Transforming spanning trees and pseudo-triangulations

Let TS be the set of all crossing-free straight line spanning trees of a planar n-point set S. Consider the graph TS where two members T and T′ of TS are adjacent if T intersects T′ only in points of S or in common edges. We prove that the diameter of TS is O(logk), where k denotes the number of convex layers of S. Based on this result, we show that the flip graph PS of pseudo-triangulations of S (where two pseudo-triangulations are adjacent if they differ in exactly one edge—either by replacement or by removal) has a diameter of O(nlogk). This sharpens a known O(nlogn) bound. Let be the induced subgraph of pointed pseudo-triangulations of PS. We present an example showing that the distance between two nodes in is strictly larger than the distance between the corresponding nodes in PS.