Segment Intersection Representations, Level Planarity and Constrained Ordering Problems

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.