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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References
Alon, N., Spencer, J.: The probabilistic method, 2nd edn. John Wiley & Sons, New York (2000)
Bollobás, B.: Random Graphs, 2nd edn. Cambridge Studies in Advanced Mathmatics, 73 (2001)
Bollobás, B., Erdös, P.: Cliques in random graphs. Math. Proc. Cam. Phil. Soc. 80, 419–427 (1976)
Bourgeois, N., Della Croce, F., Escoffier, B., Paschos, V.T.: Exact Algorithms for Dominating Clique Problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 4–13. Springer, Heidelberg (2009)
Culberson, J.C., Gao, Y., Anton, C.: Phase Transitions of Dominating Clique Problem and Their Implicátions to Heuristics in Satisfiability Search. In: Proc. 19th Int. Joint Conf. on Artificial Intelligence, IJCAI 2005, pp. 78–83, 2205–2222 (2005)
Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, New York (1979)
Gross, J.L., Yellen, J.: Handbook of Graph Theory. CRC Press (2003)
Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)
Janson, S., Luczak, T., Rucinski, A.: Random Graphs. John Wiley & Sons, New York (2000)
Kalbfleisch, J.G.: Complete subgraphs of random hypergraphs and bipartite graphs. In: Proc. 3rd Southeastern Conf. of Combinatorics, Graph Theory and Computing, pp. 297–304. Florida Atlantic University (1972)
Kratsch, D., Liedloff, M.: An exact algorithm for the minimum dominating clique problem. Theoretical Computer Science 385, 226–240 (2007)
Matula, D.W.: The largest clique size in a random graph, Technical report CS 7608, Dept. of Comp. Sci. Southern Methodist University, Dallas (1976)
Nehéz, M., Olejár, D.: An Improved Interval Routing Scheme for Almost All Networks Based on Dominating Cliques. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 524–532. Springer, Heidelberg (2005)
Nehéz, M., Olejár, D.: On Dominating Cliques in Random Graphs, Research Report, KAM-Dimatia Series 2005-750, Charles University, Prague (2005)
Nehéz, M., Olejár, D., Demetrian, M.: On Emergence of Dominating Cliques in Random Graphs In: LUMS 2nd Int. Conference on Mathematics and its Applications in Inform. Technology, Lahore, Pakistan, p. 59. Book of Abstracts (2008)
Olejár, D., Toman, E.: On the Order and the Number of Cliques in a Random Graph. Math. Slovaca 47(5), 499–510 (1997)
Palmer, E.M.: Graphical Evolution. John Wiley & Sons, Inc., New York (1985)
Ramras, D., Greenberg, S., Godbole, A.P.: Cliques and Independent Neighbor Sets in Random Graphs. Congressus Numerantium 153, 113–128 (2001)
Wieland, B., Godbole, A.P.: On the Domination Number of a Random Graph. Electronic Journal of Combinatorics 8(1), #R37(2001)
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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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