Lower bounds on the number of crossing-free subgraphs of KN
Computational Geometry: Theory and Applications
A better upper bound on the number of triangulations of a planar point set
Journal of Combinatorial Theory Series A
Matching Theory (North-Holland mathematics studies)
Matching Theory (North-Holland mathematics studies)
On the Number of Plane Geometric Graphs
Graphs and Combinatorics
Computational Geometry: Theory and Applications
On the number of spanning trees a planar graph can have
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Triangulations and Applications
Triangulations and Applications
Triangulations: Structures for Algorithms and Applications
Triangulations: Structures for Algorithms and Applications
On the number of cycles in planar graphs
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn's technique
Proceedings of the twenty-eighth annual symposium on Computational geometry
Computational geometry column 54
ACM SIGACT News
Counting plane graphs: cross-graph charging schemes
GD'12 Proceedings of the 20th international conference on Graph Drawing
A simple aggregative algorithm for counting triangulations of planar point sets and related problems
Proceedings of the twenty-ninth annual symposium on Computational geometry
| Hi-index | 0.00 |
We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane into so called pseudo-simultaneously flippable edges. We prove a worst-case tight lower bound for the number of pseudosimultaneously 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) N) that any N-element point set admits at most 6.9283N ċ tr(N) N crossing-free straight-edge graphs, O(4.8795N) ċ tr(N) = O(146.39N) spanning trees, and O(5.4723N) ċ tr(N) = O(164.17N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have fewer than cN or more than cN edges, for a constant parameter c, in terms of c and N.