TY - JOUR

T1 - Transforming spanning trees: A lower bound

AU - Buchin, K.

AU - Razen, A.

AU - Uno, T.

AU - Wagner, U.

PY - 2009

Y1 - 2009

N2 - 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 O(logn/loglogn) 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(logn). 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 O(logk) which matches the known upper bound of O(logk). So far only constant lower bounds were known.

AB - 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 O(logn/loglogn) 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(logn). 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 O(logk) which matches the known upper bound of O(logk). So far only constant lower bounds were known.

U2 - 10.1016/j.comgeo.2008.03.005

DO - 10.1016/j.comgeo.2008.03.005

M3 - Article

SN - 0925-7721

VL - 42

SP - 724

EP - 730

JO - Computational Geometry

JF - Computational Geometry

IS - 8

ER -