  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.
