# Classification of Efficient Total Domination Sets of Circulant Graphs of Degree 5

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Theorem**

**1**

**Theorem**

**2.**

- (1)
- G has a type-I efficient total dominating set if and only if $\{\pm a,\pm b\}\equiv \{1,2,3,4\}$$\left(\mathrm{mod}\phantom{\rule{4.pt}{0ex}}5\right)$. In this case, any type-I efficient total dominating set of G is the subgroup $\langle 5\rangle $ of ${\mathbb{Z}}_{10m}$ or its coset.
- (2)
- There is no type-II efficient total dominating set of G.
- (3)
- G has a type-III efficient total dominating set if and only if m is odd and $\{\pm a,\pm b\}\equiv \{1,3,7,9\}\left(\mathrm{mod}\phantom{\rule{4.pt}{0ex}}10\right)$. In this case, any type-III efficient total dominating set of G is $(\langle 10\rangle +i)\cup (\langle 10\rangle +j)$, where i is even and j is odd with $j\ne i+5\left(\mathrm{mod}\phantom{\rule{4.pt}{0ex}}10\right)$.

## 3. Constructions

## 4. Proof of Main Theorem

**Proposition**

**1**

**Remark**

**1.**

**Lemma**

**1.**

**Proof of**

**Lemma 1.**

**Proof of**

**Theorem 2(1).**

**Lemma**

**2.**

- (1)
- ${N}_{2}\left(x\right)\cap D=\varnothing $,
- (2)
- $x+2\alpha +2\beta \notin D$,
- (3)
- $x+3\alpha +5m\notin D$, and
- (4)
- $x+4\alpha \notin D$.

**Proof of**

**Lemma 2.**

**Corollary**

**1.**

- (1)
- $\{x\pm 2a\pm 2b\}\cap D=\varnothing $,
- (2)
- $\{x\pm 3a+5m,x\pm 3b+5m\}\cap D=\varnothing $ and
- (3)
- $\{x\pm 4a,x\pm 4b\}\cap D=\varnothing $.

**Proof of**

**Corollary 1.**

**Lemma**

**3.**

**Proof of**

**Lemma 3.**

**Lemma**

**4.**

**Proof of**

**Lemma 4.**

**Lemma**

**5.**

**Proof of**

**Lemma 5.**

**Lemma**

**6.**

- (1)
- all of $x+\alpha +2\beta +5m,x+3\alpha +\beta ,x+2\alpha -\beta +5m$ belong to ${D}_{2}$.
- (2)
- all of $x\pm (2\alpha +\beta ),x\pm (3\alpha +\beta ),x\pm (2\alpha -\beta )+5m,x\pm (3\alpha -\beta )+5m,x\pm (\alpha -2\beta ),x\pm (\alpha -3\beta ),x\pm (\alpha +2\beta )+5m,x\pm (\alpha +3\beta )+5m$ belong to ${D}_{2}$. Furthermore, these elements are all elements in $\bigcup _{i=1}^{4}\left(({N}_{i}\left(x\right)\cup {N}_{i}(x+5m))\cap {D}_{2}\right)\setminus \{x,x+5m\}$.
- (3)
- All of $x\pm 5\alpha ,x\pm 5\alpha +5m,x\pm 5\beta ,x\pm 5\beta +5m$ belong to ${D}_{2}$.

**Proof of**

**Lemma 6.**

**Lemma**

**7.**

**Proof of**

**Lemma 7.**

**Proof of**

**Theorem 2(2).**

**Lemma**

**8.**

**Proof of**

**Lemma 8.**

**Lemma**

**9.**

**Proof of**

**Lemma 9.**

**Lemma**

**10.**

- (1)
- $x+3\alpha +\beta +5m,x+4\alpha +2\beta \in {D}_{3}$,
- (2)
- $x+\alpha -2\beta +5m,x+2\alpha -2\beta +5m,x+3\alpha -\beta \in {D}_{3}$.
- (3)
- for any integers $r,s$, $x+r(2\alpha +\beta )+s(\alpha -2\beta )+5(r+s)m,x+r(2\alpha +\beta )+s(\alpha -2\beta )+\alpha +5(r+s)m\in \phantom{\rule{3.33333pt}{0ex}}{D}_{3}$.

**Proof of**

**Lemma 10.**

**Lemma**

**11.**

- (1)
- the set $H\cup (H+a)$ is a subset of ${D}_{3}$ and $H\cap (H+a)=\varnothing $,
- (2)
- ${\mathbb{Z}}_{10m}$ is generated by a and b, and the index of H in ${\mathbb{Z}}_{10m}$ is exactly 10.

**Proof of**

**Lemma 11.**

**Proof of**

**Theorem 2(3).**

## Author Contributions

## Funding

## Conflicts of Interest

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Soo Kwon, Y.; Young Sohn, M.
Classification of Efficient Total Domination Sets of Circulant Graphs of Degree 5. *Symmetry* **2020**, *12*, 1944.
https://doi.org/10.3390/sym12121944

**AMA Style**

Soo Kwon Y, Young Sohn M.
Classification of Efficient Total Domination Sets of Circulant Graphs of Degree 5. *Symmetry*. 2020; 12(12):1944.
https://doi.org/10.3390/sym12121944

**Chicago/Turabian Style**

Soo Kwon, Young, and Moo Young Sohn.
2020. "Classification of Efficient Total Domination Sets of Circulant Graphs of Degree 5" *Symmetry* 12, no. 12: 1944.
https://doi.org/10.3390/sym12121944