The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.

Abstract. We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 n O(n −6) straight-edge triangulations of its convex hull. This improves the previous upper bound of 276.75 n+O(log(n)).

We show that a set of n points in the plane has at most O(10.05n) perfect matchings with crossing-free straight-line embedding. The expected number of perfect crossing-free matchings of a set of n points drawn i.i.d. from an arbitrary distribution in the plane is at most

We show that the number of straight-edge triangulations exhibited by any set of n points in general position in the plane is bounded from below by 4 :33 ).

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de

Abstract. We give a brief account of results concerning the number of triangulations on finite point sets in the plane, both for arbitrary sets and for specific sets such as the n×n integer lattice. Given a finite point set P in the plane, a geometric graph is a straight line embedded graph with vertex set P where no segment realizing an edge contains points from P other than its endpoints. We are interested in crossing-free geometric graphs on a given planar point set, i.e. segments are not allowed to share points other than common endpoints. A maximal crossing-free geometric graph on a point set P is called a triangulation of P. Fig.1. All triangulations of five points in convex position. If, as it is the case in Fig. 1, the points are in convex position, i.e. vertices of a convex polygon, then every triangulation clearly must contain all edges of “its” convex polygon, and we are left with choosing a triangulation for this polygon. Euler was the first to consider how many choices there are for a convex n-gon, � but it was proven only later that this number is Cn−2, where Cm: = 1

We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30 n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of 43^n for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, we derive new upper bounds for the number of planar graphs (O ∗ (239.4 n)), spanning cycles (O ∗ (70.21 n)), spanning trees (160 n), and cycle-free graphs (O ∗ (202.5 n)).

Graph augmentation problems are motivated by network design, and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems.

We study the expected number of interior vertices of degree i in a triangulation of a point set S, drawn uniformly at random from the set of all triangulations of S, and derive various bounds and inequalities for these expected values. One of our main results is: For any set S of N points in general position, and for any fixed i, the expected number of vertices of degree i in a random triangulation is at least γiN, for some fixed positive constant γi (assuming that N> i and that at least some fixed fraction of the points are interior). We also present a new application for these expected values, using upper bounds on the expected number of interior vertices of degree 3 to get a new lower bound, Ω(2.4317 N), for the minimal number of triangulations any N-element planar point set in general position must have. This improves the previously best known lower bound of Ω(2.33 N).