Near-Optimal dominating sets in dense random graphs in polynomial expected time

Abstract

The existence and efficient finding of small dominating sets in dense random graphs is examined in this work. We show, for the model Gn,p with p=1/2, that: 1. The probability of existence of dominating sets of size less than log n tends to zero as n tends to infinity. 2. Dominating sets of size [log n] exist almost surely. 3. We provide two algorithms which construct small dominating sets in Gn,1/2 run in O (n alog n) time (on the average and also with high probability). Our algorithms almost surely construct a dominating set of size at most (1+ε) log n, for any fixed ε > 0. Our results extend to the case Gn,p with p fixed to any constant < 1.