Geometric tree graphs of points in convex position

Abstract

Given a set P of points in the plane, the geometric tree graph of P is defined as the graph T(P) whose vertices are non-crossing spanning with straight edges trees of P, and where two trees T1 and T2 are adjacent if T2 = T1 − e + f for some edges e and f. In this paper we concentrate on the geometric tree graph of a set of n points in convex position, denoted by Gn. We prove several results about Gn, among them the existence of Hamiltonian cycles and the fact that they have maximum connectivity.