# Near-optimal dominating sets in dense random graphs in polynomial expected time

Conference paper

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## Abstract

The existence and efficient finding of small dominating sets in dense random graphs is examined in this work. We show, for the model

*G*_{ n,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

*G*_{ n, 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 G_{n,p} with *p* fixed to any constant < 1.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1994