Computing crossing numbers with topological and geometric restrictions

TL;DR

This work introduces a general, topologically aware framework for partially predrawn crossing-number problems by formalizing topological crossing patterns and associated constraints. It provides an algorithmic pipeline that reduces instances to bounded treewidth and then encodes the existence of low-crossing drawings as MSO-formulas, enabling fixed-parameter tractability via Courcelle's theorem. The framework unifies and extends prior results, enabling tractability for fan-planar and pseudolinear variants under partial predrawings, and proves hardness for partially predrawn rectilinear crossing numbers, highlighting the limits of geometric restrictions. Beyond tractability, it also contributes a robust method to handle topological and geometric restrictions in crossing-number problems, with potential applicability to a broad class of drawing-extension problems.

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.

Figures (5)

• Figure 1: The obstructions for pseudolinear drawing, figure based on DBLP:journals/jocg/ArroyoBR21.
• Figure 2: The two obstructions characterizing fan-planar drawings.
• Figure 3: The two basic forbidden patterns for fan-planar crossing number (left). The two other versions how the two patterns might appear in a planarized drawing.
• Figure 4: Basic patterns derived from the obstructions in \ref{['fig:plobstructions']}. Cyan squares are crossings. Brown edges form the cycle and squares are crossings or vertices, both are predrawn in the pattern.
• Figure 5: Illustration for embedding choices distinguished by the selection of rotation markers. Black disks are 1- and 2-vertex cuts which separate 3-connected components of a prospective planarization. Special vertices in 3-connected components are blue and indexed from $1$ through $9$. $w$ is the only important vertex for $v$. Components comprising important groups for $v$ are highlighted. To encode (a), all selected rotation markers for edges incident to $v$ have $\{\{1\}, \{2,3,4\}, \{5,6\},\{7\}\}$ as first label entry and second label entry would specify the permutation in the written order. The exemplary colored edges incident to $v$ would have the following third label entries in (a): green $\rightarrow$ 1, red $\rightarrow$ 1, yellow $\rightarrow$ 0, brown $\rightarrow$ 3. Here, the third entries of the labels of the selected rotation markers cannot allow us to distinguish the order of the three components between $v$ and $w$; they will be 1 for all edges leading into these components. Similarly, we cannot distinguish (a) and (b). However, our encoding distinguishes (a) and (b) from (c) by selecting a rotation marker with third label entry 4 instead of 3 for the brown edge. Encoding (d) corresponds to selecting rotation markers with the same first label entries as for (a), (b) and (c) but different second label entries.

Theorems & Definitions (13)

• Theorem 1: DBLP:journals/jct/RobertsonS86
• Theorem 2
• Remark 3
• Remark 4
• Corollary 5
• Corollary 6
• Lemma 7
• Theorem 8
• Lemma 9
• Lemma 10