Monophonic eccentric domination in graphs

For any two vertices $u$ and $v$ in a connected graph $G,$ the monophonic distance $d_m(u,v)$ from $u$ to $v$ is defined as the length of a longest $u-v$ monophonic path in $G$. The monophonic eccentricity $e_m(v)$ of a vertex $v$ in $G$ is the maximum monophonic distance from $v$ to a vertex of $G$.  A vertex $v$ in $G$ is a monophonic eccentric vertex of a vertex $u$ in $G$ if $e_m(u) = d_m(u,v)$. A set $S \subseteq V$  is a  monophonic eccentric  dominating $set$ if every vertex in $V-S$ has a monophonic eccentric vertex in $S$. The monophonic eccentric  domination number $\gamma_{me}(G)$ is the  cardinality of a minimum monophonic eccentric  dominating set of $G$. We investigate some properties of monophonic eccentric  dominating sets. Also, we determine the bounds of monophonic eccentric  domination number and find the same for some standard graphs.