Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Given a planar straight-line graph G=(V,E) in ℝ^2 , a 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 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 Ω (√(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 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.