# On 3-Restricted Edge Connectivity of Replacement Product Graphs

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3**

## 2. Bounds on 3-Restricted Edge Connectivity

**Lemma**

**1**

**Lemma**

**2.**

**Lemma**

**3**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Case 1.**$X\subset {X}_{i}$. In this case, $\overline{X}=(V\left(G\right)\backslash {X}_{i})\cup ({X}_{i}\backslash X)$. Thus,

**Case 2.**$X\not\subset {X}_{i}$. Since $|\overline{X}|\ge |X|$, it follows that $\overline{X}\not\subset {X}_{i}$. Thus, ${X}_{i}\cap X\ne \varnothing $, ${X}_{i}\cap \overline{X}\ne \varnothing $ and there are at least two sets of ${X}_{j}$ and ${X}_{k}$ other than ${X}_{i}$ such that ${X}_{j}\subset X$ and ${X}_{k}\subset \overline{X}$. Since $\kappa ({G}_{1}-{x}_{i})\ge \kappa \left({G}_{1}\right)-1$, there exists at least $\kappa \left({G}_{1}\right)-1$ internally vertex-disjoint paths between any two vertices u and v, where $u,v\in V({G}_{1}-{x}_{i})$. By the definition of G, there exists at least $\kappa \left({G}_{1}\right)-1$ internally vertex-disjoint paths between ${X}_{j}$ and ${X}_{k}$ in $G-{X}_{i}$. Each of these paths has at least one edge of F. Hence

**Case 3.**$X\subset \left({X}_{i}\cup {X}_{j}\right)$ or $\overline{X}\subset \left({X}_{i}\cup {X}_{j}\right)$, say $X\subset \left({X}_{i}\cup {X}_{j}\right)$. In this case, $\overline{X}=(V\left(G\right)\backslash ({X}_{i}\cup {X}_{j}))\cup ({X}_{j}\backslash ({X}_{j}\cap X))\cup ({X}_{i}\backslash ({X}_{i}\cap X))$ and

**Case 4.**$X\not\subset \left({X}_{i}\cup {X}_{j}\right)$ and $\overline{X}\not\subset \left({X}_{i}\cup {X}_{j}\right)$. In this case, there are at least two sets of ${X}_{k}$ and ${X}_{l}$ other than ${X}_{i}$ and ${X}_{j}$ such that ${X}_{k}\subset X$ and ${X}_{l}\subset \overline{X}$. Since $\kappa ({G}_{1}-{x}_{i}-{x}_{j})\ge \kappa \left({G}_{1}\right)-2$, ${G}_{1}-\{{x}_{i},{x}_{j}\}$ has at least $\kappa \left({G}_{1}\right)-2$ internally vertex-disjoint paths between vertices ${x}_{k}$ and ${x}_{l}$. These paths corresponds to the same number of internally vertex-disjoint paths between ${X}_{k}$ and ${X}_{l}$ in $G-{X}_{i}-{X}_{j}$, each of which contains at least one edge of F. Therefore,

**Lemma**

**8.**

**Proof.**

**Case 1.**$X\subset {X}_{i}$. In this case, we have $\overline{X}=(V\left(G\right)\backslash {X}_{i})\cup ({X}_{i}\backslash X)$. Since every vertex of X is incident with exactly one edge not contained in ${x}_{i}{G}_{2}$, it follows that

**Case 2.**$X\not\subset {X}_{i}$. Since $|\overline{X}|\ge |X|$, it follows that $\overline{X}\not\subset {X}_{i}$. So, ${X}_{i}\cap X\ne \varnothing $, ${X}_{i}\cap \overline{X}\ne \varnothing $ and there are at least two sets of ${X}_{j}$ and ${X}_{k}$ other than ${X}_{i}$ such that ${X}_{j}\subset X$ and ${X}_{k}\subset \overline{X}$. Since $\kappa ({G}_{1}-{x}_{i})\ge \kappa \left({G}_{1}\right)-1$, there exists at least $\kappa \left({G}_{1}\right)-1$ internally vertex-disjoint paths between any two vertices ${x}_{k}$ and ${x}_{l}$. Correspondingly, G has at least $\kappa \left({G}_{1}\right)-1$ internally vertex-disjoint paths between ${X}_{j}$ and ${X}_{k}$ in $G-{X}_{i}$, and each of these paths contains at least one edge of F. Hence,

**Case 3.**$X\subset \left({X}_{i}\cup {X}_{j}\right)$ or $\overline{X}\subset \left({X}_{i}\cup {X}_{j}\right)$, say $X\subset \left({X}_{i}\cup {X}_{j}\right)$. In this case, $\overline{X}=(V\left(G\right)\backslash ({X}_{i}\cup {X}_{j}))\cup ({X}_{j}\backslash ({X}_{j}\cap X))\cup ({X}_{i}\backslash ({X}_{i}\cap X))$. For simplicity, we assume without loss of generality that $|{X}_{i}\cap X|\le |{X}_{j}\cap X|$. Then,

**Case 4.**$X\not\subset \left({X}_{i}\cup {X}_{j}\right)$ and $\overline{X}\not\subset \left({X}_{i}\cup {X}_{j}\right)$. So, there are two sets of ${X}_{k}$ and ${X}_{l}$ other than ${X}_{i}$ and ${X}_{j}$ such that ${X}_{k}\subset X$ and ${X}_{l}\subset \overline{X}$. Since $\kappa ({G}_{1}-{x}_{i}-{x}_{j})\ge \kappa \left({G}_{1}\right)-2$, ${G}_{1}$ has at least $\kappa \left({G}_{1}\right)-2$ internally vertex-disjoint paths between vertices ${x}_{k}$ and ${x}_{l}$. Correspondingly, G has at least $\kappa \left({G}_{1}\right)-2$ internally vertex-disjoint paths between ${X}_{k}$ and ${X}_{l}$ in $G-{X}_{i}-{X}_{j}$, and each of these paths contains at least one edge of F. Hence,

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 3. Optimization of 3-Restricted Edge Connectivity

**Theorem**

**5.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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$\mathit{g}\left({\mathit{G}}_{2}\right)=3$ | $\mathit{g}\left({\mathit{G}}_{2}\right)\ge 4$ | |
---|---|---|

${k}_{1}=3$ | ${\lambda}_{3}({G}_{1}\circledR {G}_{2})=\lambda \left({G}_{1}\right)$ | $--$ |

$4\le {k}_{1}\le 5$ | ${\lambda}_{3}({G}_{1}\circledR {G}_{2})=\lambda \left({G}_{1}\right)$ | $\begin{array}{c}min\{\lambda \left({G}_{1}\right),\kappa \left({G}_{1}\right)+1\}\le \\ {\lambda}_{3}({G}_{1}\circledR {G}_{2})\le \lambda \left({G}_{1}\right)\end{array}$ |

${k}_{1}\ge 6$ | $\begin{array}{c}min\{\lambda \left({G}_{1}\right),{\lambda}_{3}\left({G}_{2}\right)+3,{k}_{1}+\lambda \left({G}_{2}\right)-\\ 2,\kappa \left({G}_{1}\right)+\lambda \left({G}_{2}\right)-1,\lambda \left({G}_{2}\right)+{\lambda}_{2}\left({G}_{2}\right)+\\ 1,3\lambda \left({G}_{2}\right)\}\le {\lambda}_{3}({G}_{1}\circledR {G}_{2})\le \\ min\{\lambda \left({G}_{1}\right),3{k}_{2}-3\}\end{array}$ | $\begin{array}{c}min\{\lambda \left({G}_{1}\right),{\lambda}_{3}\left({G}_{2}\right)+3,{k}_{1}+\lambda \left({G}_{2}\right)-\\ 2,\kappa \left({G}_{1}\right)+\lambda \left({G}_{2}\right)-1,\lambda \left({G}_{2}\right)+{\lambda}_{2}\left({G}_{2}\right)+\\ 1,3\lambda \left({G}_{2}\right)\}\le {\lambda}_{3}({G}_{1}\circledR {G}_{2})\le \\ min\{\lambda \left({G}_{1}\right),3{k}_{2}-1\}\end{array}$ |

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**MDPI and ACS Style**

Cui, Y.; Ou, J.; Liu, S.
On 3-Restricted Edge Connectivity of Replacement Product Graphs. *Axioms* **2023**, *12*, 504.
https://doi.org/10.3390/axioms12050504

**AMA Style**

Cui Y, Ou J, Liu S.
On 3-Restricted Edge Connectivity of Replacement Product Graphs. *Axioms*. 2023; 12(5):504.
https://doi.org/10.3390/axioms12050504

**Chicago/Turabian Style**

Cui, Yilan, Jianping Ou, and Saihua Liu.
2023. "On 3-Restricted Edge Connectivity of Replacement Product Graphs" *Axioms* 12, no. 5: 504.
https://doi.org/10.3390/axioms12050504