Rectilinear Planarity Testing of Plane Series-Parallel Graphs in Linear Time

TL;DR

This paper presents an optimal linear-time algorithm for testing rectilinear planarity of plane series-parallel graphs and constructing such drawings if they exist, based on a new characterization involving orthogonal spirality intervals.

Contribution

It introduces a novel characterization of rectilinear planar series-parallel graphs and provides an efficient linear-time algorithm for testing and constructing rectilinear drawings.

Findings

Rectilinear planarity testing is solvable in O(n) time for plane series-parallel graphs.

An embedding-preserving rectilinear drawing can be constructed in O(n) time if the graph is rectilinear planar.

The characterization involves intervals of orthogonal spirality of graph components.

Abstract

A plane graph is rectilinear planar if it admits an embedding-preserving straight-line drawing where each edge is either horizontal or vertical. We prove that rectilinear planarity testing can be solved in optimal $O(n)$ time for any plane series-parallel graph $G$ with $n$ vertices. If $G$ is rectilinear planar, an embedding-preserving rectilinear planar drawing of $G$ can be constructed in $O(n)$ time. Our result is based on a characterization of rectilinear planar series-parallel graphs in terms of intervals of orthogonal spirality that their components can have, and it leads to an algorithm that can be easily implemented.