Graph domination in distance two

Let G = (V,E) be a graph, and k ≥ 1 an integer. A subgraph D is said to be k-dominating in G if every vertex of G-D is at distance at most k from some vertex of D. For a given class of graphs, Domₖ is the set of those graphs G in which every connected induced subgraph H has some k-dominating induced subgraph D ∈ which is also connected. In our notation, Dom coincides with Dom₁. In this paper we prove that ${\displaystyle DomDo{m}_u=Dom{₂}_u}$ holds for ${\displaystyle \_u}$ = {all connected graphs without induced ${\displaystyle P_u}$} (u ≥ 2). (In particular, ₂ = {K₁} and ₃ = {all complete graphs}.) Some negative examples are also given.