# Drawing outerplanar graphs using thirteen edge lengths

## TL;DR

This paper proves that all outerplanar graphs can be embedded in the plane with at most thirteen distinct edge lengths, avoiding overlaps and intersections, thus solving a previously open geometric graph drawing problem.

## Contribution

It extends prior work by allowing no vertex-edge overlaps and establishes a universal embedding method for outerplanar graphs with limited edge length diversity.

## Key findings

- Every outerplanar graph can be embedded with at most thirteen distinct edge lengths.
- The embedding avoids vertex-edge overlaps and edge intersections.
- It resolves a problem posed by Carmi, Dujmović, Morin, and Wood.

## Abstract

We show that every outerplanar graph $G$ can be linearly embedded in the plane such that the number of distinct distances between pairs of adjacent vertices is at most thirteen and there is no intersection between the image of a vertex and that of an edge not containing it. This extends the work of Alon and the second author, where only overlap between vertices was disallowed, thus settling a problem posed by Carmi, Dujmovi\'{c}, Morin and Wood.