# On the Outer-Independent Roman Domination in Graphs
## 2 Bounds and Relationships with Other Parameters
For example, the lower bound is achieved for the complete bipartite graphs \(K_{r,r}\), where \(\gamma_{oiR}(K_{r,r})=r+1=\left\lceil\frac{r^{2}}{r}\right\rceil+1=\left\lceil \frac{i(K_{r,r})\delta(K_{r,r})}{\Delta(K_{r,r})}\right\rceil+1\). In addition, the upper bound is achieved for the case of complete graphs, and in connection with this fact, we pose the following question. **Open question:** Is it the case that \(\gamma_{oiR}(G)=n-i(G)+\gamma(G)\) if and only if \(G\) is a complete graph? Next, we give new bounds for the outer-independent Roman domination number of triangle-free graphs. Recall that in these graphs, no pair of adjacent vertices can have a common neighbor. For this purpose, we shall need to introduce the following definitions. A set \(S\subseteq V(G)\) is a 3-packing if the distance between \(u\) and \(v\) is greater than three for every pair of different vertices \(u,v\in S\). The 3-packing number of \(G\), denoted by \(\rho_{3}(G)\), is the maximum cardinality among all 3-packings of \(G\). We also define  
\[\alpha(G)+\left\lceil\frac{n-\alpha(G)}{\Delta}\right\rceil\leq\gamma_{oiR}( G)\leq\alpha(G)+\gamma(G).\]6  
**Theorem 7**. : _For any triangle-free graph \(G\) of order \(n\),_  
\[\alpha(G)+\left\lceil\frac{n-\alpha(G)}{\Delta}\right\rceil\leq\gamma_{oiR}( G)\leq\alpha(G)+\gamma(G).\]7  
Proof. : Let \(S\in\mathcal{P}_{3}(G)\). As \(G\) is triangle-free, it follows that \(N(v)\) is an independent set of \(G\) for every \(v\in V(G)\). Hence, \(N(S)\) is an independent set of \(G\), which implies that the function \(f(V_{0},V_{1},V_{2})\), defined by \(V_{2}=S\), \(V_{0}=N(S)\) and \(V_{1}=V(G)\setminus N[S]\), is an OIRDF on \(G\). Thus, \(\gamma_{oiR}(G)\leq 2|V_{2}|+|V_{1}|=2|S|+(n-|N[S]|)=n-\sum_{v\in S}(\deg(v)-1)\). Since the inequality holds for every \(S\in\mathcal{P}_{3}(G)\), the result follows. **Corollary 2**. : _For any triangle-free graph \(G\) of order \(n\) and minimum degree \(\delta\),_  
\[\alpha(G)+\left\lceil\frac{n-\alpha(G)}{\Delta}\right\rceil\leq\gamma_{oiR}( G)\leq\alpha(G)+\gamma(G).\]8In [8], the bound \(\gamma_{oiR}(G)\leq n-\Delta(G)+1\) was given for the case of triangle-free graph. Next, we state a result which improve the bound above for the triangle-free graphs \(G\) that satisfy the condition \(diam(G)(\delta(G)-1)\geq 4(\Delta(G)-1)\). Proposition 3: _Let \(G\) be a connected triangle-free graph of order \(n\), minimum degree \(\delta\) and maximum degree \(\Delta\). If \(diam(G)\geq 4\), then_  
\[\alpha(G)+\left\lceil\frac{n-\alpha(G)}{\Delta}\right\rceil\leq\gamma_{oiR}( G)\leq\alpha(G)+\gamma(G).\]9  
Proof: Assume that \(diam(G)\geq 4\). Let \(P=v_{0}v_{1}\cdots v_{k}\) be a diametrical path of \(G\) (notice that \(k=diam(G)\)), and \(S=\{v_{0},v_{4},\ldots,v_{4\lfloor k/4\rfloor}\}\). It is easy to see that \(S\in\mathcal{P}_{3}(G)\), and so, by Theorem 7 we deduce that \(\gamma_{oiR}(G)\leq n-\sum_{v\in S}(\deg(v)-1)\leq n-\left\lceil\frac{diam(G)} {4}\right\rceil(\delta-1)\) which completes the proof. The bounds given in Corollary 2 and Proposition 3 are tight. For instance, they are achieved for the cycle \(C_{10}\).