Computing crossing numbers with topological and geometric restrictions

TL;DR

This paper introduces a new framework for analyzing the computational complexity of crossing number problems with topological and geometric constraints, advancing understanding of their tractability and hardness.

Contribution

It proposes a general framework for crossing number variants with restrictions and demonstrates its effectiveness, including new hardness results for geometric cases.

Findings

Framework generalizes previous results on crossing numbers.

Shows tractability for certain restricted crossing number variants.

Establishes W-hardness for rectilinear crossing number extension.

Abstract

Computing the crossing number of a graph is one of the most classical problems in computational geometry. Both it and numerous variations of the problem have been studied, and overcoming their frequent computational difficulty is an active area of research. Particularly recently, there has been increased effort to show and understand the parameterized tractability of various crossing number variants. While many results in this direction use a similar approach, a general framework remains elusive. We suggest such a framework that generalizes important previous results, and can even be used to show the tractability of deciding crossing number variants for which this was stated as an open problem in previous literature. Our framework targets variants that prescribe a partial predrawing and some kind of topological restrictions on crossings. Additionally, to provide evidence for the non-generalizability of previous approaches for the partially crossing number problem to allow for geometric restrictions, we show a new more constrained hardness result for partially predrawn rectilinear crossing number. In particular, we show W-hardness of deciding Straight-Line Planarity Extension parameterized by the number of missing edges.