A detailed study of the dominating cliques phase transition in random graphs

Authors:
Martin Nehéz;Daniel Olejár;Michal Demetrian
Affiliations:
Gratex International, a.s., Bratislava, Slovak Republic;Department of Computer Science, FMPI, Comenius University, Bratislava, Slovak Republic;Department of Mathematical and Numerical Analysis, FMPI, Comenius University, Bratislava, Slovak Republic
Venue:
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Year:
2012

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Abstract

A subset of nodes S ⊆V of a graph G =(V , E ) is a dominating clique if S is a dominating set and a clique of G . The phase transition of dominating cliques in Erdös-Rényi random graph model is investigated in this paper. Lower and upper bounds on the edge probability p for the existence of an r -node dominating clique are established in this paper. We prove therein that given an n -node random graph G from for r =c log1/p n with 1≤c ≤2 it holds: (1) if p 1/2 then an r -clique is dominating in G with a high probability and, (2) if $p \leq ( 3 - \sqrt{5})/2$ then an r -clique is not dominating in G with a high probability. The remaining range of the probability p is discussed with more attention. Within such a range, we provide intervals of r where a dominating clique existence probability is zero, positive but less than one, and one, respectively.