The Parameterized Complexity of Geometric 1-Planarity

2026-02-10

AI Summary

This study investigates the parameterized complexity of recognizing geometric 1-planar graphs—those admitting a straight-line drawing in which each edge is crossed at most once. By integrating Thomassen’s characterization of straight-line 1-planar embeddings with the framework of Bannister, Cabello, and Eppstein, the authors present the first fixed-parameter tractable (FPT) algorithm parameterized by tree-depth. Furthermore, they derive a kernel of size $O(\ell \cdot 8^\ell)$ parameterized by the feedback edge number $\ell$, significantly improving upon existing kernel bounds for $k$-planarity recognition. The work also establishes that the problem remains NP-complete even when restricted to graphs of bounded pathwidth, feedback vertex set size, or bandwidth, thereby revealing its inherent computational hardness despite various structural restrictions.

Technology Category

Planning, Routing, and Scheduling: Activity and Plan RecognitionKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A graph is geometric 1-planar if it admits a straight-line drawing where each edge is crossed at most once. We provide the first systematic study of the parameterized complexity of recognizing geometric 1-planar graphs. By substantially extending a technique of Bannister, Cabello, and Eppstein, combined with Thomassen's characterization of 1-planar embeddings that can be straightened, we show that the problem is fixed-parameter tractable when parameterized by treedepth. Furthermore, we obtain a kernel for Geometric 1-Planarity parameterized by the feedback edge number $\ell$. As a by-product, we improve the best known kernel size of $O((3\ell)!)$ for 1-Planarity and $k$-Planarity under the same parameterization to $O(\ell \cdot 8^{\ell})$. Our approach naturally extends to Geometric $k$-Planarity, yielding a kernelization under the same parameterization, albeit with a larger kernel. Complementing these results, we provide matching lower bounds: Geometric 1-Planarity remains \NP-complete even for graphs of bounded pathwidth, bounded feedback vertex number, and bounded bandwidth.

Problem

• Geometric 1-Planarity
• parameterized complexity
• graph drawing
• straight-line embedding
• edge crossings

Innovation

• parameterized complexity
• geometric 1-planarity
• fixed-parameter tractability
• kernelization
• treedepth