Transforming spanning trees: A lower bound

Abstract

For a planar point set we consider the graph whose vertices are the crossing-free straight-line spanning trees of the point set, and two such spanning trees are adjacent if their union is crossing-free. An upper bound on the diameter of this graph implies an upper bound on the diameter of the flip graph of pseudo-triangulations of the underlying point set. We prove a lower bound of Ω ( log n / log log n ) for the diameter of the transformation graph of spanning trees on a set of n points in the plane. This nearly matches the known upper bound of O ( log n ) . If we measure the diameter in terms of the number of convex layers k of the point set, our lower bound construction is tight, i.e., the diameter is in Ω ( log k ) which matches the known upper bound of O ( log k ) . So far only constant lower bounds were known.