# Circumscribing Polygons and Polygonizations for Disjoint Line Segments

## TL;DR

This paper investigates the existence and computational complexity of circumscribing polygons for disjoint line segments, providing new bounds, exploring geometric relations, and proving NP-completeness results.

## Contribution

It proves a new lower bound on the subset size for circumscribing polygons, explores their relation to other geometric problems, and establishes NP-completeness for related decision problems.

## Key findings

- Every arrangement of n disjoint segments has an $oldsymbol{ ext{ extonehalf} ext{-} ext{ extonequarter} ext{-} ext{ extonehalf}}$ subset with a circumscribing polygon.
- Deciding if a graph admits a circumscribing polygon is NP-complete, even for 2-regular graphs.
- Determining if a geometric matching admits a polygonization is NP-complete, settling a 30-year-old conjecture.

## Abstract

Given a planar straight-line graph $G=(V,E)$ in $\mathbb{R}^2$, a \emph{circumscribing polygon} of $G$ is a simple polygon $P$ whose vertex set is $V$, and every edge in $E$ is either an edge or an internal diagonal of $P$. A circumscribing polygon is a \emph{polygonization} for $G$ if every edge in $E$ is an edge of $P$. We prove that every arrangement of $n$ disjoint line segments in the plane has a subset of size $\Omega(\sqrt{n})$ that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to $\mathbb{R}^3$. We show that it is NP-complete to decide whether a given graph $G$ admits a circumscribing polygon, even if $G$ is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.