# Total Roman {3}-domination in Graphs

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## Abstract

**:**

## 1. Introduction

- (a)
- if $f\left(v\right)=0$, then the vertex v must have at least two neighbors in ${V}_{2}$ or one neighbor in ${V}_{3}$.
- (b)
- if $f\left(v\right)=1$, then the vertex v must have at least one neighbor in ${V}_{2}\cup {V}_{3}$.

**Definition**

**1.**

## 2. Total Roman $\left\{3\right\}$-domination of Some Graphs

**Proposition**

**1.**

**Corollary**

**1.**

**Observation**

**1.**

**Proof.**

**Observation**

**2.**

**Observation**

**3.**

**Proof.**

**Corollary**

**2.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**3.**

**Proposition**

**3.**

- ${\gamma}_{t\left\{R3\right\}}\left({K}_{1,n}\right)=4$,
- ${\gamma}_{t\left\{R3\right\}}\left({K}_{m,n}\right)=5$ for $m\in \{2,3\}$ and $n\ge 3$.
- ${\gamma}_{t\left\{R3\right\}}\left({K}_{m,n}\right)={\gamma}_{dR}\left({K}_{m,n}\right)=6$ for $m,n\ge 4$.

**Proof.**

- (i)
- Let $U=\{{u}_{1},{u}_{2}\}$ and $W=\{{w}_{1},{w}_{2},\dots ,{w}_{n}\}$. Let f be a TR$\left\{3\right\}$DF of ${K}_{2,n}$. If $f\left(W\right)=2$, then $f\left(U\right)\ge 3$. If $f\left(W\right)=3$, then $f\left(U\right)\ge 2$. If $f\left(W\right)\ge 4$, since $f\left(U\right)$ is positive, then $f\left(V\right)\ge 5$. Therefore $f\left(V\right)\ge 5$. Assigning $f\left({u}_{1}\right)=2,\phantom{\rule{4pt}{0ex}}f\left({u}_{2}\right)=1$ and $f\left({w}_{1}\right)=2$, shows that ${\gamma}_{t\left\{R3\right\}}\left({K}_{2,n}\right)\le 5$.
- (ii)
- Let $U=\{{u}_{1},{u}_{2},{u}_{3}\}$ and $W=\{{w}_{1},{w}_{2},\dots ,{w}_{n}\}$. Using sketch of the proof of item 2, ${\gamma}_{t\left\{R3\right\}}\left({K}_{3,n}\right)\ge 5$. If we assign value 1 to the vertices ${u}_{1},{u}_{2},{u}_{3}$, weight 2 to ${w}_{1}$ and 0 to ${w}_{j},$ for $j\ge 2$, then ${\gamma}_{t\left\{R3\right\}}\left({K}_{3,n}\right)\le 5$.

**Proposition**

**4.**

- If ${n}_{1}={n}_{2}=1$, then ${\gamma}_{t\left\{R3\right\}}\left(G\right)=3$.
- If ${n}_{1}=1$ and ${n}_{2}\ge 2$, then ${\gamma}_{t\left\{R3\right\}}\left(G\right)=4$.
- If ${n}_{1}=2$ or ${n}_{1}\ge 3$ and $r\ge 4$, then ${\gamma}_{t\left\{R3\right\}}\left(G\right)=4$.
- If $r=3$ and ${n}_{1}\ge 3$, then ${\gamma}_{t\left\{R3\right\}}\left(G\right)=5$.

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Theorem**

**6.**

## 3. Total Roman $\left\{3\right\}$-domination and Total Domination

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

- If $k\equiv 0\phantom{\rule{4pt}{0ex}}\left(mod\phantom{\rule{4pt}{0ex}}3\right)$, then we say $G=\frac{k-3}{3}{P}_{5}\cup {G}_{1}$.
- If $k\equiv 1\phantom{\rule{4pt}{0ex}}\left(mod\phantom{\rule{4pt}{0ex}}3\right)$, then we say $G=\frac{k-4}{3}{P}_{5}\cup {G}_{2}$.
- If $k\equiv 2\phantom{\rule{4pt}{0ex}}\left(mod\phantom{\rule{4pt}{0ex}}3\right)$, then we say $G=\frac{k-5}{3}{P}_{5}\cup {G}_{3}$.

- If $k\equiv 1\phantom{\rule{4pt}{0ex}}\left(mod\phantom{\rule{4pt}{0ex}}3\right)$, then we say $G=\frac{k-4}{3}{P}_{5}\cup {G}_{1}^{\prime}$.
- If $k\equiv 2\phantom{\rule{4pt}{0ex}}\left(mod\phantom{\rule{4pt}{0ex}}3\right)$, then we say $G=\frac{k-5}{3}{P}_{5}\cup {G}_{2}^{\prime}$.
- If $k\equiv 0\phantom{\rule{4pt}{0ex}}\left(mod\phantom{\rule{4pt}{0ex}}3\right)$, then we say $G=\frac{k-6}{3}{P}_{5}\cup {G}_{3}^{\prime}$.

## 4. Total Roman $\left\{3\right\}$ and Total Roman $\left\{2\right\}$-domination

**Observation**

**7.**

**Proof.**

**Proposition**

**8.**

- ${\gamma}_{t\left\{R3\right\}}\left(G\right)=3$ if and only if ${\gamma}_{t\left\{R2\right\}}\left(G\right)=2$.
- If ${\gamma}_{t\left\{R3\right\}}\left(G\right)=4$, then ${\gamma}_{t\left\{R2\right\}}\left(G\right)=3$.
- If ${\gamma}_{t\left\{R2\right\}}\left(G\right)=3$, then $4\le {\gamma}_{t\left\{R3\right\}}\left(G\right)\le 5$.

**Proof.**

- 2.1.
- There exist 4 vertices $v,u,w,z$ with label 1 for which the induced subgraph by them is the cycle ${C}_{4}$, the graph $K={K}_{4}-e$ or the complete graph ${K}_{4}$. In any induced subgraph, there are no two vertices of them for which any vertex with label 0 is adjacent to them. Thus in the case of a TR$\left\{2\right\}$-DF we change one of the labels 1 to the label 0. Therefore ${\gamma}_{t\left\{R2\right\}}\left(G\right)=3$.
- 2.2.
- There exist 2 vertices $v,u$ with label 1 and one vertex w with label 2, for which the induced subgraph by them is the cycle ${C}_{3}$, or the path ${P}_{3}=v-w-u$. In any of the two cases each vertex with label 0 is adjacent to $v,w$ or $u,w$ or three of them. Now we change the label of w to 1, and we obtain a ${\gamma}_{t\left\{R2\right\}}$-function for G with weight 3.
- 2.3.
- There exist 2 vertices $v,u$ with label 3 and label 1, respectively, for which the induced subgraph by $v,u$ is ${K}_{2}$. By this assumption each vertex with label 0 is adjacent to v, but there maybe exist some vertices (none of them) which are adjacent to u. Now we change the label v to 2, and we obtain a ${\gamma}_{t\left\{R2\right\}}$-function for G with weight 3.

- 3.1.
- There exist 3 vertices $v,u,w$ with label 1 for which the induced subgraph by $v,u,w$ is the cycle ${C}_{3}$ or a path ${P}_{3}$. If each vertex with label 0 is adjacent to $v,w$ or $u,w$, then by changing the label w to 2, we obtain a ${\gamma}_{t\left\{R3\right\}}$-function for G with weight 4.If some vertices with label 0 are adjacent to $v,u$, some of them are adjacent to $v,w$ and the other are adjacent $u,w$, then by changing two vertices of $v,u,w$ to label 2, we obtain a ${\gamma}_{t\left\{R3\right\}}$-function for G with weight 5.
- 3.2.
- There exist 2 vertices $v,u$ with label 2 and label 1, respectively, for which the induced subgraph by $v,u$ is ${K}_{2}$. By this assumption each vertex with label 0 is adjacent to v, but there maybe exist some vertices (none of them) which are adjacent to u. Now we change the label v to 3, and we obtain a ${\gamma}_{t\left\{R3\right\}}$-function for G with weight 4. Therefore $4\le {\gamma}_{t\left\{R3\right\}}\left(G\right)\le 5$. □

**Observation**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

## 5. Large Total Roman $\left\{3\right\}$-domination Number

**Theorem**

**9.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Observation**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

## 6. Complexity

**Total Roman $\left\{3\right\}$-domination problem TR3DP**.**Instance**: Graph $G=(V,E)$, and a positive integer $k\le \left|V\right|$.**Question**: Does G have a total Roman $\left\{3\right\}$-domination of weight at most k?

**Theorem**

**13.**

**Proof.**

## 7. Open Problems

**Problems**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Shao, Z.; Mojdeh, D.A.; Volkmann, L. Total Roman {3}-domination in Graphs. *Symmetry* **2020**, *12*, 268.
https://doi.org/10.3390/sym12020268

**AMA Style**

Shao Z, Mojdeh DA, Volkmann L. Total Roman {3}-domination in Graphs. *Symmetry*. 2020; 12(2):268.
https://doi.org/10.3390/sym12020268

**Chicago/Turabian Style**

Shao, Zehui, Doost Ali Mojdeh, and Lutz Volkmann. 2020. "Total Roman {3}-domination in Graphs" *Symmetry* 12, no. 2: 268.
https://doi.org/10.3390/sym12020268