$G$, denoted $\gamma(G)$, is the minimum cardinality of a dominating set. A dominating set of $G$ of cardinality $\gamma(G)$ is called a $\gamma(G)$-set.

Let $G$ be a graph that is not the empty graph. For $k \ge 1$ an integer, a *$k$-fair dominating set*, abbreviated kFD-set, in $G$ is a dominating set $D$ such that $|N(v) \cap D| = k$ for every vertex $v \in V \setminus D$. We note that the set $D = V$ is a kFD-set since vacuously every vertex in $V \setminus D = \emptyset$ satisfies the desired property. The *$k$-fair domination number* of $G$, denoted by ${{\rm fd}_k}(G)$, is the minimum cardinality of a kFD-set. A kFD-set of $G$ of cardinality ${{\rm fd}_k}(G)$ is called a ${{\rm fd}_k}(G)$-set. With this definition in mind, we point to a related problem on the so called $(k, \tau)$-regular sets discussed in [@CKV] (and all the references given there). Some reminiscent of this approach appears more explicitly in Proposition \[l:line1\].

A *fair dominating set*, abbreviated FD-set, in $G$ is a kFD-set for some integer $k \ge 1$. Thus a dominating set $D$ is a FD-set in $G$ if $D = V$ or if $D \ne V$ and all vertices 