# Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Complexity

**Total Roman {3}-Domination Problem TR3DP.****Instance:**Graph $G=(V,E)$, and a positive integer m.**Question:**Does G have a total Roman {3}-function with weight at most m?

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof of Lemma 3.**

**Lemma**

**4.**

**Proof of Lemma 4.**

**Claim**

**1.**

**Proof of Claim 1.**

**Claim**

**2.**

**Proof of Claim 2.**

**Lemma**

**5.**

**Proof of Lemma 5.**

**Theorem**

**1.**

## 3. A Linear-Time Algorithm for Total Roman {3}-Domination in Trees

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Lemma**

**6.**

**Lemma**

**7.**

**Proof of Lemma 7.**

Algorithm 1 Counting ${\gamma}_{t\{R3\}}$ in trees. |

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DF | Dominating function |

DSP | Dominating set problem |

TRDF | Total Roman dominating function |

R3DF | Roman $\{3\}$-domination |

TR3DF | Total Roman $\{3\}$-domination |

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**MDPI and ACS Style**

Liu, X.; Jiang, H.; Wu, P.; Shao, Z.
Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees. *Mathematics* **2021**, *9*, 293.
https://doi.org/10.3390/math9030293

**AMA Style**

Liu X, Jiang H, Wu P, Shao Z.
Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees. *Mathematics*. 2021; 9(3):293.
https://doi.org/10.3390/math9030293

**Chicago/Turabian Style**

Liu, Xinyue, Huiqin Jiang, Pu Wu, and Zehui Shao.
2021. "Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees" *Mathematics* 9, no. 3: 293.
https://doi.org/10.3390/math9030293