# On the Outer-Independent Roman Domination in Graphs
## 1 Introduction  
Throughout this paper, we consider \(G=(V(G),E(G))\) as a simple graph with no isolated vertex. Given a vertex \(v\) of \(G\), \(N(v)\) and \(N[v]\) represent the open neighbourhood and the closed neighbourhood of \(v\), respectively. We also denote by \(\deg(v)=|N(v)|\) the degree of vertex \(v\). For a set \(D\subseteq V(G)\), its open neighbourhood and closed neighbourhood are \(N(D)=\cup_{v\in D}N(v)\) and \(N[D]=N(D)\cup D\), respectively. Moreover, the subgraph of \(G\) induced by \(D\subseteq V(G)\) will be denoted by \(G[D]\). Domination theory is an interesting topic in the theory of graphs, as well as one of the most active topic of research in this area. A set \(D\subseteq V(G)\) is a dominating set of \(G\) if \(N[D]=V(G)\). The domination number of \(G\), denoted by \(\gamma(G)\), is the minimum cardinality amongst all dominating sets of \(G\). Numerous results on this issue obtained in the previous century are shown in [1,2]. We define a \(\gamma(G)\)-set as a dominating set of cardinality \(\gamma(G)\). The same terminology will be assumed for optimal parameters associated with other sets or functions defined in the paper. Moreover, in the last two decades, the interest in the domination theory in graphs has increased. In that sense, a very high number of variants of domination parameters have been studied, many of which are combinations of two or more parameters. Next, we expose some of them. * A set \(S\subseteq V(G)\) is an independent set of \(G\) if the subgraph induced by \(S\) is edgeless. The maximum cardinality among all independent sets of \(G\) is the independence number of \(G\), and is denoted by \(\beta(G)\). In some kind of \"opposed\" side of an independent set, we find a vertex cover, which is a set \(D\subseteq V(G)\) such that \(V(G)\setminus D\) is an independent set of \(G\). The vertex cover number of \(G\), denoted by \(\alpha(G)\), is the minimum cardinality among all vertex covers of \(G\). It is well-known that for any graph \(G\) of order \(n\), \(\alpha(G)+\beta(G)=n\) (see [3]). * A set \(S\subseteq V(G)\) is an independent dominating set of \(G\) if \(S\) is an independent and dominating set at the same time. The independent domination number of \(G\) is the minimum cardinality among all independent dominating sets of \(G\) and is denoted by \(i(G)\). Independent domination in graphs was formally introduced in [4; 5]. However, a fairly complete survey on this topic was recently published in [6]. * A function \(f:V(G)\rightarrow\{0,1,2\}\) is called a Roman dominating function on \(G\), if every \(v\in V(G)\) for which \(f(v)=0\) is adjacent to at least one vertex \(u\in V(G)\) for which \(f(u)=2\). The Roman domination number of \(G\), denoted by \(\gamma_{R}(G)\), is the minimum weight \(\omega(f)=\sum_{v\in V(G)}f(v)\) among all Roman dominating functions \(f\) on \(G\). This parameter was introduced in [7]. Let \(V_{i}=\{v\in V(G):f(v)=i\}\) for \(i\in\{0,1,2\}\). We will identify a Roman dominating function \(f\) with the subsets \(V_{0}\), \(V_{1}\), \(V_{2}\) of \(V(G)\) associated with it, and so we will use the unified notation \(f(V_{0},V_{1},V_{2})\) for the function and these associated subsets. * A Roman dominating function \(f(V_{0},V_{1},V_{2})\) is called an outer-independent Roman dominating function, abbreviated OIRDF, if \(V_{0}\) is an independent set of \(G\). Notice that then \(V_{1}\cup V_{2}\) is a vertex cover of \(G\). The outer-independent Roman domination number of \(G\) is the minimum weight among all outer-independent Roman dominating functions on \(G\), and is denoted by \(\gamma_{oiR}(G)\). This parameter was introduced in [8] and also studied in [9; 10; 11]. All the previous parameters are, in one way or another, related to each other. Next, we show the most natural relationships that exist between them, which are easily deductible by definition. **Remark 1**.