# Computing Open Locating-Dominating Number of Some Rotationally-Symmetric Graphs

## Abstract

**:**

## 1. Introduction and Preliminaries

**Theorem**

**1**

## 2. Main Results

#### Cycle Graphs

**Lemma**

**1.**

**Proof.**

**Case 1:**When $n=3q$, and $q\ge 2$; let $S=\{{e}_{3p+1},{e}_{3p+2}\mid p=0,\dots ,q-1\}$.

**Case 2:**When $n=3q+1$, and $q\ge 2$; let $S=\{{e}_{3p},{e}_{3p+1}\mid p=0,\dots ,q-1\}\cup \{{e}_{3q-1},{e}_{3q}\}$.

**Case 3:**When $n=3q+2$, and $q\ge 2$; let $S=\{{e}_{3p},{e}_{3p+1}\mid p=0,\dots ,q-1\}\cup \{{e}_{3q},{e}_{3q+1}\}$.

**Conjecture**

**1.**

## 3. Exact Values

#### 3.1. The Graph of Prism ${\mathcal{D}}_{N}$

**Theorem**

**2.**

**Proof.**

#### 3.2. The Prism Related Graph ${\mathcal{D}}_{N}^{*}$

**Theorem**

**3.**

**Proof.**

## 4. Exact Values of Convex Polytopes

#### 4.1. The Graph of Convex Polytopes ${\mathcal{R}}_{n}$

**Theorem**

**4.**

**Proof.**

#### 4.2. The Graph of Convex Polytope ${\mathcal{H}}_{n}$

**Theorem**

**5.**

**Proof.**

**Remark.**

## 5. Upper Bounds

#### 5.1. The Graph of Convex Polytope ${\mathcal{S}}_{n}$

**Theorem**

**6.**

**Proof.**

**Case 1:**When $n=3q$, and $q\ge 2$. Let $S=\{{f}_{p}\mid p=0,\dots ,n-1\}\cup \{{h}_{3p+1},{h}_{3p+2}\mid p=0,$$\dots ,q-1\}$.

**Case 2:**When $n=3q+1$, and $q\ge 2$. Let $S=\{{f}_{p}\mid p=0,\dots ,n-1\}\cup \{{h}_{3p},{h}_{3p+1}\mid p=0,$$\dots ,q-1\}\cup \{{h}_{3q-1},{h}_{3q}\}$.

**Case 3:**When $n=3q+2$, and $q\ge 2$. Let $S=\{{f}_{p}\mid p=0,\dots ,n-1\}\cup \{{h}_{3p},{h}_{3p+1}\mid p=0,$$\dots ,q-1\}\cup \{{h}_{3q},{h}_{3q+1}\}$.

#### 5.2. The Graph of Convex Polytope ${\mathcal{R}}_{n}^{\prime}$

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 11.**(

**a**) The graph of convex polytope ${\mathcal{A}}_{n}$. (

**b**) The graph of convex polytope ${\mathcal{Q}}_{n}$. (

**c**) The web graph ${\mathbb{W}}_{n}$. (

**d**) The graph of convex polytope ${\mathcal{U}}_{n}$.

n | v | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ | v | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ |
---|---|---|---|---|

$3q$ | ${e}_{3p+1}$ | $\left\{{e}_{3p+2}\right\}$ | ${e}_{3p+2}$ | $\left\{{e}_{3p+1}\right\}$ |

${e}_{3p+3}(p=0,\dots ,q-2)$ | $\{{e}_{3p+2},{e}_{3p+4}\}(p=0,\dots ,q-2)$ | ${e}_{0}$ | $\{{e}_{1},{e}_{3q-1}\}$ | |

$3q+1$ | ${e}_{3p+1}(p=0,\dots ,q-2)$ | $\left\{{e}_{3p}\right\}(p=0,\dots ,q-2)$ | ${e}_{3p+2}$ | $\{{e}_{3p+1},{e}_{3p+3}\}$ |

${e}_{3p+3}$ | $\left\{{e}_{3p+4}\right\}$ | ${e}_{3q-2}$ | $\{{e}_{3q-3},{e}_{3q-1}\}$ | |

${e}_{3q}$ | $\{{e}_{3q-1},{e}_{0}\}$ | ${e}_{0}$ | $\{{e}_{1},{e}_{3q}\}$ | |

$3q+2$ | ${e}_{3p+1}$ | $\left\{{e}_{3p}\right\}$ | ${e}_{3p+2}$ | $\{{e}_{3p+1},{e}_{3p+3}\}$ |

${e}_{3p+3}(p=0,\dots ,q-2)$ | $\left\{{e}_{3p+4}\right\}(p=0,\dots ,q-2)$ | ${e}_{0}$ | $\{{e}_{1},{e}_{3q+1}\}$ | |

${e}_{3q+1}$ | $\{{e}_{3q},{e}_{0}\}$ |

n | $\mathit{v}\in \mathit{V}$ | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ | $\mathit{v}\in \mathit{V}$ | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ |
---|---|---|---|---|

${C}_{13}$ | ${e}_{1}$ | $\left\{{e}_{2}\right\}$ | ${e}_{2}$ | $\left\{{e}_{1}\right\}$ |

${e}_{3}$ | $\{{e}_{2},{e}_{4}\}$ | ${e}_{4}$ | $\left\{{e}_{5}\right\}$ | |

${e}_{5}$ | $\left\{{e}_{4}\right\}$ | ${e}_{6}$ | $\{{e}_{5},{e}_{7}\}$ | |

${e}_{7}$ | $\left\{{e}_{8}\right\}$ | ${e}_{8}$ | $\left\{{e}_{7}\right\}$ | |

${e}_{9}$ | $\{{e}_{8},{e}_{10}\}$ | ${e}_{10}$ | $\left\{{e}_{11}\right\}$ | |

${e}_{11}$ | $\{{e}_{10},{e}_{12}\}$ | ${e}_{12}$ | $\left\{{e}_{11}\right\}$ | |

${e}_{13}$ | $\{{e}_{12},{e}_{1}\}$ |

v | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ |
---|---|

${e}_{p}$ | $\{{e}_{p-1},{e}_{p+1}\}$ |

${f}_{p}$ | $\left\{{e}_{p}\right\}$ |

${g}_{p}$ | $\left\{{h}_{p}\right\}$ |

${h}_{p}$ | $\{{i}_{p},{i}_{p+1}\}$ |

${i}_{p}$ | $\{{h}_{p-1},{h}_{p}\}$ |

${j}_{p}$ | $\left\{{i}_{p}\right\}$ |

v | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ |
---|---|

${e}_{p}$ | $\{{e}_{p-1},{e}_{p+1}\}$ |

${f}_{p}$ | $\left\{{e}_{p}\right\}$ |

${g}_{p}$ | $\left\{{h}_{p}\right\}$ |

${h}_{p}$ | $\{{i}_{p},{i}_{p+1}\}$ |

${i}_{p}$ | $\{{h}_{p-1},{h}_{p}\}$ |

${j}_{p}$ | $\left\{{i}_{p}\right\}$ |

${k}_{p}$ | $\left\{{l}_{p}\right\}$ |

${l}_{p}$ | $\{{l}_{p+1},{l}_{p-1}\}$ |

n | v | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ |
---|---|---|

$3q$ | ${e}_{p}$ | $\{{f}_{p-1},{f}_{p}\}(p=0,\dots ,n-1$) |

${f}_{p}$ | $\{{f}_{p-1},{f}_{p+1}\}(p=0,\dots ,n-1$) | |

${g}_{3p}$ | $\left\{{f}_{3p}\right\}$ | |

${g}_{3p+1}$ | $\{{f}_{3p+1},{h}_{3p+1}\}$ | |

${g}_{3p+2}$ | $\{{f}_{3p+2},{h}_{3p+2}\}$ | |

${h}_{3p+1}$ | $\left\{{h}_{3p+2}\right\}$ | |

${h}_{3p+2}$ | $\left\{{h}_{3p+1}\right\}$ | |

${h}_{3p+3}(p=0,\dots ,q-2)$ | $\{{h}_{3p+2},{h}_{3p+4}\}(p=0,\dots ,q-2)$ | |

${h}_{0}$ | $\{{h}_{1},{h}_{3q-1}\}$ | |

$3q+1$ | ${e}_{p}$ | $\{{f}_{p-1},{f}_{p}\}(p=0,\dots ,n-1$) |

${f}_{p}$ | $\{{f}_{p-1},{f}_{p+1}\}(p=0,\dots ,n-1$) | |

${g}_{3p}(p=0,\dots ,q)$ | $\{{f}_{3p},{h}_{3p}\}(p=0,\dots ,q)$ | |

${g}_{3p+1}$ | $\{{f}_{3p+1},{h}_{3p+1}\}$ | |

${g}_{3p+2}(p=0,\dots ,q-2)$ | $\left\{{f}_{3p+2}\right\}(p=0,\dots ,q-2)$ | |

${h}_{3p+1}(p=0,\dots ,q-2)$ | $\left\{{h}_{3p}\right\}(p=0,\dots ,q-2)$ | |

${h}_{3p+2}$ | $\{{h}_{3p+1},{h}_{3p+3}\}$ | |

${h}_{3p+3}$ | $\left\{{h}_{3p+4}\right\}$ | |

${g}_{3q-1}$ | $\{{f}_{3q-1},{h}_{3q-1}\}$ | |

${h}_{3q-2}$ | $\{{h}_{3q-3},{h}_{3q-1}\}$ | |

${h}_{3q}$ | $\{{h}_{3q-1},{h}_{0}\}$ | |

${h}_{0}$ | $\{{h}_{1},{h}_{3q}\}$ | |

$3q+2$ | ${e}_{p}$ | $\{{f}_{p-1},{f}_{p}\}(p=0,\dots ,n-1$) |

${f}_{p}$ | $\{{f}_{p-1},{f}_{p+1}\}(p=0,\dots ,n-1$) | |

${g}_{3p}(p=0,\dots ,q)$ | $\{{f}_{3p},{h}_{3p}\}(p=0,\dots ,q)$ | |

${g}_{3p+1}(p=0,\dots ,q)$ | $\{{f}_{3p+1},{h}_{3p+1}\}(p=0,\dots ,q)$ | |

${g}_{3p+2}$ | $\left\{{f}_{3p+2}\right\}$ | |

${h}_{3p+1}$ | $\left\{{h}_{3p}\right\}$ | |

${h}_{3p+2}$ | $\{{h}_{3p+1},{h}_{3p+3}\}$ | |

${h}_{3p+3}$ | $\left\{{h}_{3p+4}\right\}$ | |

${h}_{0}$ | $\{{h}_{3q+1},{h}_{1}\}$ | |

${h}_{3q+1}$ | $\{{h}_{3q},{h}_{0}\}$ |

v | $\mathit{S}\cap \mathit{N}\left(\mathit{v}\right)$ |
---|---|

${e}_{p}$ | $\{{e}_{p-1},{e}_{p+1}\}$ |

${f}_{p}$ | $\{{e}_{p},{e}_{p+1}\}$ |

${g}_{p}$ | $\{{h}_{p},{h}_{p+1}\}$ |

${h}_{p}$ | $\{{h}_{p-1},{h}_{p+1}\}$ |

${i}_{p}$ | $\left\{{h}_{p}\right\}$ |

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Raza, H.
Computing Open Locating-Dominating Number of Some Rotationally-Symmetric Graphs. *Mathematics* **2021**, *9*, 1415.
https://doi.org/10.3390/math9121415

**AMA Style**

Raza H.
Computing Open Locating-Dominating Number of Some Rotationally-Symmetric Graphs. *Mathematics*. 2021; 9(12):1415.
https://doi.org/10.3390/math9121415

**Chicago/Turabian Style**

Raza, Hassan.
2021. "Computing Open Locating-Dominating Number of Some Rotationally-Symmetric Graphs" *Mathematics* 9, no. 12: 1415.
https://doi.org/10.3390/math9121415