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 Recognition
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible 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