# On the Outer-Independent Roman Domination in Graphs

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## Abstract

**:**

## 1. Introduction

- A set $S\subseteq V\left(G\right)$ 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 $\beta \left(G\right)$. In some kind of “opposed” side of an independent set, we find a vertex cover, which is a set $D\subseteq V\left(G\right)$ such that $V\left(G\right)\backslash D$ is an independent set of G. The vertex cover number of G, denoted by $\alpha \left(G\right)$, is the minimum cardinality among all vertex covers of G. It is well-known that for any graph G of order n, $\alpha \left(G\right)+\beta \left(G\right)=n$ (see [3]).
- A set $S\subseteq V\left(G\right)$ 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 $i\left(G\right)$. 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 $f:V\left(G\right)\to \{0,1,2\}$ is called a Roman dominating function on G, if every $v\in V\left(G\right)$ for which $f\left(v\right)=0$ is adjacent to at least one vertex $u\in V\left(G\right)$ for which $f\left(u\right)=2$. The Roman domination number of G, denoted by ${\gamma}_{R}\left(G\right)$, is the minimum weight $\omega \left(f\right)={\sum}_{v\in V\left(G\right)}f\left(v\right)$ among all Roman dominating functions f on G. This parameter was introduced in [7]. Let ${V}_{i}=\{v\in V\left(G\right):f\left(v\right)=i\}$ for $i\in \{0,1,2\}$. We will identify a Roman dominating function f with the subsets ${V}_{0}$, ${V}_{1}$, ${V}_{2}$ of $V\left(G\right)$ 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 ${\gamma}_{oiR}\left(G\right)$. This parameter was introduced in [8] and also studied in [9,10,11].

**Remark**

**1.**

- (i)
- $\gamma \left(G\right)\le i\left(G\right)\le \beta \left(G\right)=n-\alpha \left(G\right)$.
- (ii)
- $\gamma \left(G\right)\le {\gamma}_{R}\left(G\right)\le {\gamma}_{oiR}\left(G\right)$.

- $\gamma \left({G}_{1}\right)=2<i\left({G}_{1}\right)<4=\alpha \left({G}_{1}\right)={\gamma}_{R}\left({G}_{1}\right)<\beta \left({G}_{1}\right)<{\gamma}_{oiR}\left({G}_{1}\right)=6$.
- $\gamma \left({G}_{2}\right)=i\left({G}_{2}\right)=\alpha \left({G}_{2}\right)=2<{\gamma}_{R}\left({G}_{2}\right)={\gamma}_{oiR}\left({G}_{2}\right)=3<\beta \left({G}_{2}\right)=5$.

## 2. Bounds and Relationships with Other Parameters

**Theorem**

**1**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**3.**

- (i)
- ${\gamma}_{oiR}\left(G\right)=\alpha \left(G\right)+1$.
- (ii)
- There exist an $\alpha \left(G\right)$-set S and a vertex $v\in S$ such that $V\left(G\right)\backslash S\subseteq N\left(v\right)$.

**Proof.**

**Lemma**

**1.**

- (i)
- $diam\left(T\right)\le 4$.
- (ii)
- $V\left(T\right)=L\left(T\right)\cup S\left(T\right)$.

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Proposition**

**1.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Open question:**Is it the case that ${\gamma}_{oiR}\left(G\right)=n-i\left(G\right)+\gamma \left(G\right)$ if and only if G is a complete graph?

**Theorem**

**7.**

**Proof.**

**Corollary**

**2.**

**Proposition**

**3.**

**Proof.**

## 3. Rooted Product Graphs

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

- (i)
- If $g\left(v\right)=0$ for some ${\gamma}_{oiR}\left(H\right)$-function g, then ${\gamma}_{oiR}\left(G{\circ}_{v}H\right)\le \alpha \left(G\right)+n{\gamma}_{oiR}\left(H\right).$
- (ii)
- If $g\left(v\right)>0$ for some ${\gamma}_{oiR}\left(H\right)$-function g, then ${\gamma}_{oiR}\left(G{\circ}_{v}H\right)\le n{\gamma}_{oiR}\left(H\right).$
- (iii)
- If there exists a ${\gamma}_{oiR}(H-v)$-function g such that $g\left(x\right)>0$ for every $x\in N\left(v\right)$, then ${\gamma}_{oiR}\left(G{\circ}_{v}H\right)\le {\gamma}_{oiR}\left(G\right)+n{\gamma}_{oiR}(H-v).$

**Proof.**

**Lemma**

**4.**

- (i)
- $\omega \left({f}_{x}\right)\ge {\gamma}_{oiR}\left(H\right)-1$.
- (ii)
- If $\omega \left({f}_{x}\right)={\gamma}_{oiR}\left(H\right)-1$, then $x\in {V}_{0}$ and $N\left(x\right)\cap V\left({H}_{x}\right)\subseteq {V}_{1}$.

**Proof.**

**Theorem**

**8.**

**Proof.**

**Example**

**1.**

- ${\gamma}_{oiR}\left(G{\circ}_{v}H\right)=\alpha \left(G\right)+n{\gamma}_{oiR}\left(H\right).$
- ${\gamma}_{oiR}\left(G{\circ}_{w}H\right)=n{\gamma}_{oiR}\left(H\right).$
- ${\gamma}_{oiR}\left(G{\circ}_{{v}^{\prime}}H\right)={\gamma}_{oiR}\left(G\right)+n({\gamma}_{oiR}\left(H\right)-1).$
- ${\gamma}_{oiR}\left(G{\circ}_{{w}^{\prime}}H\right)=\alpha \left(G\right)+n({\gamma}_{oiR}\left(H\right)-1).$

**Theorem**

**9.**

- (i)
- ${\gamma}_{oiR}\left(G{\circ}_{v}H\right)=\alpha \left(G\right)+n{\gamma}_{oiR}\left(H\right)$.
- (ii)
- $g\left(v\right)=0$ for every ${\gamma}_{oiR}\left(H\right)$-function g.

**Proof.**

**Theorem**

**10.**

- (i)
- ${\gamma}_{oiR}\left(G{\circ}_{v}H\right)=\alpha \left(G\right)+n({\gamma}_{oiR}\left(H\right)-1)$.
- (ii)
- There exist two ${\gamma}_{oiR}\left(H\right)$-functions ${g}_{1}$ and ${g}_{2}$ such that ${g}_{1}\left(x\right)=1$ for every $x\in N\left[v\right]$ and ${g}_{2}\left(v\right)=2$.

**Proof.**

**Theorem**

**11.**

- (i)
- ${\gamma}_{oiR}\left(G{\circ}_{v}H\right)={\gamma}_{oiR}\left(G\right)+n({\gamma}_{oiR}\left(H\right)-1)$.
- (ii)
- $g\left(v\right)\le 1$ for every ${\gamma}_{oiR}\left(H\right)$-function g and also, there exists a ${\gamma}_{oiR}\left(H\right)$-function ${g}_{1}$ such that ${g}_{1}\left(x\right)=1$ for every $x\in N\left[v\right]$.

**Proof.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The labels of (gray and black) coloured vertices describe the positive weights of a ${\gamma}_{oiR}\left({G}_{i}\right)$-function, for $i\in \{1,2\}$.

**Figure 4.**The labels of (gray and black) coloured vertices describe the positive weights of a ${\gamma}_{oiR}\left(H\right)$-function.

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**MDPI and ACS Style**

Martínez, A.C.; García, S.C.; Carrión García, A.; Grisales del Rio, A.M.
On the Outer-Independent Roman Domination in Graphs. *Symmetry* **2020**, *12*, 1846.
https://doi.org/10.3390/sym12111846

**AMA Style**

Martínez AC, García SC, Carrión García A, Grisales del Rio AM.
On the Outer-Independent Roman Domination in Graphs. *Symmetry*. 2020; 12(11):1846.
https://doi.org/10.3390/sym12111846

**Chicago/Turabian Style**

Martínez, Abel Cabrera, Suitberto Cabrera García, Andrés Carrión García, and Angela María Grisales del Rio.
2020. "On the Outer-Independent Roman Domination in Graphs" *Symmetry* 12, no. 11: 1846.
https://doi.org/10.3390/sym12111846