Computing Bend-Minimum Orthogonal Drawings of Plane Series-Parallel Graphs in Linear Time

TL;DR

This paper presents a linear-time algorithm for computing bend-minimum orthogonal drawings of plane series-parallel graphs, solving a 30-year-old open problem in graph drawing.

Contribution

It introduces a characterization based on orthogonal spirality, enabling efficient computation of bend-minimum drawings for this class of graphs.

Findings

Linear-time algorithm for bend-minimum drawings

Characterization using orthogonal spirality

Applicable to plane series-parallel graphs

Abstract

A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of $G$ to a sequence of horizontal and vertical segments between its end-points. A longstanding open question in Graph Drawing, dating back over 30 years, is whether there exists a linear-time algorithm to compute an orthogonal drawing of a plane 4-graph with the minimum number of bends. The term "plane" indicates that the input graph comes together with a planar embedding, which must be preserved by the drawing (i.e., the drawing must have the same set of faces as the input graph). In this paper, we positively answer the question above for the widely-studied class of series-parallel graphs. Our linear-time algorithm is based on a characterization of the planar series-parallel graphs that admit an orthogonal drawing without bends. This characterization is given in terms of the orthogonal spirality that each type of triconnected component of the graph can take; the orthogonal spirality of a component measures how much that component is "rolled-up" in an orthogonal drawing of the graph.