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The random division of faces in a planar graph

Part of: Designs and configurations Distribution theory Low-dimensional topology Real and complex geometry Markov processes

Published online by Cambridge University Press:  01 July 2016

Richard Cowan  and
Simone Chen
Richard Cowan*
Affiliation:
University of Hong Kong
Simone Chen*
Affiliation:
University of Hong Kong
*
∗ Postal address: Department of Statistics, The University of Hong Kong, Pokfulam Road, Hong Kong.
∗ Postal address: Department of Statistics, The University of Hong Kong, Pokfulam Road, Hong Kong.
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Abstract

A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

Keywords

STOCHASTIC GEOMETRYPLANAR GRAPH

MSC classification

Primary: 62E20: Asymptotic distribution theory
Secondary: 51M20: Polyhedra and polytopes; regular figures, division of spaces 57M15: Relations with graph theory 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05B45: Tessellation and tiling problems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 377 - 383
Copyright
Copyright © Applied Probability Trust 1996 

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References

Cowan, R. (1989) The division of space and the Poisson distribution. Adv. Appl. Prob. 21, 233–234.CrossRefGoogle Scholar
Cowan, R. and Chen, S. (1996) The random topology of polygon splitting. In preparation.Google Scholar
Cowan, R. and Morris, V. B. (1988) Division rules for polygonal cells. J. Theor. Biol. 131, 33–42.Google Scholar
Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar