On -tuple domination of random graphs

In graph G = ( V , E ) , a vertex set D ⊆ V is called a domination set if any vertex u ∈ V ∖ D is connected to at least one vertex in D . Generally, for any natural number k , the k -tuple domination set D is a vertex set such that any vertex u ∈ V ∖ D is connected to at least k vertices in D . The k -tuple domination number is the minimum size of k -tuple domination sets. It is known that the 1-tuple domination number (i.e. domination number) of classical random graphs G ( n , p ) with fixed p ∈ ( 0 , 1 ) asymptotically almost surely ( a . a . s . ) has a two-point concentration [B. Wieland, A.P. Godbole, On the domination number of a random graph, Electron. J. Combin. 8 (2001) R37]. In this work, we prove that the 2-tuple domination number of G ( n , p ) with fixed p ∈ ( 0 , 1 ) a . a . s . has a two-point concentration.