# Shortest Path Solution of Trapezoidal Fuzzy Neutrosophic Graph Based on Circle-Breaking Algorithm

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis

#### 2.1. NS

**Definition 1.**

#### 2.2. TrFNN

**Definition 2.**

**Definition 3.**

**Definition 4.**

#### 2.3. Ranking Function

**Definition 5.**

**Definition 6.**

**Definition 7.**

- if
- $S\left({\tilde{n}}_{1}\right)>S\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}>{\tilde{n}}_{2}$
- if
- $S\left({\tilde{n}}_{1}\right)<S\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}<{\tilde{n}}_{2}$
- if
- $S\left({\tilde{n}}_{1}\right)=S\left({\tilde{n}}_{2}\right)$, then
- ①
- if$H\left({\tilde{n}}_{1}\right)>H\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}>{\tilde{n}}_{2}$
- ②
- if$H\left({\tilde{n}}_{1}\right)<H\left({\tilde{n}}_{2}\right)$, then${\tilde{n}}_{1}<{\tilde{n}}_{2}$

## 3. Neutrosophic Graph Theory

## 4. Method for Solving SPP of Trapezoidal Fuzzy Neutrosophic Graph Based on Circle-Breaking Algorithm

- Step 1:
- Arbitrarily define a closed circle in the trapezoidal fuzzy neutrosophic graph, and find the two paths p
_{1}and p_{2}surrounding the closed circle, whereby p_{1}and p_{2}have a common starting node recorded as N_{0}and a common ending node recorded as N_{1}. - Step 2:
- According to Equation (5), all edges of each path are summed. The trapezoidal fuzzy numbers ${\tilde{n}}_{p1}$ and ${\tilde{n}}_{p2}$ are then obtained, which represent the two paths.
- Step 3:
- Obtain the score function value $S({\tilde{n}}_{p1})$ and exact function value $H({\tilde{n}}_{p1})$ of ${\tilde{n}}_{p1}$, as well as the score function value $S({\tilde{n}}_{p2})$ and exact function value $H({\tilde{n}}_{p2})$ of ${\tilde{n}}_{p2}$.
- Step 4:
- Compare the sizes of ${\tilde{n}}_{p1}$ and ${\tilde{n}}_{p2}$ according to the ranking function, and find and delete the edge in the larger path whose vertex is N1.
- Step 5:
- Determine whether a closed circle still exists on the map. If so, go to Step 1; if not, the algorithm terminates. At this time, only one path exists from the starting node to the ending node in the neutrosophic graph, which is the shortest path.

Algorithm 1 Circle-breaking Algorithm. |

for i = N to 1 while(in-degree _{[i]} > 1)if(${\tilde{n}}_{p1}>{\tilde{n}}_{p2}$) Delete the last edge in p _{1}else Delete the last edge in p _{2}end while end for |

## 5. Case Study and Comparative Analysis

#### 5.1. Case Analysis

**Example 1.**

_{1}, the neutrosophic graph in Figure 4 can be obtained.

_{3}, the neutrosophic graph depicted in Figure 6 can be obtained.

_{5}is deleted, and the neutrosophic graph is obtained, as depicted in Figure 8.

**Example 2.**

#### 5.2. Comparative Analysis of Different Algorithms

_{s}represents the start point and V

_{e}represents the end point.

_{2}is the smallest, and the shortest path is $\u2780\text{}\to \text{}\u2781\text{}\to \text{}\u2784\text{}\to \text{}\u2785$.

_{(1369)}.

Algorithm 2 Multi-threaded realization of circle-breaking algorithm |

Class Circle-breaking extends Thread{ public Circle-breaking(int k){ for(int i = (k − 1)*N/10, i < k*N/10, i++) { while(in-degree[i] > 1) { if(${\tilde{n}}_{p1}>{\tilde{n}}_{p2}$) { Delete the last edge in p _{1}} else { Delete the last edge in p _{2}} } } } } public static void main(String[] args) { Circle-breaking[] c = new Circle-breaking [10]; for(int i = 0; i < c.length; i++) { c[i] = new Circle-breaking(i + 1); c[i].start(); } } |

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Edges | Trapezoidal Fuzzy Neutrosophic Distances |
---|---|

(1, 2) | <(0.1, 0.2, 0.3, 0.5), (0.2, 0.3, 0.5, 0.6), (0.4, 0.5, 0.6, 0.8)> |

(1, 3) | <(0.2, 0.4, 0.5, 0.7), (0.3, 0.5, 0.6, 0.9), (0.1, 0.2, 0.3, 0.4)> |

(2, 4) | <(0.3, 0.4, 0.6, 0.7), (0.1, 0.2, 0.3, 0.5), (0.3, 0.5, 0.7, 0.9)> |

(2, 5) | <(0.1, 0.3, 0.4, 0.5), (0.3, 0.4, 0.5, 0.7), (0.2, 0.3, 0.6, 0.7)> |

(3, 4) | <(0.2, 0.3, 0.5, 0.6), (0.2, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.8)> |

(3, 5) | <(0.3, 0.6, 0.7, 0.8), (0.1, 0.2, 0.3, 0.4), (0.1, 0.4, 0.5, 0.6)> |

(4, 6) | <(0.4, 0.6, 0.8, 0.9), (0.2, 0.4, 0.5, 0.6), (0.1, 0.3, 0.4, 0.5)> |

(5, 6) | <(0.2, 0.3, 0.4, 0.5), (0.3, 0.4, 0.5, 0.6), (0.1, 0.3, 0.5, 0.6)> |

Edges | Trapezoidal Fuzzy Neutrosophic Distance |
---|---|

(1, 2) | <(0.1, 0.3, 0.4, 0.5), (0.3, 0.5, 0.6, 0.7), (0.1, 0.3, 0.4, 0.6)> |

(1, 3) | <(0.2, 0.4, 0.5, 0.7), (0.3, 0.5, 0.6, 0.9), (0.1, 0.2, 0.3, 0.4)> |

(1, 4) | <(0.2, 0.3, 0.4, 0.6), (0.2, 0.4, 0.5, 0.6), (0.4, 0.5, 0.7, 0.8)> |

(2, 4) | <(0.1, 0.3, 0.4, 0.5), (0.3, 0.4, 0.5, 0.7), (0.2, 0.3, 0.6, 0.7)> |

(2, 5) | <(0.4, 0.5, 0.7, 0.8), (0.2, 0.3, 0.5, 0.6), (0.1, 0.2, 0.3, 0.5)> |

(3, 4) | <(0.3, 0.6, 0.7, 0.8), (0.1, 0.2, 0.3, 0.4), (0.1, 0.4, 0.5, 0.6)> |

(3, 6) | <(0.1, 0.2, 0.3, 0.4), (0.2, 0.4, 0.5, 0.6), (0.4, 0.5, 0.6, 0.7)> |

(4, 5) | <(0.2, 0.3, 0.4, 0.5), (0.3, 0.4, 0.5, 0.6), (0.1, 0.3, 0.5, 0.6)> |

(4, 6) | <(0.4, 0.6, 0.7, 0.9), (0.1, 0.2, 0.4, 0.5), (0.1, 0.3, 0.4, 0.6)> |

(4, 7) | <(0.1, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.7), (0.4, 0.5, 0.7, 0.8)> |

(5, 7) | <(0.2, 0.3, 0.5, 0.6), (0.2, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.8)> |

(5, 8) | <(0.3, 0.5, 0.6, 0.7), (0.2, 0.3, 0.5, 0.6), (0.1, 0.2, 0.4, 0.5)> |

(6, 7) | <(0.3, 0.4, 0.6, 0.7), (0.1, 0.2, 0.3, 0.5), (0.3, 0.5, 0.7, 0.9)> |

(6, 9) | <(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.4, 0.5), (0.5, 0.6, 0.7, 0.9)> |

(7, 8) | <(0.1, 0.2, 0.3, 0.5), (0.2, 0.3, 0.5, 0.6), (0.4, 0.5, 0.6, 0.8)> |

(7, 9) | <(0.4, 0.6, 0.8, 0.9), (0.2, 0.4, 0.5, 0.6), (0.1, 0.3, 0.4, 0.5)> |

(8, 9) | <(0.2, 0.3, 0.5, 0.6), (0.1, 0.2, 0.4, 0.5), (0.5, 0.6, 0.7, 0.8)> |

Nodes | Distance of Shortest Path | Shortest Path |
---|---|---|

② | $\langle \left(0.1,0.2,0.3,0.5\right),\text{}\left(0.2,0.3,0.5,0.6\right),\text{}\left(0.4,0.5,0.6,0.8\right)\rangle $ | $\u2780\text{}\to \text{}\u2781$ |

③ | $\langle \left(0.2,0.4,0.5,0.7\right),\text{}\left(0.3,0.5,0.6,0.9\right),\text{}\left(0.1,0.2,0.3,0.4\right)\rangle $ | $\u2780\text{}\to \text{}\u2782$ |

④ | $\langle \left(0.37,0.52,0.72,0.85\right),\text{}\left(0.02,0.06,0.15,0.30\right),\text{}\left(0.12,0.25,0.42,0.72\right)\rangle $ | $\u2780\text{}\to \text{}\u2781\text{}\to \text{}\u2783$ |

⑤ | $\langle \left(0.19,0.44,0.58,0.75\right),\text{}\left(0.06,0.12,0.25,0.42\right),\text{}\left(0.08,0.15,0.36,0.56\right)\rangle $ | $\u2780\text{}\to \text{}\u2781\text{}\to \text{}\u2784$ |

⑥ | $\langle \begin{array}{l}\left(0.352,0.608,0.748,0.875\right),\text{}\left(0.018,0.048,0.125,0.294\right),\text{}\\ \left(0.008,0.045,0.180,0.336\right)\end{array}\rangle $ | $\u2780\text{}\to \text{}\u2781\text{}\to \text{}\u2784\text{}\to \text{}\u2785$ |

Nodes | Distance of Shortest Path | Shortest Path |
---|---|---|

② | $<(0.1,0.3,0.4,0.5),(0.3,0.5,0.6,0.7),(0.1,0.3,0.4,0.6)>$ | $\u2780\text{}\to \text{}\u2781$ |

③ | $<(0.2,0.4,0.5,0.7),(0.3,0.5,0.6,0.9),(0.1,0.2,0.3,0.4)>$ | $\u2780\text{}\to \text{}\u2782$ |

④ | $<(0.2,0.3,0.4,0.6),(0.2,0.4,0.5,0.6),(0.4,0.5,0.7,0.8)>$ | $\u2780\text{}\to \text{}\u2783$ |

⑤ | $\begin{array}{l}<(0.36,0.51,0.64,0.8),(0.06,0.16,0.25,0.36),\\ (0.04,0.15,0.35,0.48)>\end{array}$ | $\u2780\text{}\to \text{}\u2783\text{}\to \text{}\u2784$ |

⑥ | $\begin{array}{l}<\left(0.28,0.52,0.65,0.82\right),\left(0.06,0.2,0.3,0.54\right),\\ \left(0.04,0.1,0.18,0.28\right)>\end{array}$ | $\u2780\text{}\to \text{}\u2782\text{}\to \text{}\u2785$ |

⑦ | $\begin{array}{l}<\left(0.28,0.51,0.7,0.84\right),\left(0.04,0.12,0.25,0.42\right),\\ \left(0.16,0.25,0.49,0.64\right)>\end{array}$ | $\u2780\text{}\to \text{}\u2783\text{}\to \text{}\u2786$ |

⑧ | $\begin{array}{l}<\left(0.352,0.608,0.79,0.92\right),\left(0.008,0.036,0.125,0.252\right),\\ \left(0.064,0.125,0.294,0.512\right)>\end{array}$ | $\u2780\text{}\to \text{}\u2783\text{}\to \text{}\u2786\text{}\to \text{}\u2787$ |

⑨ | $\begin{array}{l}<\left(0.424,0.664,0.825,0.928\right),\left(0.012,0.06,0.12,0.27\right),\\ \left(0.02,0.06,0.126,0.252\right)>\end{array}$ | $\u2780\text{}\to \text{}\u2782\text{}\to \text{}\u2785\text{}\to \text{}\u2788$ |

Optional Path | Score Function | Exact Function |
---|---|---|

$p1:\text{}\u2780\text{}\to \text{}\u2781\text{}\to \text{}\u2783\text{}\to \text{}\u2785$ | 0.872 | 0.686 |

$p2:\text{}\u2780\text{}\to \text{}\u2781\text{}\to \text{}\u2784\text{}\to \text{}\u2785$ | 0.798 | 0.504 |

$p3:\text{}\u2780\text{}\to \text{}\u2782\text{}\to \text{}\u2783\text{}\to \text{}\u2785$ | 0.871 | 0.780 |

$p4:\text{}\u2780\text{}\to \text{}\u2782\text{}\to \text{}\u2784\text{}\to \text{}\u2785$ | 0.889 | 0.755 |

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**MDPI and ACS Style**

Yang, L.; Li, D.; Tan, R.
Shortest Path Solution of Trapezoidal Fuzzy Neutrosophic Graph Based on Circle-Breaking Algorithm. *Symmetry* **2020**, *12*, 1360.
https://doi.org/10.3390/sym12081360

**AMA Style**

Yang L, Li D, Tan R.
Shortest Path Solution of Trapezoidal Fuzzy Neutrosophic Graph Based on Circle-Breaking Algorithm. *Symmetry*. 2020; 12(8):1360.
https://doi.org/10.3390/sym12081360

**Chicago/Turabian Style**

Yang, Lehua, Dongmei Li, and Ruipu Tan.
2020. "Shortest Path Solution of Trapezoidal Fuzzy Neutrosophic Graph Based on Circle-Breaking Algorithm" *Symmetry* 12, no. 8: 1360.
https://doi.org/10.3390/sym12081360