Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

The $\mathit{\text{planar}}\mathit{\text{slope}}\mathit{\text{number}}$ $psn(G)$ of a planar graph $G$ is the minimum number of edge slopes in a planar straight-line drawing of $G$. It is known that $psn(G)\in O(c^{\mathrm{\Delta }})$ for every planar graph $G$ of maximum degree $\mathrm{\Delta }$. This upper bound has been improved to $O({\mathrm{\Delta }}^5)$ if $G$ has treewidth three, and to $O(\mathrm{\Delta })$ if $G$ has treewidth two. In this paper we prove $psn(G)\le \max \{4,\mathrm{\Delta }\}$ 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 $O({\mathrm{\Delta }}^2)$ slopes suffice for nested pseudotrees.