# Geodetic Number of Powers of Cycles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Lemmas

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**2.**

**Proof.**

## 3. The Geodetic Number of Power of Cycles

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Case**

**1.**

**Case**

**2.**

**Lemma**

**5.**

**Proof.**

**Case**

**1.**

- (a)
- ${I}_{{C}_{n}^{k}}[{v}_{0},{v}_{qk+1}]$ covers all vertices that lie on the shortest path from ${v}_{0}$ to ${v}_{qk+1}$ in ${C}_{n}$.
- (b)
- ${I}_{{C}_{n}^{k}}[{v}_{qk+1},{v}_{2qk+2}]$ covers all vertices that lie on the shortest path from ${v}_{qk+1}$ to ${v}_{2qk+2}$ in ${C}_{n}$.
- (c)
- ${I}_{{C}_{n}^{k}}[{v}_{(2q-1)k+r-1},{v}_{0}]$ covers all vertices that lie on the shortest path from ${v}_{0}$ to ${v}_{(2q-1)k+r-1}$ in ${C}_{n}$.

**Case**

**2.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Case**

**1.**

**Case**

**2.**

**Lemma**

**10.**

**Proof.**

**Case**

**1.**

**Case**

**2.**

## 4. Conclusions

**Theorem**

**1.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Type 1: $7=3+3+1$ | Type 2: $7=2+2+3$ |
---|---|

$0-1-4-7$ | $0-2-5-7$ |

$0-3-4-7$ | $0-3-5-7$ |

$0-3-6-7$ | $0-2-4-7$ |

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**MDPI and ACS Style**

Abudayah, M.; Alomari, O.; Al Ezeh, H.
Geodetic Number of Powers of Cycles. *Symmetry* **2018**, *10*, 592.
https://doi.org/10.3390/sym10110592

**AMA Style**

Abudayah M, Alomari O, Al Ezeh H.
Geodetic Number of Powers of Cycles. *Symmetry*. 2018; 10(11):592.
https://doi.org/10.3390/sym10110592

**Chicago/Turabian Style**

Abudayah, Mohammad, Omar Alomari, and Hassan Al Ezeh.
2018. "Geodetic Number of Powers of Cycles" *Symmetry* 10, no. 11: 592.
https://doi.org/10.3390/sym10110592