On Upward Drawings of Trees on a Given Grid

TL;DR

This paper investigates the complexity of creating minimal-area upward drawings of trees within a grid, establishing NP-hardness for deciding the existence of such drawings, which advances understanding of tree drawing problems.

Contribution

It proves NP-hardness for determining if a rooted tree can be strictly-upward drawn within a specified grid, a key step in understanding area minimization complexity.

Findings

NP-hardness of upward tree drawing decision problem

Advances understanding of tree drawing complexity

Highlights open problems in area minimization for trees

Abstract

Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known for straight-line drawings of trees under various restrictions such as edge length or slope constraints. On the other hand, there exist polynomial-time algorithms for computing minimum-width (resp., minimum-height) upward drawings of trees, where the height (resp., width) is unbounded. In this paper we take a major step in understanding the complexity of the area minimization problem for strictly-upward drawings of trees, which is one of the most common styles for drawing rooted trees. We prove that given a rooted tree $T$ and a $W\times H$ grid, it is NP-hard to decide whether $T$ admits a strictly-upward (unordered) drawing in the given grid.