the average degree in $G$. For a set $S\subseteq V$, we denote the number of vertices of $S$ adjacent to $v$ in $G$ by $d_S(v)$. In particular, $d_V(v) = d_G(v)$. For a set $S \subseteq V$, the subgraph induced by $S$ is denoted by $G[S]$. We denote by ${{\rm span}}(G)$, the *span* of $G$, the number of distinct values in the degree sequence of $G$ and by ${{\rm rep}}(G)$, the *repetition number* of $G$, the maximum multiplicity in the list of vertex degrees. The parameter $\alpha(G)$ denotes the (vertex) *independence number* of $G$, while $\chi(G)$ denotes the *chromatic number* of $G$.

Further, we denote the complete graph on $n$ vertices by $K_n$ and the empty graph on $n$ vertices by ${{\overline{K}}}_n$. Moreover, $P_n$, $C_n$ and $K_{m,n}$ denote, respectively, the path on $n$ vertices, the cycle on $n$ vertices and the complete bipartite graph with one partite set of cardinality $m$ and the other of cardinality $n$.

Preliminary Results and Observations
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It is a simple exercise to determine the fair domination number of certain well-studied families of graphs. Recall that for $n \ge 3$, $\gamma(P_n) = \gamma(C_n) = \lceil n/3 \rceil$, while for $m \ge n
\ge 2$, $\gamma(K_{m,n}) = 