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 \(G\).