Drawing Two Posets
Guido Br\"uckner, Vera Chekan

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 and two sets with , each of which can be interpreted as a partial order of . The task is to find a planar drawing of such that each directed edge in is drawn as an -monotone edge, and each directed edge in is drawn as a -monotone edge. Such a drawing is called an -planar drawing. Testing whether a graph admits an -planar drawing is NP-complete in general. We consider the case that the planar embedding of is fixed and the subgraph of induced by the edges in is a connected spanning subgraph of whose upward embedding is fixed. For this case we present a linear-time algorithm that determines whether admits…
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Taxonomy
TopicsComputational Geometry and Mesh Generation
