# On the Outer-Independent Roman Domination in Graphs
## 3 Rooted Product Graphs
The following statements are equivalent._  
* \(\gamma_{oiR}(G\circ_{v}H)=\alpha(G)+n(\gamma_{oiR}(H)-1)\)_._
* _There exist two_ \(\gamma_{oiR}(H)\)_-functions_ \(g_{1}\) _and_ \(g_{2}\) _such that_ \(g_{1}(x)=1\) _for every_ \(x\in N[v]\) _and_ \(g_{2}(v)=2\)_._  
Proof. : We first assume that (i) holds, i.e., \(\gamma_{oiR}(G\circ_{v}H)=\alpha(G)+n(\gamma_{oiR}(H)-1)\). Let \(f(V_{0},V_{1},V_{2})\) be a \(\gamma_{oiR}(G\circ_{v}H)\)-function. As \(\alpha(G)<n\), it follows that \(\mathcal{C}_{f}\
eq\emptyset\), and so, by Lemma 4 (ii) we can obtain a \(\gamma_{oiR}(H)\)-function \(g_{1}\) such that \(g_{1}(x)=1\) for every \(x\in N[v]\). Moreover, if \(g(v)\leq 1\) for every \(\gamma_{oiR}(H)\)-function \(g\), then, by proceeding analogously to Subcase 2 in the proof of Theorem 8 we deduce  
\[\mathcal{A}_{f} =\{x\in V(G):\omega(f_{x})>\gamma_{oiR}(H)\},\]5 The labels of (gray and black) coloured vertices describe the positive weights of a \(\gamma_{oiR}(H)\)-function. that \(\gamma_{oiR}(G\circ_{v}H)\geq\gamma_{oiR}(G)+n(\gamma_{oiR}(H)-1)\), which is a contradiction as \(\gamma_{oiR}(G)>\alpha(G)\). Therefore, there exists a \(\gamma_{oiR}(H)\)-function \(g_{2}\) such that \(g_{2}(v)=2\), and (ii) follows. On the other hand, we assume that there exist two \(\gamma_{oiR}(H)\)-functions \(g_{1}\) and \(g_{2}\) such that \(g_{1}(x)=1\) for every \(x\in N[v]\) and \(g_{2}(v)=2\). Let \(D\) be an \(\alpha(G)\)-set and let \(g_{1}^{\prime}\) be a function on \(H\) such that \(g_{1}^{\prime}(v)=0\) and \(g_{1}^{\prime}(x)=g_{1}(x)\) whenever \(x\in V(H)\setminus\{v\}\). From \(D\), \(g_{1}^{\prime}\) and \(g_{2}\), we define a function \(h\) on \(G\circ_{v}H\) as follows. For every \(x\in D\), the restriction of \(h\) to \(V(H_{x})\) is induced from \(g_{2}\). Moreover, if \(x\in V(G)\setminus D\), then the restriction of \(h\) to \(V(H_{x})\) is induced from \(g_{1}^{\prime}\). Notice that \(h\) is an OIRDF on \(G\circ_{v}H\), and so \(\gamma_{oiR}(G\circ_{v}H)\leq\omega(h)=|D|\gamma_{oiR}(H)+|V(G)\setminus D|( \gamma_{oiR}(H)-1)=\alpha(G)+n(\gamma_{oiR}(H)-1)\). Therefore, Theorem 8 leads to \(\gamma_{oiR}(G\circ_{v}H)=\alpha(G)+n(\gamma_{oiR}(H)-1)\), which completes the proof. Next we proceed to characterize the graphs with \(\gamma_{oiR}(G\circ_{v}H)=\gamma_{oiR}(G)+n(\gamma_{oiR}(H)-1)\). Notice that it is excluded the case \(\gamma_{oiR}(G)=n\), since then \(\gamma_{oiR}(G\circ_{v}H)=n\gamma_{oiR}(H)\). Theorem 11: _Let \(G\) be a graph of order \(n\) with no isolated vertex such that \(\gamma_{oiR}(G)<n\) and let \(H\) be a graph with no isolated vertex and \(v\in V(H)\). The following statements are equivalent._  
1. \(\gamma_{oiR}(G\circ_{v}H)=\gamma_{oiR}(G)+n(\gamma_{oiR}(H)-1)\)_._
2. \(g(v)\leq 1\) _for every_ \(\gamma_{oiR}(H)\)_-function_ \(g\) _and also, there exists a_ \(\gamma_{oiR}(H)\)_-function_ \(g_{1}\) _such that_ \(g_{1}(x)=1\) _for every_ \(x\in N[v]\)_._  
Proof: We first assume that (i) holds, i.e., \(\gamma_{oiR}(G\circ_{v}H)=\gamma_{oiR}(G)+n(\gamma_{oiR}(H)-1)\). Let \(f(V_{0},V_{1},V_{2})\) be a \(\gamma_{oiR}(G\circ_{v}H)\)-function. Since \(\gamma_{oiR}(G)<n\), it follows that \(\mathcal{C}_{f}\
eq\emptyset\), and so, by Lemma 4 (ii) we can obtain a \(\gamma_{oiR}(H)\)-function \(g_{1}\) such that \(g_{1}(x)=1\) for every \(x\in N[v]\). Moreover, if there exists a \(\gamma_{oiR}(H)\)-function \(g_{2}\) such that \(g_{2}(v)=2\), then by Theorem 10 we deduce that \(\gamma_{oiR}(G\circ_{v}H)=\alpha(G)+n(\gamma_{oiR}(H)-1)\), which is a contradiction as \(\gamma_{oiR}(G)>\alpha(G)\). Therefore, \(g(v)\leq 1\) for every \(\gamma_{oiR}(H)\)-function \(g\), which implies that (ii) follows. On the other side, we assume that \(g(v)\leq 1\) for every \(\gamma_{oiR}(H)\)-function \(g\) and also, that there exists a \(\gamma_{oiR}(H)\)-function \(g_{1}\) such that \(g_{1}(x)=1\) for every \(x\in N[v]\). Under these assumptions, observe that the function \(g_{1}\) restricted to \(V(H)\setminus\{v\}\), namely \(g_{1}^{\prime}\), is an OIRDF on \(H-v\).