A subgraph \(D\) of a graph \(G\) is \(k\)-dominating in \(G\) if every vertex of \(G-D\) is of distance \(\leq k\) from a vertex of \(D\). If \(\mathcal D\) is a specified class of graphs, Dom\(_k \mathcal D\) consists of all those graphs \(G\) in which every connected induced subgraph \(H\) has a \(k\)-dominating induced subgraph \(D \in {\mathcal D}\) which is also connected. The authors rely on a result from a 1993 paper of theirs to show that \(\operatorname{Dom} \operatorname{Dom}\mathcal D _u = \operatorname{Dom}_2 \mathcal D _u\) where \(\mathcal D _u\) is the class of all connected graphs without induced \(P_u\) (\(u \geq 2\)).