A quadratic distance bound on sliding between crossing-free spanning trees

A quadratic distance bound on sliding between crossing-free spanning trees

Let S be a set of n points in the plane and let TS be the set of all crossing-free spanning trees of S. We show that any two trees in TS can be transformed into each other by O(n ) local and constant-size edge slide operations. No polynomial upper bound for this task has been known, but in [1] a bound of O(n log n) operations was conjectured.

On Spanning Trees with Low Crossing Numbers

Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O( √ n) intersections with the tree (where the edges are embedded as straight line segments). We review the proof of this result (originally proved by Bernard Chazelle and the author in a more general setting), point at some methods for constructing such a tree, and describe some algorithmic...

Edge Operations on Non-Crossing Spanning Trees

Crossing-Free Spanning Trees in Visibility Graphs of Points between Monotone Polygonal Obstacles

We consider the problem of deciding whether or not a geometric graph has a crossing-free spanning tree. This problem is known to be NP-hard even for very restricted types of geometric graphs. In this paper, we present an O(n) time algorithm to solve this problem for the special case of geometric graphs that arise as visibility graphs of a finite set of n points between two monotone polygonal ob...

Transforming spanning trees: A lower bound

For a planar point set we consider the graph of crossing-free straight-line spanning trees where two spanning trees are adjacent in the graph 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 diamet...