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

A random recursive tree on n vertices is either a single isolated vertex (for n = 1) or is a vertex vn connected to a vertex chosen uniformly at random from a random recursive tree on n− 1 vertices. Such trees have been studied before (see [11]) as models of boolean circuits. More recently, modifications of such models [2], have been used to model for the web and other “power-law” networks. A smallest dominating set in a tree can be found in linear time using the algorithm of Cockayne, Goodman and Hedetniemi [4]. We prove that there exists a constant d ' 0.3745... such that the size of a smallest 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. ∗This research was supported by EPSRC grant EP/DO59372/1. The first author was also supported in this work by a LaBRI ENSIERB visiting research fellowship.