Abstract
Let T S be the set of all crossing-free straight line spanning trees of a planar n-point set S. Consider the graph T S where two members T and T ′ of T S are adjacent if T intersects T ′ 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 pseudo-triangulations of S (where two pseudo-triangulations 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 P ˆ S be the induced subgraph of pointed pseudo-triangulations of P S . We present an example showing that the distance between two nodes in P ˆ S is strictly larger than the distance between the corresponding nodes in P S .
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