Bitonic st-orderings for Upward Planar Graphs

TL;DR

This paper introduces bitonic st-orderings for directed planar graphs, providing a linear-time recognition algorithm and a method to modify graphs for upward planar drawings with minimal bends and quadratic area.

Contribution

It extends the concept of bitonic st-orderings to directed graphs, characterizes those that admit such orderings, and offers algorithms for recognition and modification.

Findings

Linear-time algorithm for recognition of bitonic st-orderings

Method to minimally modify graphs to admit bitonic st-orderings

Upward planar drawings with at most one bend per edge within quadratic area

Abstract

Canonical orderings serve as the basis for many incremental planar drawing algorithms. All these techniques, however, have in common that they are limited to undirected graphs. While $st$-orderings do extend to directed graphs, especially planar $st$-graphs, they do not offer the same properties as canonical orderings. In this work we extend the so called bitonic $st$-orderings to directed graphs. We fully characterize planar $st$-graphs that admit such an ordering and provide a linear-time algorithm for recognition and ordering. If for a graph no bitonic $st$-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. With this new technique we are able to draw every upward planar graph on $n$ vertices by using at most one bend per edge, at most $n - 3$ bends in total and within quadratic area.