Planar Octilinear Drawings with One Bend Per Edge

TL;DR

This paper investigates octilinear drawings of planar graphs with minimal bends, proving that 4-planar and 5-planar graphs can be drawn with at most one bend per edge, while some 6-planar graphs require at least two bends.

Contribution

It establishes bounds on the number of bends needed for octilinear drawings of k-planar graphs, including constructions and limitations for 4-, 5-, and 6-planar graphs.

Findings

4-planar graphs admit 1-bend octilinear drawings on polynomial area.

5-planar graphs also admit 1-bend drawings, but on super-polynomial area.

Some 6-planar graphs require at least 2 bends per edge.

Abstract

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A $k$-planar graph is a planar graph in which each vertex has degree less or equal to $k$. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size $O(n^2) \times O(n)$. For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge.