On k-connected restrained domination in graphs

Let G = (V, E) be a graph. A k-connected restrained dominating set is a set S ⊆ V , where S is a restrained dominating set and G[S] has at most k components. The k-connected restrained domination number of G, denoted by γ k r (G), is the smallest cardi-nality of a k-connected restrained dominating set of G. In this talk, I will give some exact values and sharp bounds for γ k r (G). Then the necessary and sufficient conditions for γ r (G) = γ 1 r (G) = γ 2 r (G) are given if G is a tree or a unicyclic graph. Finally, I will show that if T is a tree of order n, then γ k r (T) ≥ max{⌈ n+2 3 ⌉, n − 2(k − 1)}. Moreover, I will constructively characterize the extremal trees T of order n achieving this lower bound.