# Total Weak Roman Domination in Graphs

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Notation

## 3. General Results

**Proposition**

**1.**

- (i)
- $\gamma \left(G\right)\le {\gamma}_{r}\left(G\right)\le {\gamma}_{tr}\left(G\right)\le {\gamma}_{tR}\left(G\right)\le 2{\gamma}_{t}\left(G\right)$.
- (ii)
- ${\gamma}_{t}\left(G\right)\le {\gamma}_{tr}\left(G\right)\le {\gamma}_{st}\left(G\right)$.

**Proof.**

**Problem**

**1.**

- (i)
- ${\gamma}_{tr}\left(G\right)={\gamma}_{t}\left(G\right)$.
- (ii)
- ${\gamma}_{tr}\left(G\right)={\gamma}_{r}\left(G\right)$.
- (iii)
- ${\gamma}_{tr}\left(G\right)={\gamma}_{st}\left(G\right)$.
- (iv)
- ${\gamma}_{tr}\left(G\right)={\gamma}_{tR}\left(G\right)$.

**Theorem**

**1.**

- (a)
- ${\gamma}_{tr}\left(G\right)={\gamma}_{r}\left(G\right)$.
- (b)
- There exists a ${\gamma}_{r}\left(G\right)$-function $f({V}_{0},{V}_{1},{V}_{2})$ such that ${V}_{1}=\varnothing $ and ${V}_{2}$ is a total dominating set.
- (c)
- ${\gamma}_{r}\left(G\right)=2{\gamma}_{t}\left(G\right)$.

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

- (i)
- ${\gamma}_{tr}\left(G\right)=\gamma \left(G\right)+1$.
- (ii)
- ${\gamma}_{st}\left(G\right)=\gamma \left(G\right)+1$.

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Remark**

**1.**

- (i)
- ${\gamma}_{tr}\left({P}_{n}\right)={\gamma}_{st}\left({P}_{n}\right)=\u2308\frac{5(n-2)}{7}\u2309+2.$
- (ii)
- ${\gamma}_{tr}\left({C}_{n}\right)={\gamma}_{st}\left({C}_{n}\right)=\u2308\frac{5n}{7}\u2309$.

**Proposition**

**2.**

**Proof.**

**Corollary**

**2.**

- If G is a Hamiltonian graph, then ${\gamma}_{tr}\left(G\right)\le \u2308\frac{5n}{7}\u2309.$
- If G has a Hamiltonian path, then ${\gamma}_{tr}\left(G\right)\le \u2308\frac{5(n-2)}{7}\u2309+2.$

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Theorem**

**7.**

**Theorem**

**8.**

**Theorem**

**9**

**.**If G is a graph with no isolated vertex, then ${\gamma}_{tR}\left(G\right)\le 3\gamma \left(G\right)$. Furthermore, if ${\gamma}_{tR}\left(G\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}3\gamma \left(G\right)$, then every $\gamma \left(G\right)$-set is a 2-packing of G.

**Theorem**

**10.**

**Theorem**

**11.**

**Proof.**

- (a)
- For every $x\in {V}_{2}\cap S$, choose a vertex $u\in ({V}_{0}\cap N\left(x\right))\backslash S$ if it exists, and label it as ${f}^{\prime}\left(u\right)=1$.
- (b)
- For every $x\in {V}_{1}\cap S$, choose a vertex $u\in epn(x,{V}_{1}\cup {V}_{2})\backslash S$ if it exists, otherwise choose a vertex $u\in ({V}_{0}\cap N\left(x\right))\backslash S$ (if exists) and label it as ${f}^{\prime}\left(u\right)=1$.
- (c)
- For every vertex $x\in {V}_{0}\cap S$, ${f}^{\prime}\left(x\right)=1$.
- (d)
- For any other vertex u not previously labelled, ${f}^{\prime}\left(u\right)=f\left(u\right)$.

**Corollary**

**5.**

**Theorem**

**12**

**.**For any connected graph $G\ncong {C}_{5}$ of order n and minimum degree $\delta \left(G\right)\ge 2$,

**Theorem**

**13.**

**Theorem**

**14.**

**Proof.**

- (a)
- For every $x\in {V}_{2}$, choose a vertex $u\in {V}_{0}\cap N\left(x\right)$ and label it as ${g}^{\prime}\left(u\right)=1$.
- (b)
- For every $x\in {V}_{1}$, choose a vertex $u\in epn(x,{V}_{1}\cup {V}_{2})$ if it exists, otherwise choose a vertex $u\in {V}_{0}\cap N\left(x\right)$ (if exists) and label it as ${g}^{\prime}\left(u\right)=1$.
- (c)
- For any other vertex u not previously labelled, ${g}^{\prime}\left(u\right)=g\left(u\right)$.

**Corollary**

**6.**

**Theorem**

**15**

**.**If G is a graph of order n with no isolated vertex, then ${\gamma}_{tR}\left(G\right)\le 2{\gamma}_{R}\left(G\right)-1$. Furthermore, ${\gamma}_{tR}\left(G\right)=2{\gamma}_{R}\left(G\right)-1$ if and only if $\Delta \left(G\right)=n-1$.

**Theorem**

**16.**

**Proof.**

## 4. General Bounds

**Theorem**

**17.**

**Proof.**

**Theorem**

**18**

**.**Let G be a graph of order n. Then ${\gamma}_{st}\left(G\right)=n$ if and only if $V\left(G\right)\backslash \left(L\right(G)\cup S(G\left)\right)$ is an independent set.

**Theorem**

**19**

- ${\gamma}_{st}\left(G\right)={\gamma}_{t}\left(G\right)$.
- ${\gamma}_{st}\left(G\right)=2$.
- G has two universal vertices.

**Theorem**

**20.**

- (i)
- The following statements are equivalent.
- (a)
- ${\gamma}_{tr}\left(G\right)=2$.
- (b)
- ${\gamma}_{tr}\left(G\right)={\gamma}_{t}\left(G\right)$.
- (c)
- ${\gamma}_{st}\left(G\right)={\gamma}_{t}\left(G\right)$.
- (d)
- G has two universal vertices.

- (ii)
- ${\gamma}_{tr}\left(G\right)=n$ if and only if G is ${K}_{1,(n-1)/2}^{*}$ or $H\odot {N}_{1}$ for some connected graph H.

**Proof.**

**Theorem**

**21.**

**Proof.**

## 5. Rooted Product Graphs and Computational Complexity

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

- (i)
- $f\left(N\right(v\left)\right)=0$ for every ${\gamma}_{tr}(H-\left\{v\right\})$-function f.
- (ii)
- There exists a ${\gamma}_{tr}\left(H\right)$-function ${h}_{0}$ such that ${h}_{0}\left(v\right)=0$.
- (iii)
- There exists a ${\gamma}_{tr}\left(H\right)$-function ${h}_{1}$ such that ${h}_{1}\left(v\right)=1$.

**Proof.**

**Corollary**

**7.**

- If $g\left(v\right)=0$ for every ${\gamma}_{tr}\left(H\right)$-function g, then ${\gamma}_{tr}(H-\left\{v\right\})\in \{{\gamma}_{tr}\left(H\right),{\gamma}_{tr}\left(H\right)-1\}$.
- If $h\left(v\right)>0$ for every ${\gamma}_{tr}\left(H\right)$-function h, then ${\gamma}_{tr}(H-\left\{v\right\})\ge {\gamma}_{tr}\left(H\right)-1$.

**Proposition**

**3.**

**Proof.**

**Theorem**

**22.**

**Proposition**

**4.**

**Theorem**

**23.**

- (a)
- ${\mathcal{C}}_{f}\ne \varnothing $ for any ${\gamma}_{tr}\left(G{\circ}_{v}H\right)$-function f.
- (b)
- ${\gamma}_{tr}(H-\left\{v\right\})={\gamma}_{tr}\left(H\right)-2$.

**Proof.**

**Theorem**

**24.**

- (i)
- ${\gamma}_{tr}\left(G{\circ}_{v}H\right)=n{\gamma}_{tr}\left(H\right)$.
- (ii)
- $n({\gamma}_{tr}\left(H\right)-1)\le {\gamma}_{tr}\left(G{\circ}_{v}H\right)\le {\gamma}_{tr}\left(G\right)+n({\gamma}_{tr}\left(H\right)-1)$.
- (iii)
- $2\gamma \left(G\right)+n({\gamma}_{tr}\left(H\right)-2)\le {\gamma}_{tr}\left(G{\circ}_{v}H\right)\le {\gamma}_{tr}\left(G\right)+n({\gamma}_{tr}\left(H\right)-2)$.
- (iv)
- ${\gamma}_{t}\left(G\right)+n({\gamma}_{tr}\left(H\right)-2)\le {\gamma}_{tr}\left(G{\circ}_{v}H\right)\le {\gamma}_{tr}\left(G\right)+n({\gamma}_{tr}\left(H\right)-2)$.

**Proof.**

**Theorem**

**25.**

**Proof.**

**Theorem**

**26.**

**Proof.**

**Conjecture.**

**Theorem**

**27.**

**Proof.**

**Corollary**

**8.**

**Proof.**

**Theorem**

**28.**

- (i)
- If $g\left(v\right)=0$ for every ${\gamma}_{tr}\left(H\right)$-function g, then ${\gamma}_{tr}\left(G{\circ}_{v}H\right)=n{\gamma}_{tr}\left(H\right)$.
- (ii)
- If $g\left(v\right)>0$ for every ${\gamma}_{tr}\left(H\right)$-function g, then ${\gamma}_{tr}\left(G{\circ}_{v}H\right)\in \{n{\gamma}_{tr}\left(H\right),n({\gamma}_{tr}\left(H\right)-1)\}$.

**Proof.**

**Corollary**

**9.**

**Theorem**

**29.**

**Proof.**

**Theorem**

**30.**

**Theorem**

**31.**

**Proof.**

## 6. Conclusions and Open Problems

- The work proved several new theorems, thanks to which we have shown the close relationship that exists between the total weak Roman domination number and other domination parameters such as the (total) domination number, secure (total) domination number, weak Roman domination number, (total) Roman domination number and 2-packing number.
- We obtained general bounds and discussed some extreme cases.
- In a specific section of the paper, we focused on the case of rooted product graphs and we obtained closed formulas and tight bounds for the total weak Roman domination number of these graphs.
- Through the results obtained on rooted product graphs, we have shown that the problem of finding the total weak Roman domination number of a graph is NP-hard.

- (a)
- We have shown that if G is a $\{{K}_{1,3},{K}_{1,3}+e\}$-free graph with no isolated vertex, then ${\gamma}_{tr}\left(G\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\gamma}_{st}\left(G\right)$. We conjecture that these two parameters also coincide for lexicographic product graphs, and we propose the general problem of characterizing all graphs for which the equality holds.
- (b)
- We have shown that ${\gamma}_{tr}\left(G\right)=\gamma \left(G\right)+1$ if and only if ${\gamma}_{st}\left(G\right)=\gamma \left(G\right)+1$. Therefore, the problem of characterizing all graphs with ${\gamma}_{st}\left(G\right)=\gamma \left(G\right)+1$ is an open problem, which is a particular case of problem (a).
- (c)
- We have shown that ${\gamma}_{tr}\left(G\right)\le {\gamma}_{t}\left(G\right)+\gamma \left(G\right)$ and ${\gamma}_{tr}\left(G\right)\le {\gamma}_{r}\left(G\right)+\gamma \left(G\right).$ We propose the problem of characterizing all graphs for which these equalities hold; or providing necessary or sufficient conditions for achieving them.
- (d)
- Since the problem of finding ${\gamma}_{tr}\left(G\right)$ is NP-hard, we consider the following question. Is there a polynomial-time algorithm for finding ${\gamma}_{tr}\left(T\right)$ for any tree T of order n?

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Graph G which satisfies ${\gamma}_{t}\left(G\right)=4$ (

**a**), ${\gamma}_{r}\left(G\right)=5$ (

**b**), ${\gamma}_{R}\left(G\right)=6$ (

**c**), ${\gamma}_{tr}\left(G\right)=7$ (

**d**), ${\gamma}_{tR}\left(G\right)=8$ (

**e**) and ${\gamma}_{st}\left(G\right)=9$ (

**f**).

**Figure 4.**The graph ${C}_{3}\square {P}_{3}$ satisifies ${\gamma}_{tr}({C}_{3}\square {P}_{3})=5>3={\gamma}_{r}({C}_{3}\square {P}_{3})$, while ${\gamma}_{t}({C}_{3}\square {P}_{3})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\gamma ({C}_{3}\square {P}_{3})\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}3$.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cabrera Martínez, A.; Montejano, L.P.; Rodríguez-Velázquez, J.A.
Total Weak Roman Domination in Graphs. *Symmetry* **2019**, *11*, 831.
https://doi.org/10.3390/sym11060831

**AMA Style**

Cabrera Martínez A, Montejano LP, Rodríguez-Velázquez JA.
Total Weak Roman Domination in Graphs. *Symmetry*. 2019; 11(6):831.
https://doi.org/10.3390/sym11060831

**Chicago/Turabian Style**

Cabrera Martínez, Abel, Luis P. Montejano, and Juan A. Rodríguez-Velázquez.
2019. "Total Weak Roman Domination in Graphs" *Symmetry* 11, no. 6: 831.
https://doi.org/10.3390/sym11060831