Drawing Outer 1-planar Graphs with Few Slopes
A graph is outer 1-planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1-planar graphs are a superclass of the outerplanar graphs and a subclass of the partial 3-trees. We show that an outer 1-planar graph G of bounded degree Δ admits an outer 1-planar straight-line drawing that uses O(Δ) different slopes, which extends a previous result by Knauer et al. about the planar slope number of outerplanar graphs (CGTA, 2014). We also show that O(Δ2) slopes suffice to construct a crossing-free straight-line drawing of G; the best known upper bound on the planar slope number of planar partial 3-trees of bounded degree Δ is O(Δ5) and is proved by Jelínek et al. (Graphs and Combinatorics, 2013).
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