Smooth Orthogonal Drawings of Planar Graphs

TL;DR

This paper investigates smooth orthogonal layouts of planar graphs, establishing bounds on edge complexity and area requirements for different classes of graphs, and demonstrating both possibilities and limitations of such drawings.

Contribution

It proves that all 4-planar graphs have SC_2-layouts, and 3-planar graphs have cubic area SC_2-layouts, advancing understanding of smooth orthogonal graph drawings.

Findings

Every 4-planar graph has an SC_2-layout.

3-planar graphs admit cubic area SC_2-layouts.

Some 4-planar graphs require exponential area for SC_1-layouts.

Abstract

In \emph{smooth orthogonal layouts} of planar graphs, every edge is an alternating sequence of axis-aligned segments and circular arcs with common axis-aligned tangents. In this paper, we study the problem of finding smooth orthogonal layouts of low \emph{edge complexity}, that is, with few segments per edge. We say that a graph has \emph{smooth complexity} k---for short, an SC_k-layout---if it admits a smooth orthogonal drawing of edge complexity at most $k$. Our main result is that every 4-planar graph has an SC_2-layout. While our drawings may have super-polynomial area, we show that, for 3-planar graphs, cubic area suffices. Further, we show that every biconnected 4-outerplane graph admits an SC_1-layout. On the negative side, we demonstrate an infinite family of biconnected 4-planar graphs that requires exponential area for an SC_1-layout. Finally, we present an infinite family of biconnected 4-planar graphs that does not admit an SC_1-layout.