Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

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.Comment: Extended version of "Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees" appeared in the Proceedings of the 17th Algorithms and Data Structures Symposium (WADS 2021

Graph Drawings with One Bend and Few Slopes

International audienceWe consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that ∆ 2 edge slopes suffice for outerplanar drawings of outerpla-nar graphs with maximum degree ∆ 3. This matches the obvious lower bound. We also show that ∆ 2 + 1 edge slopes suffice for drawings of general graphs, improving on the previous bound of ∆ + 1. Furthermore, we improve previous upper bounds on the number of slopes needed for planar drawings of planar and bipartite planar graphs