Triangulations without pointed spanning trees

Problem 50 in the Open Problems Project asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph. We provide a counterexample. As a consequence we show that there exist triangulations which require a linear number of edge flips to become Hamiltonian.Acciones Integradas 2003-2004Austrian Fonds zur Förderung der Wissenschaftlichen Forschun

Towards compatible triangulations

AbstractWe state the following conjecture: any two planar n-point sets that agree on the number of convex hull points can be triangulated in a compatible manner, i.e., such that the resulting two triangulations are topologically equivalent. We first describe a class of point sets which can be triangulated compatibly with any other set (that satisfies the obvious size and shape restrictions). The conjecture is then proved true for point sets with at most three interior points. Finally, we demonstrate that adding a small number of extraneous points (the number of interior points minus three) always allows for compatible triangulations. The linear bound extends to point sets of arbitrary size and shape

Convexity Minimizes Pseudo-Triangulations

AbstractThe number of minimum pseudo-triangulations is minimized for point sets in convex position

Robot Kinematics INDUSTRIAL Classical Geometry Computer Vision GEOMETRY Computer Aided Geometric Design Image Processing Abstract Transforming Spanning Trees and Pseudo-Triangulations ∗

Let TS be the set of all crossing-free straight line spanning trees of a planar n-point set S. Consider the graph TS where two members T and T ′ of TS are adjacent if T intersects T ′ only in points of S or in common edges. We prove that the diameter of TS is O(log k), where k denotes the number of convex layers of S. Based on this result, we show that the flip graph PS 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 � PS be the induced subgraph of pointed pseudo-triangulations of PS. We present an example showing that the distance between two nodes in � PS is strictly larger than the distance between the corresponding nodes in PS

Convexity Minimizes Pseudo-Triangulations

The number of minimum pseudo-triangulations is minimized for point sets in convex position

On the number of plane graphs

We investigate the number of plane geometric, i.e., straight-line, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extremal configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ ∗ ( √ 72 n) = Θ ∗ (8.4853 n) triangulations and Θ ∗ (41.1889 n) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples