  In this paper, we consider dominating sets $D$ and $D'$ such that $D$ and
$D'$ are disjoint and there exists a perfect matching between them. Let
$DD_{\textrm{m}}(G)$ denote the cardinality of smallest such sets $D, D'$ in
$G$ (provided they exist, otherwise $DD_{\textrm{m}}(G) = \infty$). This
concept was introduced in [Klostermeyer et al., Theory and Application of
Graphs, 2017] in the context of studying a certain graph protection problem. We
characterize the trees $T$ for which $DD_{\textrm{m}}(T)$ equals a certain
graph protection parameter and for which $DD_{\textrm{m}}(T) = \alpha(T)$,
where $\alpha(G)$ is the independence number of $G$. We also further study this
parameter in graph products, e.g., by giving bounds for grid graphs, and in
graphs of small independence number.
