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

We study two variants of the well-known orthogonal graph drawing model: (1) the smooth orthogonal, and (2) the octilinear. Both models are extensions of the orthogonal one, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of maximum vertex degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {NP}$$\end{document}-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher vertex 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.