On the total $(k, r)$-domination number of random graphs

A subset S of a vertex set of a graphG is a total (k, r)-dominating set if every vertex u ∈ V (G) is within distance k of at least r vertices in S. The minimum cardinality among all total (k, r)-dominating sets ofG is called the total (k, r)domination number of G, denoted by γ (k,r)(G). We previously gave an upper bound on γ t (2,r)(G(n, p)) in random graphs with non-fixed p ∈ (0, 1). In this paper we generalize this result to give an upper bound on γ (k,r)(G(n, p)) in random graphs with non-fixed p ∈ (0, 1) for k ≥ 3 as well as present an upper bound on γ (k,r)(G) in graphs with large girth.