Upward Point Set Embeddings of Paths and Trees

TL;DR

This paper investigates upward planar straight-line embeddings of directed trees on point sets, providing characterizations for paths, NP-completeness results, and guarantees for caterpillars with extra points.

Contribution

It offers new results on UPSE existence for paths and trees, including exact counts, special case conditions, complexity proofs, and embedding guarantees with additional points.

Findings

Number of UPSEs for one-sided convex point sets equals maximal monotone paths in a path.

Certain monotone path configurations always admit UPSEs in general position.

Deciding UPSE existence for directed trees is NP-complete, even with fixed vertex positions.

Abstract

We study upward planar straight-line embeddings (UPSE) of directed trees on given point sets. The given point set $S$ has size at least the number of vertices in the tree. For the special case where the tree is a path $P$ we show that: (a) If $S$ is one-sided convex, the number of UPSEs equals the number of maximal monotone paths in $P$. (b) If $S$ is in general position and $P$ is composed by three maximal monotone paths, where the middle path is longer than the other two, then it always admits an UPSE on $S$. We show that the decision problem of whether there exists an UPSE of a directed tree with $n$ vertices on a fixed point set $S$ of $n$ points is NP-complete, by relaxing the requirements of the previously known result which relied on the presence of cycles in the graph, but instead fixing position of a single vertex. Finally, by allowing extra points, we guarantee that each directed caterpillar on $n$ vertices and with $k$ switches in its backbone admits an UPSE on every set of $n 2^{k-2}$ points.