An analysis of the size of the minimum dominating sets in random recursive trees, using the Cockayne-Goodman-Hedetniemi algorithm

  • Authors:

  • Colin Cooper;Michele Zito

  • Affiliations:

  • Department of Computer Science, King's College, London WC2R 2LS, UK;Department of Computer Science, University of Liverpool, Ashton Street, Liverpool L69 3BX, UK

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

A random recursive tree on n vertices is either a single isolated vertex (for n=1) or is a vertex v"n connected to a vertex chosen uniformly at random from a random recursive tree on n-1 vertices. Such trees have been studied before [R. Smythe, H. Mahmoud, A survey of recursive trees, Theory of Probability and Mathematical Statistics 51 (1996) 1-29] as models of boolean circuits. More recently, Barabasi and Albert [A. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509-512] have used modifications of such models to model for the web and other ''power-law'' networks. A minimum (cardinality) dominating set in a tree can be found in linear time using the algorithm of Cockayne et al. [E. Cockayne, S. Goodman, S. Hedetniemi, A linear algorithm for the domination number of a tree, Information Processing Letters 4 (1975) 41-44]. We prove that there exists a constant d~0.3745... such that the size of a minimum dominating set in a random recursive tree on n vertices is dn+o(n) with probability approaching one as n tends to infinity. The result is obtained by analysing the algorithm of Cockayne, Goodman and Hedetniemi.