Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

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