On kk-tuple domination of random graphs

In graph G=(V,E)G=(V,E), a vertex set D⊆VD⊆V is called a domination set if any vertex u∈V∖Du∈V∖D is connected to at least one vertex in DD. Generally, for any natural number kk, the kk-tuple domination set DD is a vertex set such that any vertex u∈V∖Du∈V∖D is connected to at least kk vertices in DD. The kk-tuple domination number is the minimum size of kk-tuple domination sets. It is known that the 1-tuple domination number (i.e. domination number) of classical random graphs G(n,p)G(n,p) with fixed p∈(0,1)p∈(0,1) asymptotically almost surely (a.a.s.)(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)G(n,p) with fixed p∈(0,1)a.a.s. has a two-point concentration.