Counting Plane Graphs: Flippability and Its Applications

Abstract

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S.We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let tr(N ) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N ) < 30 N ) that any N -element point set admits at most 6. forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN , fewer than cN , or more than cN edges, for any constant parameter c, in terms of c and N .

Counting Plane Graphs: Cross-Graph Charging Schemes

“…Razen, Snoeyink and Welzl [19] were the first to address this inefficiency, deriving the slightly improved inequality pg(N) = O(7.9792 N ) • tr(N). A more significant improvement of pg(N) < 6.9283 N • tr(N) was recently obtained by Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14]. This implies the bound pg(N) < 207.85 N .…”

“…The first exponential bound, 10 13N , on the number of such graphs was proved by Ajtai, Chvátal, Newborn and Szemerédi [4] back in 1982. Since then, progressively (and significantly) smaller upper bounds have been derived (see, e.g., [14,19,23]). Upper bounds on numbers of more specific types of crossing-free straight-edge graphs, such as Hamiltonian cycles, spanning trees, perfect matchings, and triangulations, were also studied (see, e.g., [7,6,20,21,24]).…”

“…More precisely, an initial, more direct application of this method implies only a bound of 3207.42 N . On the other hand, by combining this method with the current bound on the number of triangulations (indirectly, by using an upper bound on the maximum number of plane graphs with at least cN edges, which is derived in [14] and relies on tr(N)), we obtain pg(N) = O * (187.53 N ).…”

“…at most cN edges) that can be embedded in S, for some parameter 0 < c < 3 (by Euler's formula, c = 3 is the largest meaningful parameter). To obtain our improved bound for pg(N), we rely on the following theorem, established by Hoffmann, Schulz, Sharir, Sheffer, Tóth and Welzl [14].…”

Disjoint Compatible Geometric Matchings

“…The union of two disjoint compatible perfect geometric matchings is a collection of disjoint polygons in the plane. This simple fact is crucial for the best current upper bound on the number of simple polygons on a fixed set of n points in the plane [6,12,19]. The maximum number of perfect geometric matchings on a set of n points in the plane is known to be between Ω(3 n ) and O(10.05 n ), as shown by García et al [10] and Sharir and Welzl [18], respectively.…”