Bitonic st-Orderings for Upward Planar Graphs: Splits and Bends in the Variable Embedding Scenario

• Published in: Algorithmica Vol. 85; no. 9; pp. 2667 - 2692
• Main Authors: Angelini, Patrizio, Bekos, Michael A., Förster, Henry, Gronemann, Martin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-09-2023; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Bends; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Embedding; Graph theory; Graphs; Lower bounds; Mathematics of Computing; Optimization; Polynomials; Theory of Computation; Upper bounds; Bend minimization; Bitonic st-orderings; Planar polyline drawings; Upward planar graphs
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-023-01111-5

Bitonic
st
-orderings for
st
-planar graphs were 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 in polynomial area. 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 in polynomial area, 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. We provide analogous bounds for graphs with small degree. Finally, we further investigate the relationship between splits in bitonic st-orderings and bends in upward planar polyline drawings with polynomial area, by providing bounds on the number of bends in such drawings.