On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

TL;DR

This paper explores two extended orthogonal drawing models, smooth orthogonal and octilinear, analyzing their relationships, computational complexity, and providing algorithms for specific graph classes and constraints.

Contribution

It introduces the study of smooth orthogonal and octilinear drawings, establishing their relationships, proving NP-hardness for bendless variants, and presenting algorithms for higher-degree planar graphs.

Findings

NP-hardness of bendless drawing problems in both models

Relationships between classes of graphs drawable bendless in each model

Algorithm for bi-monotone smooth orthogonal drawings with limited segments

Abstract

We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.