  Guarding a polygon with few guards is an old and well-studied problem in
computational geometry. Here we consider the following variant: We assume that
the polygon is orthogonal and thin in some sense, and we consider a point $p$
to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned
by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is
NP-hard on orthogonal polygons with holes, even if the polygon is thin. If
there are no holes, then a thin polygon becomes a tree polygon in the sense
that the so-called dual graph of the polygon is a tree. It was known that
finding the minimum set of $r$-guards is polynomial for tree polygons, but the
run-time was $\tilde{O}(n^{17})$. We show here that with a different approach
the running time becomes linear, answering a question posed by Biedl et al.
(SoCG 2011). Furthermore, the approach is much more general, allowing to
specify subsets of points to guard and guards to use, and it generalizes to
polygons with $h$ holes or thickness $K$, becoming fixed-parameter tractable in
$h+K$.
