An improved upper bound for the fair domination number of maximal outerplanar graphs

A dominating set $D$ of a graph $G$ is a \emph{fair dominating set} if every two vertices outside $D$ have the same number of neighbors in $D$, and the \emph{fair domination number} $\mathrm{fd}(G)$ is the minimum cardinality of such a set. Caro, Hansberg and Henning, who introduced this parameter,