TY - GEN
T1 - Counting plane graphs
T2 - 28th Annual Symposuim on Computational Geometry, SCG 2012
AU - Sharir, Micha
AU - Sheffer, Adam
AU - Welzl, Emo
N1 - Funding Information:
✩ Work on this paper by the first two authors was partially supported by Grant 338/09 from the Israel Science Fund and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Micha Sharir was also supported by NSF Grant CCF-08-30272, by Grant 2006/194 from the US–Israel Binational Science Foundation, and by the Hermann Minkowski– MINERVA Center for Geometry at Tel Aviv University. Emo Welzl acknowledges support from the EuroCores/EuroGiga/ComPoSe SNF grant 20GG21_134318/1. Part of the work on this paper was done at the Centre Interfacultaire Bernoulli (CIB), EPFL, Lausanne, during the Special Semester on Discrete and Computational Geometry, Fall 2010, and supported by the Swiss National Science Foundation. A preliminary version of this paper has appeared in Proc. 28th ACM Symp. on Computational Geometry (2012). * Fax: +972 36409357. E-mail addresses: [email protected] (M. Sharir), [email protected] (A. Sheffer), [email protected] (E. Welzl).
PY - 2012
Y1 - 2012
N2 - 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(1.8181 N) for cycles and O(1.1067 N) for matchings. These imply a new upper bound of O(54.543 N) 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(68.664 N)). Our analysis is based on a weighted variant of Kasteleyn's linear algebra technique.
AB - 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(1.8181 N) for cycles and O(1.1067 N) for matchings. These imply a new upper bound of O(54.543 N) 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(68.664 N)). Our analysis is based on a weighted variant of Kasteleyn's linear algebra technique.
KW - Kasteleyn's Technique
KW - Perfect matchings
KW - Spanning cycles
KW - Triangulations
UR - http://www.scopus.com/inward/record.url?scp=84863949808&partnerID=8YFLogxK
U2 - 10.1145/2261250.2261277
DO - 10.1145/2261250.2261277
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AN - SCOPUS:84863949808
SN - 9781450312998
T3 - Proceedings of the Annual Symposium on Computational Geometry
SP - 189
EP - 198
BT - Proceedings of the 28th Annual Symposuim on Computational Geometry, SCG 2012
Y2 - 17 June 2012 through 20 June 2012
ER -