $mathcal{S}$ contains no crossing type from $mathcal{S}_{ m bad}$, 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 $mathcal{S}_{ m bad}$, 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.

Problem

• Classifying which crossing types make 1-planar graph recognition tractable
• Determining complexity of restricted 1-planar drawings by treewidth parameterization
• Identifying NP-hard crossing type subsets for constant pathwidth graphs

Innovation

• Classified six crossing types for 1-planar drawings
• Identified three forbidden crossing types for tractability
• Applied treewidth parameterization for complexity characterization