Hamiltonian Triangulations and Circumscribing Polygons

Let Σ = { S1 , . . . , Sn } be a finite set of disjoint line segments in the plane. We conjecture that its visibility graph, Vis(Σ), is hamiltonian. In fact, we make the stronger conjecture that Vis(Σ) has a hamiltonian cycle whose embedded version is a simple polygon (i.e., its boundary edges are non-crossing visibility segments). We call such a simple polygon a spanning polygon of Σ. Existence of a spanning polygon of Σ is equivalent to the existence of a hamiltonian triangulation of Σ. A spanning polygon P is said to be a circumscribing polygon of Σ, if it has the additional property that no segment in Σ lies in the exterior of P. We prove circumscribing polygons exist for the special case when Σ is extremally situated, i.e., when each segment Si touches the convex hull boundary of Σ. Furthermore, for this special case we give an algorithm that constructs a circumscribing polygon in O(n log n) time and this is optimal.