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

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees