The Parametrized Complexity of the Segment Number

TL;DR

This work investigates the parameterized complexity of the segment number for planar graphs, a measure of geometric drawing complexity, and shows FPT results with respect to the natural parameter, the line cover number, and the vertex cover number. It combines Renegar’s existential theory of the reals with careful graph reductions, planarity- and embedding-aware constructions, and ILP techniques to obtain tractable algorithms despite the underlying $\exists\mathbb{R}$-hardness in general. The paper also develops a banana-graph warmup, extends to colored variants, and provides detailed method-ical steps (core graph construction, boomerang gadgets, and routing) to encode alignment constraints. These results advance understanding of when optimal geometric representations can be computed efficiently and establish a framework for further parameterized explorations in geometric graph drawing. The methods have potential implications for algorithmic graph drawing and the design of FPT approaches to related line- and segment-based representations.

Abstract

Given a straight-line drawing of a graph, a segment is a maximal set of edges that form a line segment. Given a planar graph $G$, the segment number of $G$ is the minimum number of segments that can be achieved by any planar straight-line drawing of $G$. The line cover number of $G$ is the minimum number of lines that support all the edges of a planar straight-line drawing of $G$. Computing the segment number or the line cover number of a planar graph is $\exists\mathbb{R}$-complete and, thus, NP-hard. We study the problem of computing the segment number from the perspective of parameterized complexity. We show that this problem is fixed-parameter tractable with respect to each of the following parameters: the vertex cover number, the segment number, and the line cover number. We also consider colored versions of the segment and the line cover number.

Figures (6)

• Figure 1: The dodecahedron graph has 30 edges, segment number 13, line cover number 10, arc number 10, and circle cover number 5 krw-dgfcf-JGAA19.
• Figure 2: Banana graphs
• Figure 3: Segment minimum drawings of banana cycles.
• Figure 5: We vary over all pairings $\pi$ of edges and boomerangs around each vertex. (a) The dashed edges are inserted into $G_2(\mathcal{C},\mathcal{E})$ in order to represent alignment requirements with boomerangs. If the alignment requirements of the thus constructed graph $G_2(\mathcal{C},\mathcal{E},\pi)$ can be geometrically realized, we use an ILP to distribute the remaining edges. (b) How to geometrically insert the 2-classes into a boomerang.
• Figure 6: Example situation for $G_2(\mathcal{C},\mathcal{E},\pi)$ as a basis for the description of the ILP.

Theorems & Definitions (14)

• lemma 1
• Theorem 1: Renegar Renegar92aRenegar92bRenegar92c
• lemma 2
• Corollary 2
• lemma 3: dujmovic_etal:compGeo07
• Theorem 3
• Theorem 4
• lemma 4
• Theorem 5
• Theorem 6