Paper detail

Outerplanar graph drawings with few slopes

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that $Δ-1$ edge slopes suffice for every outerplanar graph with maximum degree $Δ\ge 4$. This improves on the previous bound of $O(Δ^5)$, which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every $Δ\ge 4$ there is an outerplanar graph with maximum degree $Δ$ that requires at least $Δ-1$ distinct edge slopes in an outerplanar straight-line drawing.