# Total Domination in Rooted Product Graphs

^{*}

## Abstract

**:**

## 1. Closed Formulas for the Total Domination Number

**Lemma**

**1.**

- (i)
- ${\gamma}_{t}(H-\left\{v\right\})\ge {\gamma}_{t}\left(H\right)-1.$
- (ii)
- If ${\gamma}_{t}(H-\left\{v\right\})={\gamma}_{t}\left(H\right)-1,$ then the following statements hold.
- (a)
- $N\left(v\right)\cap S=\u2300$ for every ${\gamma}_{t}(H-\left\{v\right\})$-set S.
- (b)
- There exists a ${\gamma}_{t}\left(H\right)$-set S such that $v\notin S$.

- (iii)
- If ${\gamma}_{t}(H-\left\{v\right\})>{\gamma}_{t}\left(H\right)$, then $v\in S$ for every ${\gamma}_{t}\left(H\right)$-set S.

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

- (i)
- $|{S}_{x}|\ge {\gamma}_{t}\left(H\right)-1.$
- (ii)
- If $|{S}_{x}|={\gamma}_{t}\left(H\right)-1$, then $N\left(x\right)\cap {S}_{x}=\u2300$.

**Proof.**

**Theorem**

**1.**

**Proof.**

**Lemma**

**4.**

- (i)
- If ${\mathcal{B}}_{S}\cap S\ne \varnothing $, then ${\gamma}_{t}\left(G{\circ}_{v}H\right)=n\left(G\right)({\gamma}_{t}\left(H\right)-1).$
- (ii)
- If ${\mathcal{B}}_{S}\cap S=\u2300$, then ${\gamma}_{t}(H-\left\{v\right\})={\gamma}_{t}\left(H\right)-1$, and as a consequence,$$\gamma \left(G\right)+n\left(G\right)({\gamma}_{t}\left(H\right)-1)\le {\gamma}_{t}\left(G{\circ}_{v}H\right)\le {\gamma}_{t}\left(G\right)+n\left(G\right)({\gamma}_{t}\left(H\right)-1).$$

**Proof.**

**Theorem**

**2.**

**Proof.**

- (a)
- ${S}_{x}$ is a ${\gamma}_{t}\left({H}_{x}\right)$-set such that $x\notin {S}_{x}$.
- (b)
- ${S}_{x}$ is not a TDS of ${H}_{x}$ and $x\in {S}_{x}$.

- -
- ${\mathcal{A}}_{S}\subseteq X$.
- -
- For any $x\in {\mathcal{A}}_{S}^{\prime}$ which satisfies condition (a) and $N\left(x\right)\cap S\cap V\left(G\right)=\u2300$, we choose one vertex $y\in N\left(x\right)\cap V\left(G\right)$ and set $y\in X$.
- -
- For any $x\in {\mathcal{A}}_{S}^{\prime \prime}$ with $N\left(x\right)\cap S\cap V\left(G\right)=\u2300$, we choose one vertex $y\in N\left(x\right)\cap V\left(G\right)$ and set $y\in X$.

**Example**

**1.**

- ${\gamma}_{t}\left(G{\circ}_{{v}^{\prime}}{H}_{2}\right)=3n\left(G\right)=n\left(G\right)({\gamma}_{t}\left({H}_{2}\right)-1).$
- ${\gamma}_{t}\left(G{\circ}_{v}{H}_{2}\right)=\gamma \left(G\right)+3n\left(G\right)=\gamma \left(G\right)+n\left(G\right)({\gamma}_{t}\left({H}_{2}\right)-1).$
- ${\gamma}_{t}\left(G{\circ}_{v}{H}_{1}\right)={\gamma}_{t}\left(G\right)+2n\left(G\right)={\gamma}_{t}\left(G\right)+n\left(G\right)({\gamma}_{t}\left({H}_{1}\right)-1).$
- ${\gamma}_{t}\left(G{\circ}_{{v}^{\prime}}{H}_{1}\right)={\gamma}_{t}\left(G{\circ}_{{v}^{\prime \prime}}{H}_{1}\right)=3n\left(G\right)=n\left(G\right){\gamma}_{t}\left({H}_{1}\right).$

**Theorem**

**3.**

- (i)
- ${\gamma}_{t}\left(G{\circ}_{v}H\right)=n\left(G\right)({\gamma}_{t}\left(H\right)-1).$
- (ii)
- v is a universal vertex of H or ${\gamma}_{t}(H-N\left[v\right])={\gamma}_{t}\left(H\right)-2.$

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**4.**

- (i)
- ${\gamma}_{t}\left(G{\circ}_{v}H\right)=\gamma \left(G\right)+n\left(G\right)({\gamma}_{t}\left(H\right)-1).$
- (ii)
- ${\gamma}_{t}(H-N\left[v\right])={\gamma}_{t}(H-\left\{v\right\})={\gamma}_{t}\left(H\right)-1$, and in addition, ${\gamma}_{t}\left(G\right)=\gamma \left(G\right)$ or there exists a ${\gamma}_{t}\left(H\right)$-set D such that $v\in D$.

**Proof.**

**Theorem**

**5.**

- (i)
- ${\gamma}_{t}\left(G{\circ}_{v}H\right)={\gamma}_{t}\left(G\right)+n\left(G\right)({\gamma}_{t}\left(H\right)-1).$
- (ii)
- ${\gamma}_{t}(H-\left\{v\right\})={\gamma}_{t}\left(H\right)-1$ and $v\notin D$ for every ${\gamma}_{t}\left(H\right)$-set D.

**Proof.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 2. An Observation on the Domination Number

**Theorem**

**6.**

**Lemma**

**7.**

**Lemma**

**8.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The set of black-coloured vertices forms a ${\gamma}_{t}\left({H}_{i}\right)$-set for $i\in \{1,2\}$. The set $\{{v}^{\prime},{v}^{\prime \prime}\}$ forms a ${\gamma}_{t}({H}_{1}-\left\{v\right\})$-set, while $\{a,b,c\}$ forms a ${\gamma}_{t}({H}_{2}-\left\{v\right\})$-set.

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**MDPI and ACS Style**

Cabrera Martínez, A.; Rodríguez-Velázquez, J.A.
Total Domination in Rooted Product Graphs. *Symmetry* **2020**, *12*, 1929.
https://doi.org/10.3390/sym12111929

**AMA Style**

Cabrera Martínez A, Rodríguez-Velázquez JA.
Total Domination in Rooted Product Graphs. *Symmetry*. 2020; 12(11):1929.
https://doi.org/10.3390/sym12111929

**Chicago/Turabian Style**

Cabrera Martínez, Abel, and Juan A. Rodríguez-Velázquez.
2020. "Total Domination in Rooted Product Graphs" *Symmetry* 12, no. 11: 1929.
https://doi.org/10.3390/sym12111929