  Given a graph $G=(V,E)$, a set of vertices $D\subseteq V $ is called a
dominating set if every vertex in $V\backslash D$ is adjacent to a vertex in
$D$, and a subset $B\subseteq V $ is called a nonblocking set if $V-B$ is a
dominating set. In this paper, we introduce a graph dynamical systems detecting
vertex sets that are simultaneously dominating and nonblocking sets via
reversing the action of the system. Moreover, by using actions of multiple such
graph dynamical systems we define actions of free monoid on two letters for
which elements in the reverse action corresponds to more special dominating
sets.
