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Dominating sets of random recursive trees
Electronic Notes in Discrete Mathematics 27 (2006) 107–108 www.elsevier.com/locate/endm
Dominating sets of random recursive trees Michele Zito 1 Computer Science Liverpool Liverpool, United Kingdom
Colin Cooper Computer Science Kings College London, United Kingdom
Keywords: Random trees, dominating sets, algorithms
A random recursive tree on n vertices is either a single isolated vertex (for n = 1) or a vertex vn connected to a vertex chosen uniformly at random in a random recursive tree on n − 1 vertices. Such trees have been studied before (see [3]) as models of boolean circuits. More recently, modifications of such model [1], have been used to model for the web and other “power-law” networks. We prove that there exists a constant γ such that the size of a smallest dominating set in a random recursive tree on n vertices is γn + o(n) with probability approaching one as n tends to infinity. The result is obtained by analysing an algorithm proposed by Cockayne et al. [2]. 1
Email: [email protected]
1571-0653/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2006.08.078
108
M. Zito, C. Cooper / Electronic Notes in Discrete Mathematics 27 (2006) 107–108
References [1] A. Barab´ asi, and R. Albert. Emergence of scaling in random networks, Science, 286 (1999), 509–512. [2] E. Cockayne, S. Goodman, and S. Hedetniemi. A linear algorithm for the domination number of a tree, Information Processing Letters, 4 (1975), 41–44. [3] R. Smythe, and H. Mahmoud. A survey of recursive trees, Theory of Probability and Mathematical Statistics, 51 (1996), 1–29.
Dominating sets of random recursive trees Michele Zito 1 Computer Science Liverpool Liverpool, United Kingdom
Colin Cooper Computer Science Kings College London, United Kingdom
Keywords: Random trees, dominating sets, algorithms
A random recursive tree on n vertices is either a single isolated vertex (for n = 1) or a vertex vn connected to a vertex chosen uniformly at random in a random recursive tree on n − 1 vertices. Such trees have been studied before (see [3]) as models of boolean circuits. More recently, modifications of such model [1], have been used to model for the web and other “power-law” networks. We prove that there exists a constant γ such that the size of a smallest dominating set in a random recursive tree on n vertices is γn + o(n) with probability approaching one as n tends to infinity. The result is obtained by analysing an algorithm proposed by Cockayne et al. [2]. 1
Email: [email protected]
1571-0653/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2006.08.078
108
M. Zito, C. Cooper / Electronic Notes in Discrete Mathematics 27 (2006) 107–108
References [1] A. Barab´ asi, and R. Albert. Emergence of scaling in random networks, Science, 286 (1999), 509–512. [2] E. Cockayne, S. Goodman, and S. Hedetniemi. A linear algorithm for the domination number of a tree, Information Processing Letters, 4 (1975), 41–44. [3] R. Smythe, and H. Mahmoud. A survey of recursive trees, Theory of Probability and Mathematical Statistics, 51 (1996), 1–29.