One-Bend Drawings of Outerplanar Graphs Inside Simple Polygons

TL;DR

This paper investigates the problem of drawing outerplanar graphs inside simple polygons with at most one bend per edge, providing an algorithm to decide existence and construct such drawings efficiently.

Contribution

It introduces an $O(nm)$ time algorithm to determine the existence of and construct one-bend drawings of outerplanar graphs inside simple polygons.

Findings

Decidable in $O(nm)$ time whether such a drawing exists.

Provides a method to compute the drawing when it exists.

Applicable to graphs with up to $n-3$ interior edges.

Abstract

We consider the problem of drawing an outerplanar graph with $n$ vertices with at most one bend per edge if the outer face is already drawn as a simple polygon. We prove that it can be decided in $O(nm)$ time if such a drawing exists, where $m\le n-3$ is the number of interior edges. In the positive case, we can also compute such a drawing.