Sequences of spanning trees and a fixed tree theorem

Abstract

Let T S be the set of all crossing-free spanning trees of a planar n-point set S. We prove that T S contains, for each of its members T, a length-decreasing sequence of trees T 0,…, T k such that T 0= T, T k =MST( S), T i does not cross T i−1 for i=1,…, k, and k=O(log n). Here MST( S) denotes the Euclidean minimum spanning tree of the point set S. As an implication, the number of length-improving and planar edge moves needed to transform a tree T∈ T S into MST( S) is only O( nlog n). Moreover, it is possible to transform any two trees in T S into each other by means of a local and constant-size edge slide operation. Applications of these results to morphing of simple polygons are possible by using a crossing-free spanning tree as a skeleton description of a polygon.