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.
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Nehéz, M., Olejár, D., Demetrian, M. (2012). A Detailed Study of the Dominating Cliques Phase Transition in Random Graphs. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_55
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