Abstract
Bitonic st-orderings for st-planar graphs were recently introduced as a method to cope with several graph drawing problems. Notably, they have been used to obtain the best-known upper bound on the number of bends for upward planar polyline drawings with at most one bend per edge. For an st-planar graph that does not admit a bitonic st-ordering, one may split certain edges such that for the resulting graph such an ordering exists. Since each split is interpreted as a bend, one is usually interested in splitting as few edges as possible. While this optimization problem admits a linear-time algorithm in the fixed embedding setting, it remains open in the variable embedding setting. We close this gap in the literature by providing a linear-time algorithm that optimizes over all embeddings of the input st-planar graph.
The best-known lower bound on the number of required splits of an st-planar graph with n vertices is \(n-3\). However, it is possible to compute a bitonic st-ordering without any split for the st-planar graph obtained by reversing the orientation of all edges. In terms of upward planar polyline drawings, the former translates into \(n-3\) bends, while the latter into no bends. We show that this idea cannot always be exploited by describing an st-planar graph that needs at least \(n-5\) splits in both orientations.
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The authors would like to thank Michael Kaufmann and Antonios Symvonis for fruitful discussions.
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Angelini, P., Bekos, M.A., Förster, H., Gronemann, M. (2020). Bitonic st-Orderings for Upward Planar Graphs: The Variable Embedding Setting. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_27
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