Universal Slope Sets for Upward Planar Drawings
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We prove that every set S of Δ slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree Δ, i.e., every such digraph admits a 1-bend upward planar drawing whose edge segments use only slopes in S. This result is worst-case optimal in terms of the number of slopes, and, for a suitable choice of S, it gives rise to drawings with worst-case optimal angular resolution. In addition, we prove that every such set S can be used to construct 2-bend upward planar drawings of n-vertex planar st-graphs with at most 4n-9 bends in total. Our main tool is a constructive technique that runs in linear time.
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