Identifying domatic partitions via graph dynamical systems

For 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 $D.$ A domatic-$2$-partition of $G$ is a partition of its vertices into two disjoint dominating sets. In this paper, for a finite simple connected graph $