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.