Upward Point-Set Embeddability

TL;DR

This paper investigates the conditions under which upward planar digraphs can be embedded into point sets, proving positive results for switch trees, limitations for k-switch trees, and establishing NP-completeness of the problem.

Contribution

It proves that switch trees can always be embedded into convex point sets, shows limitations for k-switch trees, and establishes NP-completeness of the embedding problem.

Findings

Switch trees admit upward planar straight-line embeddings into any convex point set.

Not all k-switch trees (k ≥ 2) can be embedded into any convex point set.

The Upward Point-Set Embeddability problem is NP-complete.

Abstract

We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a given upward planar digraph $D$ has an upward planar embedding into a point set $S$. We show that any switch tree admits an upward planar straight-line embedding into any convex point set. For the class of $k$-switch trees, that is a generalization of switch trees (according to this definition a switch tree is a $1$-switch tree), we show that not every $k$-switch tree admits an upward planar straight-line embedding into any convex point set, for any $k \geq 2$. Finally we show that the problem of Upward Point-Set Embeddability is NP-complete.