On the Planar Edge-Length Ratio of Planar Graphs

TL;DR

This paper investigates the minimum possible ratio between the longest and shortest edges in planar graph drawings, establishing bounds and properties for various classes of planar graphs.

Contribution

It proves that some planar graphs have an edge-length ratio linear in the number of vertices and provides upper bounds for specific graph families.

Findings

Existence of n-vertex planar graphs with edge-length ratio in Ω(n)

Tight bound for the edge-length ratio in planar graphs

Upper bounds for series-parallel and bipartite planar graphs

Abstract

The edge-length ratio of a straight-line drawing of a graph is the ratio between the lengths of the longest and of the shortest edge in the drawing. The planar edge-length ratio of a planar graph is the minimum edge-length ratio of any planar straight-line drawing of the graph. In this paper, we study the planar edge-length ratio of planar graphs. We prove that there exist $n$-vertex planar graphs whose planar edge-length ratio is in $\Omega(n)$; this bound is tight. We also prove upper bounds on the planar edge-length ratio of several families of planar graphs, including series-parallel graphs and bipartite planar graphs.