Segment Intersection Representations, Level Planarity and Constrained Ordering Problems

TL;DR

This paper investigates the complexity of representing graphs with axis-aligned segments under fixed vertical order, providing efficient solutions through graph drawing and combinatorial approaches, and establishing equivalences among these perspectives.

Contribution

It resolves the open problem of segment intersection representation complexity with fixed vertical order by introducing two novel, efficient solution methods and connecting geometric, graph drawing, and combinatorial viewpoints.

Findings

Relates segment intersection representation to Level Planarity with constraints

Provides quadratic-time algorithms for the problem

Establishes equivalence among geometric, graph drawing, and combinatorial formulations

Abstract

In the Segment Intersection Graph Representation Problem, we want to represent the vertices of a graph as straight line segments in the plane such that two segments cross if and only if there is an edge between the corresponding vertices. This problem is NP-hard (even $\exists\mathbb{R}$-complete [Schaefer, 2010]) in the general case [Kratochv\'il & Ne\^setril, 1992] and remains so if we restrict the segments to be axis-aligned, i.e., horizontal and vertical [Kratochv\'il, 1994]. A long standing open question for the latter variant is its complexity when the order of segments along one axis (say the vertical order of horizontal segments) is already given [Kratochv\'il & Ne\^setril, 1992; Kratochv\'il, 1994]. We resolve this question by giving efficient solutions using two very different approaches that are interesting on their own. First, using a graph-drawing perspective, we relate the problem to a variant of the well-known Level Planarity problem, where vertices have to lie on pre-assigned horizontal levels. In our case, each level also carries consecutivity constraints on its vertices; this Level Planarity variant is known to have a quadratic solution. Second, we use an entirely combinatorial approach, and show that both problems can equivalently be formulated as a linear ordering problem subject to certain consecutivity constraints. While the complexity of such problems varies greatly, we show that in this case the constraints are well-structured in a way that allows a direct quadratic solution. Thus, we obtain three different-but-equivalent perspectives on this problem: the initial geometric one, one from planar graph drawing and a purely combinatorial one.