# A Vulnerability Measure of k-Uniform Linear Hypergraphs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Rupture Degree of k-Uniform Linear Hypergraphs

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proposition**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

**Remark**

**1.**

**Corollary**

**2.**

**Remark**

**2.**

**Corollary**

**3.**

## 3. The Rupture Degree of k-Uniform Hypertrees

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Theorem**

**3.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**4.**

**Remark**

**3.**

## 4. A Recursive Algorithm for Computing the r-Set of k-Uniform Hypertrees

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Example**

**1.**

- 1.
- $X\leftarrow \{{u}_{1},{u}_{2},{u}_{6},{u}_{9},{u}_{10}\};$
- 2.
- $X\leftarrow \{{u}_{1},{u}_{2},{u}_{6},{u}_{9},{u}_{10}\}\cup \{{u}_{3},{u}_{8}\};$
- 3.
- $X\leftarrow \{{u}_{1},{u}_{2},{u}_{6},{u}_{9},{u}_{10},{u}_{3},{u}_{8}\}\cup \{{u}_{4},{u}_{7}\};$
- 4.
- $X\leftarrow \{{u}_{1},{u}_{2},{u}_{6},{u}_{9},{u}_{10},{u}_{3},{u}_{8},{u}_{4},{u}_{7}\}\cup \left\{{u}_{5}\right\}.$

**Example**

**2.**

- (1)
- $X\leftarrow \{1,9\};$
- (2)
- $X\leftarrow \{1,9\}\cup \{2,3,4,6,7,10,11\};$
- (3)
- $X\leftarrow \{1,9,2,3,4,6,7,10,11\}\cup \{5,8\}.$

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MVC | Minimum vertex cover |

NPC | NP-complete |

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**MDPI and ACS Style**

Zhao, N.; Zhao, H.; Li, Y.
A Vulnerability Measure of *k*-Uniform Linear Hypergraphs. *Symmetry* **2023**, *15*, 1187.
https://doi.org/10.3390/sym15061187

**AMA Style**

Zhao N, Zhao H, Li Y.
A Vulnerability Measure of *k*-Uniform Linear Hypergraphs. *Symmetry*. 2023; 15(6):1187.
https://doi.org/10.3390/sym15061187

**Chicago/Turabian Style**

Zhao, Ning, Haixing Zhao, and Yinkui Li.
2023. "A Vulnerability Measure of *k*-Uniform Linear Hypergraphs" *Symmetry* 15, no. 6: 1187.
https://doi.org/10.3390/sym15061187