Computational geometry in C (2nd ed.)
Computational geometry in C (2nd ed.)
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
On translating a set of rectangles
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
On the Number of Crossing-Free Matchings, Cycles, and Partitions
SIAM Journal on Computing
On the Number of Plane Geometric Graphs
Graphs and Combinatorics
Fast enumeration algorithms for non-crossing geometric graphs
Proceedings of the twenty-fourth annual symposium on Computational geometry
On degrees in random triangulations of point sets
Journal of Combinatorial Theory Series A
Counting plane graphs with exponential speed-up
Rainbow of computer science
Counting plane graphs: flippability and its applications
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
On the number of cycles in planar graphs
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Computational geometry column 54
ACM SIGACT News
A simple aggregative algorithm for counting triangulations of planar point sets and related problems
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of N points in the plane. More specifically, we bound the ratio between the number of spanning cycles (or perfect matchings) that can be embedded over a point set and the number of triangulations that can be embedded over it. The respective bounds are O(1.8181N) for cycles and O(1.1067N) for matchings. These imply a new upper bound of O(54.543N) on the number of crossing-free straight-edge spanning cycles that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound O(68.664N)). Our analysis is based on a weighted variant of Kasteleyn's linear algebra technique.