DOUBLE ROMAN DOMINATION NUMBER OF MIDDLE GRAPH

 For any graph $ G(V,\ E) $, a function $ f:V(G)\rightarrow \{0,\ 1, \ 2,\ 3 \} $ is called Double Roman dominating function (DRDF) if the following properties holds,

\begin{enumerate}

\item If $ f(v)=0 $, then there exist two vertices $ v_{1}, v_{2} \in N(v) $ for which $ f(v_{1})=f(v_{2})=2 $ or there exist one vertex $ u\in N(v) $ for which $ f(u)=3 $.

\item If $ f(v)=1 $, then there exist one vertex $ u\in N(v) $ for which $ f(u)=2 $ or $ f(u)=3 $.

\end{enumerate}

The weight of DRDF is the value $ w(f) = \sum _{v\in V(G)} f(v) $. The minimum weight among all double Roman dominating function is called double Roman domination number and is denoted by $ \gamma_{dR}(G) $. In this article we initiated research on double Roman domination number for middle graphs. We established lower and upper bounds and also we characterize the double Roman domination number of middle graphs. Later we calculated numerical value of double Roman domination number of middle graph of path, cycle, star, double star and friendship graphs.