Summary: Transforming Spanning Trees and Pseudo­Triangulations
Oswin Aichholzer  Franz Aurenhammer y Clemens Huemer z Hannes Krasser x
Abstract
Let TS be the set of all crossing­free straight line span­
ning trees of a planar n­point set S. Consider the graph
TS where two members T and T 0 of TS are adjacent if T
intersects T 0 only in points of S or in common edges. We
prove that the diameter of T S is O(log k), where k denotes
the number of convex layers of S. Based on this result,
we show that the flip graph P S of pseudo­triangulations
of S (where two pseudo­triangulations are adjacent if they
differ in exactly one edge -- either by replacement or by
removal) has a diameter of O(n log k). This sharpens a
known O(n log n) bound. Let b
PS be the induced subgraph
of pointed pseudo­triangulations of P S . We present an ex­
ample showing that the distance between two nodes in b
PS
is strictly larger than the distance between the correspond­
ing nodes in PS .

Source: Aurenhammer, Franz - Institute for Theoretical Computer Science, Technische Universität Graz

Collections: Computer Technologies and Information Sciences