Graham triangulations and triangulations with a center are hamiltonean

Abstract

Let P be a point set with n elements in general position. A triangulation T of P is a set of triangles with disjoint interiors such that their union is the convex hull of P, no triangle contains an element of P in its interior, and the vertices of the triangles of T are points of P. Given T we define a graph G(T) whose vertices are the triangles of T, two of which are adjacent if they share an edge. We say that T is hamiltonean if G(T) has a hamiltonean path. We prove that the triangulations produced by Graham's Scan are hamiltonean. Furthermore we prove that any triangulation T of a point set which has a point adjacent to all the points in P (a center of T) is hamiltonean. (C) 2004 Elsevier B.V. All rights reserved.