Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees
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The 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) ∈ O(c^Δ) for every planar graph G of degree Δ. This upper bound has been improved to O(Δ^5) if G has treewidth three, and to O(Δ) if G has treewidth two. In this paper we prove psn(G) ∈Θ(Δ) 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(Δ^2) slopes suffice for nested pseudotrees.
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