Improved upper bounds on the reflexivity of point sets

Given a set S of n points in the plane, the reflexivity of S, ρ(S), is the minimum number of reflex vertices in a simple polygonalization of S. Arkin et al. [4] proved that ρ(S) ≤ ⌈n/2 ⌉ for any set S, and conjectured that the tight upper bound is ⌊n/4⌋. We show that the reflexivity of any set of n points is at most 3 7n + O(1) ≈ 0.4286n. Using computer-aided abstract order type extension the upper bound can be further improved to 5 12n + O(1) ≈ 0.4167n. We also present an algorithm to compute polygonalizations with at most this number of reflex vertices in O(nlog n) time

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

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

Reprint of: Theta-3 is connected

In this paper, we show that the θ-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones

Playing with Triangulations

We analyze several perfect-information combinatorial games played on planar triangulations. We introduce three broad categories of such games: constructing, transforming, and marking triangulations. In various situations, we develop polynomial-time algorithms to determine who wins a given game under optimal play, and to find a winning strategy. Along the way, we show connections to existing combinatorial games such as Kayles

Games on triangulations

We analyze several perfect-information combinatorial games played on planar triangulations. We introduce three broad categories of such games - constructing, transforming, and marking triangulations - and several specific games within each category. In various situations of each game, we develop polynomial-time algorithms to determine who wins a given game position under optimal play, and to find a winning strategy. Along the way, we show connections to existing combinatorial games such as Kayles and Nimstring (a variation on Dots-and-Boxes)