Towards a Topology-Shape-Metrics Framework for Ortho-Radial Drawings

TL;DR

This paper introduces a combinatorial characterization for plane graphs that can be drawn in an ortho-radial manner without bends, extending the Topology-Shape-Metrics framework to this new class of graph drawings.

Contribution

It provides the first comprehensive combinatorial characterization for bend-free ortho-radial drawings of plane graphs, generalizing previous results and frameworks.

Findings

Characterization applies to all plane graphs with ortho-radial drawings without bends.

Extension of Tamassia's Topology-Shape-Metrics framework to ortho-radial drawings.

Framework enables systematic understanding and potential algorithms for ortho-radial graph layouts.

Abstract

Ortho-Radial drawings are a generalization of orthogonal drawings to grids that are formed by concentric circles and straight-line spokes emanating from the circles' center. Such drawings have applications in schematic graph layouts, e.g., for metro maps and destination maps. A plane graph is a planar graph with a fixed planar embedding. We give a combinatorial characterization of the plane graphs that admit a planar ortho-radial drawing without bends. Previously, such a characterization was only known for paths, cycles, and theta graphs, and in the special case of rectangular drawings for cubic graphs, where the contour of each face is required to be a rectangle. The characterization is expressed in terms of an ortho-radial representation that, similar to Tamassia's orthogonal representations for orthogonal drawings describes such a drawing combinatorially in terms of angles around vertices and bends on the edges. In this sense our characterization can be seen as a first step towards generalizing the Topology-Shape-Metrics framework of Tamassia to ortho-radial drawings.