Bitonic st-orderings for Upward Planar Graphs

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 lineartime 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.

Universal Slope Sets for Upward Planar Drawings

“…3a. We begin by describing an augmentation technique to "transform" σ into an upward canonical ordering of a suitable supergraph G of G. We start from a result by Gronemann [18], whose properties are summarized in the following lemma; see, e.g., Fig. 3b.…”

“…Let G be an n-vertex non-bitonic planar st-graph. All forbidden configurations of G can be removed in linear time by subdividing at most n − 3 edges of G [18]. Let G b be the resulting bitonic st-graph, called a bitonic subdivision of G. Let u, d, v be a directed path of G b obtained by subdividing the edge (u, v) of G with the dummy vertex d. We call (u, d) the lower stub, and (d, v) the upper stub of (u, v).…”

“…We extend the study of universal sets of slopes to upward planar drawings, and present the first constructive technique that works for all planar st-graphs. This technique exploits a linear ordering of the vertices of a planar digraph introduced by Gronemann [18], called bitonic st-ordering (see also Section 2). We show that any set S of ∆ slopes containing the horizontal slope is universal for 1-bend upward planar drawings of degree-∆ planar digraphs having a bitonic st-ordering (Section 3).…”

“…[18]). Let G = (V, E) be an n-vertex planar st-graph that admits a bitonic st-ordering σ = {v 1 , v 2 , .…”

“…Drawing Γ can be computed in O(n) time.Proof. A bitonic st-ordering σ of G can be computed in O(n) time[18], and the same time complexity suffices to compute a canonical augmentation G of G by Lemma 1. A triangulated planar st-graph G can be obtained in O(n) time by augmenting G as follows.…”