DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 25(3) (2005) 427-433
DOI: 10.7151/dmgt.1294

ON A SPHERE OF INFLUENCE GRAPH IN A ONE-DIMENSIONAL SPACE

Zbigniew Palka and Monika Sperling

Department of Algorithmics and Programming
Adam Mickiewicz University
Umultowska 87, 61-614 Poznań, Poland
e-mail: palka@amu.edu.pl
e-mail: dwight@amu.edu.pl

Abstract

A sphere of influence graph generated by a finite population of generated points on the real line by a Poisson process is considered. We determine the expected number and variance of societies formed by population of n points in a one-dimensional space.

Keywords: cluster, sphere of influence graph.

2000 Mathematics Subject Classification: Primary 60D05;
Secondary 60C05, 05C80.

References

[1] P. Avis and J. Horton, Remarks on the sphere of influence graph, in: ed. J.E. Goodman, et al. Discrete Geometry and Convexity (New York Academy of Science, New York) 323-327.
[2] T. Chalker, A. Godbole, P.  Hitczenko, J. Radcliff and O. Ruehr, On the size of a random sphere of influence graph, Adv. in Appl. Probab. 31 (1999) 596-609, doi: 10.1239/aap/1029955193.
[3] E.G. Enns, P.F. Ehlers and T. Misi, A cluster problem as defined by nearest neighbours, The Canadian Journal of Statistics 27 (1999) 843-851, doi: 10.2307/3316135.
[4] Z. Furedi, The expected size of a random sphere of influence graph, Intuitive Geometry, Bolyai Math. Soc. 6 (1995) 319-326.
[5] Z. Furedi and P.A. Loeb, On the best constant on the Besicovitch covering theorem, in: Proc. Coll. Math. Soc. J. Bolyai 63 (1994) 1063-1073.
[6] P. Hitczenko, S. Janson and J.E. Yukich, On the variance of the random sphere of influence graph, Random Struct. Alg. 14 (1999) 139-152, doi: 10.1002/(SICI)1098-2418(199903)14:2<139::AID-RSA2>3.0.CO;2-E.
[7] L. Guibas, J. Pach and M.  Sharir, Sphere of influence graphs in higher dimensions, in: Proc. Coll. Math. Soc. J. Bolyai 63 (1994) 131-137.
[8] T.S. Michael and T. Quint, Sphere of influence graphs: a survey, Congr. Numer. 105 (1994) 153-160.
[9] T.S. Michael and T. Quint, Sphere of influence graphs and the L-metric, Discrete Appl. Math. 127 (2003) 447-460, doi: 10.1016/S0166-218X(02)00246-9.
[10]  Toussaint, Pattern recognition of geometric complexity, in: Proceedings of the 5th Int. Conference on Pattern Recognition, (1980) 1324-1347.
[11] D. Warren and E. Seneta, Peaks and eulerian numbers in a random sequence, J. Appl. Prob. 33 (1996) 101-114, doi: 10.2307/3215267.

Received 9 September 2004
Revised 4 May 2005