The Parameterized Complexity of Geometric 1-Planarity

TL;DR

This paper studies the parameterized complexity of recognizing geometric 1-planar graphs, showing fixed-parameter tractability with respect to treedepth and providing kernelization results based on feedback edge number.

Contribution

It introduces the first systematic parameterized complexity analysis of geometric 1-planarity, extending existing techniques and improving kernel sizes for related problems.

Findings

Recognition is fixed-parameter tractable when parameterized by treedepth.

Provides a kernel of size O(ell * 8^ell) for geometric 1-planarity based on feedback edge number.

Establishes NP-completeness for geometric 1-planarity even under bounded pathwidth, feedback vertex number, and bandwidth.

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.