Restrained double monophonic number of a graph

For a connected graph $G$ of order at least two, a double monophonic set $S$ of a graph $G$ is a restrained double monophonic set if either $S=V$ or the subgraph induced by $V-S$ has no isolated vertices. The minimum cardinality of a restrained double monophonic set of $G$ is the restrained double monophonic number of $G$ and is denoted by $dm_{r}(G)$. The restrained double monophonic number of certain classes graphs are determined. It is shown that for any integers $a,\, b,\, c$ with $3 \leq a \leq b \leq c$, there is a connected graph $G$ with $m(G) = a$, $m_r(G) = b$ and $dm_{r}(G) = c$, where $m(G)$ is the monophonic number and $m_r(G)$ is the restrained monophonic number of a graph $G$.