The Minimum Bends in a Polyline Drawing with Fixed Vertex Locations

TL;DR

This paper proves that for almost all planar graphs, a polyline drawing with fixed vertex locations requires at least on the order of n^2 bends, establishing the optimality of previous upper bounds.

Contribution

It demonstrates that the known upper bound of O(n^2) bends for fixed vertex locations is tight by proving an Omega(n^2) lower bound for almost all planar graphs.

Findings

Omega(n^2) bends are necessary for almost all planar graphs.

The result generalizes previous bounds limited to convex point sets.

Settles two open problems in graph drawing theory.

Abstract

We consider embeddings of planar graphs in $R^2$ where vertices map to points and edges map to polylines. We refer to such an embedding as a polyline drawing, and ask how few bends are required to form such a drawing for an arbitrary planar graph. It has long been known that even when the vertex locations are completely fixed, a planar graph admits a polyline drawing where edges bend a total of $O(n^2)$ times. Our results show that this number of bends is optimal. In particular, we show that $\Omega(n^2)$ total bends is required to form a polyline drawing on any set of fixed vertex locations for almost all planar graphs. This result generalizes all previously known lower bounds, which only applied to convex point sets, and settles 2 open problems.