On the Total Number of Bends for Planar Octilinear Drawings

TL;DR

This paper investigates the bounds on the total number of bends in planar octilinear graph drawings, providing improved upper bounds for certain classes and establishing lower bounds using flow techniques.

Contribution

It improves existing upper bounds on the number of bends for specific classes of planar octilinear graphs and introduces lower bounds via flow-based methods.

Findings

Upper bound of 4n-10 bends for 8-planar graphs.

Significant improvement of bounds for triconnected 4-, 5-, and 6-planar graphs.

Established lower bounds for these classes using flow techniques.

Abstract

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at 45 degrees line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. As the problem of finding planar octilinear drawings of minimum number of bends is NP-hard, in this paper we focus on upper and lower bounds. From a recent result of Keszegh et al. on the slope number of planar graphs, we can derive an upper bound of 4n-10 bends for 8-planar graphs with n vertices. We considerably improve this general bound and corresponding previous ones for triconnected 4-, 5- and 6-planar graphs. We also derive non-trivial lower bounds for these three classes of graphs by a technique inspired by the network flow formulation of Tamassia.