Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

TL;DR

This paper establishes new bounds on the minimum number of slopes needed for planar straight-line drawings of specific graph families, including Halin graphs and nested pseudotrees, improving understanding of their geometric representations.

Contribution

It proves that Halin graphs have a planar slope number at most max{4, Δ} and provides the first polynomial bound for nested pseudotrees with O(Δ^2) slopes.

Findings

Halin graphs have psn ≤ max{4, Δ}.

Nested pseudotrees can be drawn with O(Δ^2) slopes.

First polynomial bound for graphs with treewidth four.

Abstract

The $\textit{planar slope number}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G) \in O(c^\Delta)$ for every planar graph $G$ of maximum degree $\Delta$. This upper bound has been improved to $O(\Delta^5)$ if $G$ has treewidth three, and to $O(\Delta)$ if $G$ has treewidth two. In this paper we prove $psn(G) \leq \max\{4,\Delta\}$ when $G$ is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that $O(\Delta^2)$ slopes suffice for nested pseudotrees.