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.

翻译：平面图$G$的$\textit{平面斜率数}$$psn(G)$定义为$G$的平面直线画法中所用边斜率的最小数目。已知对于任意最大度为$\Delta$的平面图$G$，有$psn(G) \in O(c^\Delta)$。当$G$的树宽为三时，该上界改进至$O(\Delta^5)$；当树宽为二时，改进至$O(\Delta)$。本文证明：若$G$为Halin图（其树宽为三），则$psn(G) \leq \max\{4,\Delta\}$。进一步，我们给出树宽为四的图族平面斜率数的首个多项式上界：对于嵌套伪树，$O(\Delta^2)$条斜率即足够。