Given a graph, $G = (V, E)$, I want to find an optimal $r$-domination for $G$. That is, I want a subset $S$ of $V$ such that all vertices in $G$ are at a distance of at most $r$ from some vertex in $S$, while minimizing the size of $S$.
From what I have checked so far, I got the following: There is this related problem of finding a $(k,r)$-center in a graph which is a subset $S$ of size at most $k$ such that all the vertices in the graph are at a distance of atmost $r$ from some vertex in $S$(here both $|S| \leq k$ and $r$ are parts of the input) for which Demaine et al. have a FPT algorithm for planar graphs. Otherwise the problem is $W$-hard for even $r = 1$.
Is anything known about the exact complexity of $r$-domination problem for bounded tree width graphs or even just trees? (Is $r$-domination MSO definable? the usual $k$-dominating set problem is MSO definable -- which then would allow one to use Courcelle's theorem to conclude that there is a linear time algorithm for the problem). Are there any conditional hardness results known regarding this problem?