On interior Roman domination in graphs

Let $G = (V(G), E(G))$ be a non-complete graph and let $\phi:V(G)\rightarrow \{0,1,2\}$ be a function on $G$. For each $i \in \{0, 1, 2\}$, let $V_i=\{w \in V(G): \phi(w)=i\}$. A function $\phi=(V_0, V_1, V_2)$ is an interior Roman dominating function (InRDF) on $G$ if $(i)$ for every $v\in V_0$, there exists $u\in V_2$ such that $uv\in E(G)$, and $(ii)$ either $V_1=V(G)$ or for every $z\in V_2$, $z$ is an interior vertex of $G$. Denoted by $\omega_G^{InR}(\phi)=\sum_{u \in V(G)}\phi(u)$ is the \textit{weight} of InRDF $\phi$; and the minimum weight of an InRDF $\phi$ on $G$, denoted by $\gamma_{InR}(G)$, is called the \textit{interior Roman domination number}. Any InRDF $\phi$ on graph $G$ with $\omega_{G}^{InR}(\phi)= \gamma_{InR}(G)$ is called a $\gamma_{InR}$-function on $G$. In this paper, we introduce a new parameter of a Roman dominating function in graphs and discuss some important combinatorial properties.