Drawing outerplanar graphs

TL;DR

This paper proves that every outerplanar graph can be embedded in the plane with at most three distinct edge lengths, resolving a previously open problem.

Contribution

It introduces a novel geometric embedding method for outerplanar graphs that limits the number of distinct edge lengths to three.

Findings

Every outerplanar graph can be embedded with at most three distinct edge lengths.

The proof combines geometric, combinatorial, algebraic, and probabilistic techniques.

It settles a longstanding open problem in graph drawing.

Abstract

It is shown that for any outerplanar graph G there is a one to one mapping of the vertices of G to the plane, so that the number of distinct distances between pairs of connected vertices is at most three. This settles a problem of Carmi, Dujmovic, Morin and Wood. The proof combines (elementary) geometric, combinatorial, algebraic and probabilistic arguments.