# Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. An Integer Linear Programming Model

## 3. The Exact Values

#### 3.1. The Graph of Convex Polytope ${H}_{n}$

#### 3.1.1. Construction

**Problem**

**1.**

- (1)
- (2)
- (3)

#### 3.1.2. Rotational Symmetry of the Convex Polytopes

**Theorem**

**2.**

#### 3.1.3. Binary Locating-Dominating Number of ${H}_{n}$

**Theorem**

**3.**

**Proof.**

- Case 1:
- When $n=3m$.In order to show S to be a binary locating-dominating set, we need to show that the neighborhoods of all vertices in $V\backslash S$ are non-empty and distinct. Table 1 shows these neighborhoods and their intersections. Although some formulas for some intersections can be somewhat similar, but they are distinct.
- Case 2:
- When $n=3m+1$.As in the previous case, the neighborhoods of all vertices in $V\backslash S$ are non-empty and distinct shown in Table 1.
- Case 3:
- When $n=3m+2$.Similar to the previous two cases, Table 1 shows that the neighborhoods of all vertices in $V\backslash S$ are non-empty and distinct.

#### 3.2. The Graph of Convex Polytope ${H}_{n}^{\prime}$

#### 3.2.1. Construction

#### 3.2.2. Binary Locating-Dominating Number of ${H}_{n}^{\prime}$

**Theorem**

**4.**

**Proof.**

## 4. Tight Upper Bounds

#### 4.1. The Graph of Convex Polytope ${S}_{n}$

**Theorem**

**5.**

**Proof.**

- Case 1:
- When $n=5m$.Table 2 depicts all vertices in $V\backslash $S and the intersections of their closed neighborhoods with S. From the second column, we can see that all these intersections are nonempty and distinct. Thus, for any two vertices $u,v\in V\backslash $S, we have $S\bigcap N[v]\ne S\bigcap N[u]\ne \xd8$. This shows that S is a binary locating-dominating set of ${S}_{n}$.
- Case 2:
- When $n=5m+1$.Similar to the argument in Case 1, we see from Table 2 that all the intersections are nonempty and distinct. This shows that S is a binary locating-dominating set for ${S}_{n}$, if $n=5m+1$.
- Case 3:
- When $n=5m+2$.Similar to the argument in Case 1 and Case 2, we see from Table 2 that all the intersections are nonempty and distinct. This shows that S is a binary locating-dominating set for ${S}_{n}$, if $n=5m+2$.Thus, from the above discussion, we can say that Case 4 and Case 5 are analogous to above mentioned cases.

#### 4.2. The Graph of Convex Polytope ${B}_{n}$

**Theorem**

**6.**

**Proof.**

#### 4.3. The Graph of Convex Polytope ${T}_{n}$

**Theorem**

**7.**

**Proof.**

- Case 1:
- When $n=5m$.In order to show S to be a binary locating-dominating set, we need to show that the neighborhoods of all vertices in $V\backslash $S are non-empty and distinct. Table 3 shows these neighborhoods and their intersections. Although some formulas for some intersections can be somewhat similar, but they are distinct.
- Case 2:
- When $n=5m+1$.As in the previous case, the the neighborhoods of all vertices in $V\backslash $S are non-empty and distinct shown in Table 3. Thus, from the above discussion, we can say that Case 3, Case 4 and Case 5 are analogous to the above-mentioned cases.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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n | $\mathit{v}\in \mathit{V}\backslash $S | $\mathit{S}\cap \mathit{N}[\mathit{v}]$ | $\mathit{v}\in \mathit{V}\backslash $S | $\mathit{S}\cap \mathit{N}[\mathit{v}]$ |
---|---|---|---|---|

$3m$ | ${s}_{3j}$ | $\{{s}_{3j+1},{t}_{3j}\}$ | ${s}_{3j+2}$ | $\{{s}_{3j+1}\}$ |

${t}_{3j+1}$ | $\{{s}_{3j+1},{u}_{3j+1}\}$ | ${t}_{3j+2}$ | $\{{u}_{3j+1}\}$ | |

${u}_{3j}$ | $\{{t}_{3j}\}$ | ${u}_{3j+2}$ | $\{{t}_{3(j+1)},{v}_{3j+2}\}$ | |

${v}_{3j}$ | $\{{w}_{3j+1}\}$ | ${v}_{3j+1}$ | $\{{w}_{3j+1},{u}_{3j+1}\}$ | |

${w}_{3j}$ | $\{{v}_{3j-1}\}$ | ${w}_{3j+2}$ | $\{{v}_{3j+2},{x}_{3j+2}\}$ | |

${x}_{3j}$ | $\{{y}_{3j}\}$ | ${x}_{3j+1}$ | $\{{y}_{3j},{w}_{3j+1}\}$ | |

${y}_{3j+1}$ | $\{{x}_{3j+2}\}$ | ${y}_{3j+2}$ | $\{{x}_{3j+2},{z}_{3j+2}\}$ | |

${z}_{3j}$ | $\{{y}_{3j},{z}_{3j-1}\}$ | ${z}_{3j+1}$ | $\{{z}_{3j+2}\}$ | |

$3m+1$ | ${s}_{3j+1}$ | $\{{s}_{3j+2}\}$ | ${s}_{3(j+1)}$ | $\{{s}_{3j+2},{t}_{3(j+1)}\}$ |

${t}_{3j+1}$ | $\{{u}_{3j+1}\}$ | ${t}_{3j+2}$ | $\{{u}_{3j+1},{s}_{3j+2}\}$ | |

${u}_{3j}$ | $\{{t}_{3j},{v}_{3j}\}$ | ${u}_{3j+2}$ | $\{{t}_{3(j+1)}\}$ | |

${v}_{3j+1}$ | $\{{u}_{3j+1},{w}_{3j+2}\}$ | ${v}_{3j+2}$ | $\{{w}_{3j+2}\}$ | |

${w}_{3j+1}$ | $\{{v}_{3j},{x}_{3j+1}\}$ | ${w}_{3(j+1)}$ | $\{{v}_{3(j+1)}\}$ | |

${x}_{3j+2}$ | $\{{w}_{3j+2},{y}_{3j+2}\}$ | ${x}_{3(j+1)}$ | $\{{y}_{3j+2}\}$ | |

${y}_{3j}$ | $\{{x}_{3j+1},{z}_{3j}\}$ | ${y}_{3j+1}$ | $\{{x}_{3j+1}\}$ | |

${z}_{3j+2}$ | $\{{y}_{3j+2},{z}_{3j+3}\}$ | ${z}_{3j+1}$ | $\{{z}_{3j}\}$ | |

${s}_{0}$ | $\{{t}_{0}\}$ | ${u}_{3m}$ | $\{{t}_{0},{t}_{3m},{v}_{3m}\}$ | |

${w}_{0}$ | $\{{v}_{0},{v}_{3m}\}$ | ${x}_{0}$ | $\{{y}_{3m}\}$ | |

${z}_{3m}$ | $\{{y}_{3m},{z}_{0}\}$ | |||

$3m+2$ | ${s}_{3j+1}$ | $\{{s}_{3j},{t}_{3j+1}\}$ | ${s}_{3j+2}$ | $\{{s}_{3(j+1)}\}$ |

${t}_{3j+2}$ | $\{{u}_{3j+2}\}$ | ${t}_{3(j+1)}$ | $\{{s}_{3(j+1)},{u}_{3j+2}\}$ | |

${u}_{3j}$ | $\{{t}_{3j+1},{v}_{3j}\}$ | ${u}_{3j+1}$ | $\{{t}_{3j+1}\}$ | |

${v}_{3j+1}$ | $\{{w}_{3j+2}\}$ | ${v}_{3j+2}$ | $\{{u}_{3j+2},{w}_{3j+2}\}$ | |

${w}_{3j}$ | $\{{v}_{3j}\}$ | ${w}_{3j+1}$ | $\{{v}_{3j},{x}_{3j+1}\}$ | |

${x}_{3j+2}$ | $\{{w}_{3j+2},{y}_{3j+2}\}$ | ${x}_{3(j+1)}$ | $\{{y}_{3j+2}\}$ | |

${y}_{3j+1}$ | $\{{x}_{3j+1},{z}_{3j+1}\}$ | ${y}_{3j}$ | $\{{x}_{3j+1}\}$ | |

${z}_{3j}$ | $\{{z}_{3j+1}\}$ | ${z}_{3j+2}$ | $\{{y}_{3j+2},{z}_{3j+1}\}$ | |

${s}_{3m+1}$ | $\{{s}_{3m},{s}_{0},{t}_{3m+1}\}$ | ${t}_{0}$ | $\{{s}_{0}\}$ | |

${u}_{3m+1}$ | $\{{t}_{3m+1}\}$ | ${u}_{3m}$ | $\{{t}_{3m+1},{v}_{3m}\}$ | |

${v}_{3m+1}$ | $\{{w}_{3m+1}\}$ | ${w}_{3m}$ | $\{{v}_{3m}\}$ | |

${x}_{0}$ | $\{{y}_{3m+1}\}$ | ${x}_{3m+1}$ | $\{{w}_{3m+1},{y}_{3m+1}\}$ | |

${y}_{3m}$ | $\{{z}_{3m}\}$ | ${z}_{3m+1}$ | $\{{y}_{3m+1},{z}_{3m}\}$ |

n | $\mathit{v}\in \mathit{V}\backslash $S | $\mathit{S}\cap \mathit{N}[\mathit{v}]$ | $\mathit{v}\in \mathit{V}\backslash $S | $\mathit{S}\cap \mathit{N}[\mathit{v}]$ |
---|---|---|---|---|

$5m$ | ${w}_{5j}$ | $\{{x}_{5j},{x}_{5(j-1)+4}\}$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ |

${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ | ${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | |

${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | ${y}_{5j}$ | $\{{x}_{5j}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | |||

$5m+1$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+4},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m}\}$ | |

${y}_{5m}$ | $\{{x}_{5m},{z}_{5m}\}$ | |||

$5m+2$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+4},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m+1}\}$ | |

${w}_{5m+1}$ | $\{{x}_{5m},{x}_{5m+1}\}$ | ${y}_{5m}$ | $\{{x}_{5m}\}$ | |

${y}_{5m+1}$ | $\{{x}_{5m+1},{z}_{5m+1}\}$ | ${z}_{5m}$ | $\{{z}_{5m+1}\}$ | |

$5m+3$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+4},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m+2}\}$ | |

${w}_{5m+1}$ | $\{{x}_{5m},{x}_{5m+1}\}$ | ${w}_{5m+2}$ | $\{{x}_{5m+1},{x}_{5m+2}\}$ | |

${y}_{5m}$ | $\{{x}_{5m},{z}_{5m}\}$ | ${y}_{5m+1}$ | $\{{x}_{5m+1}\}$ | |

${y}_{5m+2}$ | $\{{x}_{5m+2},{z}_{5m+2}\}$ | ${z}_{5m+1}$ | $\{{z}_{5m},{z}_{5m+2}\}$ | |

$5m+4$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+4},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m+3}\}$ | |

${w}_{5m+1}$ | $\{{x}_{5m},{x}_{5m+1}\}$ | ${w}_{5m+2}$ | $\{{x}_{5m+1},{x}_{5m+2}\}$ | |

${w}_{5m+3}$ | $\{{x}_{5m+2},{x}_{5m+3}\}$ | ${y}_{5m}$ | $\{{x}_{5m}\}$ | |

${y}_{5m+1}$ | $\{{x}_{5m+1},{z}_{5m+1}\}$ | ${y}_{5m+2}$ | $\{{x}_{5m+2}\}$ | |

${y}_{5m+3}$ | $\{{x}_{5m+3},{z}_{5m+3}\}$ | ${z}_{5m}$ | $\{{z}_{5m+1}\}$ | |

${z}_{5m+2}$ | $\{{z}_{5m+1},{z}_{5m+3}\}$ |

n | $\mathit{v}\in \mathit{V}\backslash $S | $\mathit{S}\cap \mathit{N}[\mathit{v}]$ | $\mathit{v}\in \mathit{V}\backslash $S | $\mathit{S}\cap \mathit{N}[\mathit{v}]$ |
---|---|---|---|---|

$5m$ | ${w}_{5j}$ | $\{{x}_{5j},{x}_{5(j-1)+4}\}$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ |

${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ | ${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | |

${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | ${y}_{5j}$ | $\{{x}_{5j},{z}_{5j+1}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2},{z}_{5j+3}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | |||

$5m+1$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+3},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j},{z}_{5j+1}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2},{z}_{5j+3}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m}\}$ | |

${y}_{5m}$ | $\{{x}_{5m},{z}_{5m}\}$ | |||

$5m+2$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+3},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j},{z}_{5j+1}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2},{z}_{5j+3}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m+1}\}$ | |

${w}_{5m+1}$ | $\{{x}_{5m},{x}_{5m+1}\}$ | ${y}_{5m}$ | $\{{x}_{5m},{z}_{5m}\}$ | |

${y}_{5m+1}$ | $\{{x}_{5m+1},{z}_{5m}\}$ | ${z}_{5m}$ | $\{{z}_{5m+1}\}$ | |

$5m+3$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+3},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j},{z}_{5j+1}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2},{z}_{5j+3}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m+2}\}$ | |

${w}_{5m+1}$ | $\{{x}_{5m},{x}_{5m+1}\}$ | ${w}_{5m+2}$ | $\{{x}_{5m+1},{x}_{5m+2}\}$ | |

${y}_{5m}$ | $\{{x}_{5m},{z}_{5m}\}$ | ${y}_{5m+1}$ | $\{{x}_{5m+1},{z}_{5m+2}\}$ | |

${y}_{5m+2}$ | $\{{x}_{5m+2},{z}_{5m+2}\}$ | ${z}_{5m+1}$ | $\{{z}_{5m},{z}_{5m+2}\}$ | |

$5m+4$ | ${w}_{5j+1}$ | $\{{x}_{5j},{x}_{5j+1}\}$ | ${w}_{5j+2}$ | $\{{x}_{5j+1},{x}_{5j+2}\}$ |

${w}_{5j+3}$ | $\{{x}_{5j+2},{x}_{5j+3}\}$ | ${w}_{5j+4}$ | $\{{x}_{5j+3},{x}_{5j+4}\}$ | |

${w}_{5(j+1)}$ | $\{{x}_{5j+3},{x}_{5(j+1)}\}$ | ${y}_{5j}$ | $\{{x}_{5j},{z}_{5j+1}\}$ | |

${y}_{5j+1}$ | $\{{x}_{5j+1},{z}_{5j+1}\}$ | ${y}_{5j+2}$ | $\{{x}_{5j+2},{z}_{5j+3}\}$ | |

${y}_{5j+3}$ | $\{{x}_{5j+3},{z}_{5j+3}\}$ | ${y}_{5j+4}$ | $\{{x}_{5j+4}\}$ | |

${z}_{5j}$ | $\{{z}_{5j+1}\}$ | ${z}_{5j+2}$ | $\{{z}_{5j+1},{z}_{5j+3}\}$ | |

${z}_{5j+4}$ | $\{{z}_{5j+3}\}$ | ${w}_{0}$ | $\{{x}_{0},{x}_{5m+2}\}$ | |

${w}_{5m+1}$ | $\{{x}_{5m},{x}_{5m+1}\}$ | ${w}_{5m+2}$ | $\{{x}_{5m+1},{x}_{5m+2}\}$ | |

${w}_{5m+3}$ | $\{{x}_{5m+2},{x}_{5m+3}\}$ | ${y}_{5m}$ | $\{{x}_{5m},{z}_{5m}\}$ | |

${y}_{5m+1}$ | $\{{x}_{5m+1},{z}_{5m+1}\}$ | ${y}_{5m+2}$ | $\{{x}_{5m+2},{z}_{5m+3}\}$ | |

${y}_{5m+3}$ | $\{{x}_{5m+3},{z}_{5m+3}\}$ | ${z}_{5m}$ | $\{{z}_{5m+1}\}$ | |

${z}_{5m+2}$ | $\{{z}_{5m+1},{z}_{5m+3}\}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Raza, H.; Hayat, S.; Pan, X.-F.
Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes. *Symmetry* **2018**, *10*, 727.
https://doi.org/10.3390/sym10120727

**AMA Style**

Raza H, Hayat S, Pan X-F.
Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes. *Symmetry*. 2018; 10(12):727.
https://doi.org/10.3390/sym10120727

**Chicago/Turabian Style**

Raza, Hassan, Sakander Hayat, and Xiang-Feng Pan.
2018. "Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes" *Symmetry* 10, no. 12: 727.
https://doi.org/10.3390/sym10120727