The Parametrized Complexity of the Segment Number

TL;DR

This paper investigates the computational complexity of determining the minimum number of segments in planar graph drawings, showing fixed-parameter tractability with respect to several parameters and exploring colored variants.

Contribution

It establishes fixed-parameter tractability results for computing the segment number based on various parameters and analyzes colored versions of the problem.

Findings

Computing the segment number is $orall ext{R}$-complete and NP-hard.

The problem is fixed-parameter tractable with respect to vertex cover, segment number, and line cover number.

Colored variants of the segment and line cover number are also studied.

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.