Drawing Two Posets

TL;DR

This paper studies the problem of drawing two partial orders on the same set of vertices as a planar graph with monotone edges, providing a linear-time algorithm for a specific connected case.

Contribution

It introduces a linear-time algorithm for testing and constructing $xy$-planar drawings when the embedding is fixed and the subgraph is connected.

Findings

NP-complete in general

Linear-time algorithm for fixed embedding case

Produces drawings with at most three bends per edge

Abstract

We investigate the problem of drawing two posets of the same ground set so that one is drawn from left to right and the other one is drawn from the bottom up. The input to this problem is a directed graph $G = (V, E)$ and two sets $X, Y$ with $X \cup Y = E$, each of which can be interpreted as a partial order of $V$. The task is to find a planar drawing of $G$ such that each directed edge in $X$ is drawn as an $x$-monotone edge, and each directed edge in $Y$ is drawn as a $y$-monotone edge. Such a drawing is called an $xy$-planar drawing. Testing whether a graph admits an $xy$-planar drawing is NP-complete in general. We consider the case that the planar embedding of $G$ is fixed and the subgraph of $G$ induced by the edges in $Y$ is a connected spanning subgraph of $G$ whose upward embedding is fixed. For this case we present a linear-time algorithm that determines whether $G$ admits an $xy$-planar drawing and, if so, produces an $xy$-planar polyline drawing with at most three bends per edge.