Bitonic st-Orderings for Upward Planar Graphs: The Variable Embedding Setting

摘要

Bitonic si-orderings for si-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 si-planar graph that does not admit a bitonic si-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 si-planar graph. The best-known lower bound on the number of required splits of an si-planar graph with n vertices is n-3. However, it is possible to compute a bitonic si-ordering without any split for the si-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 si-planar graph that needs at least n-5 splits in both orientations.