# An Improved Algorithm for Identification of Dominating Vertex Set in Intuitionistic Fuzzy Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**Let $\sigma $ be a fuzzy subset of a set $X$ and ${\alpha}_{\mathcal{N}}:X\times X\to \left[0,1\right]$ be a fuzzy MF on $X\times X$. Then, ${\alpha}_{\mathcal{N}}$ is called a fuzzy relation on $X$ if

**Definition**

**2**

**.**The pair ${G}^{*}=\left(V,U\right)$ represents a simple crisp graph with vertices in $V$ and edges in $U$. If two vertices $a$ and $b$ are adjacent through an edge, then we write the edge as $\left(a,b\right)$. A fuzzy graph $G=\left(V,\sigma ,\alpha \right)$ correlating to a crisp graph ${G}^{*}$ is a triplet containing a non-empty set $V$ of vertices together with a pair of functions $\sigma :V\to \left[0,1\right]$ and $\alpha :V\times V\to \left[0,1\right]$ such that for all $a,b\in V,$

**Definition**

**3**

**.**Consider a nonempty set $X$. An “intuitionistic fuzzy (IF) set” $\mathcal{N}$ in $X$ is represented in the form:

**Definition**

**4**

**.**The “intuitionistic fuzzy relation” $R$ on the set $X$ is an intuitionistic fuzzy set over $X\times X$ of the form:

**Definition**

**5**

**.**Let $a=\left(\mu \left(a\right),\nu \left(a\right)\right)$ and $b=\left(\mu \left(b\right),\nu \left(b\right)\right)$ be intuitionistic fuzzy variables. Here, $\mu \left(a\right)+\nu \left(a\right)\le 1$ , and $\mu \left(b\right)+\nu \left(b\right)\le 1.$ Then the operators & and ∨ are described as below:

**Definition**

**6**

**.**An IFG $\tilde{G}=\left(V,U\right)$ is said to be first kind if $V$ is crisp set of vertices and $U$ is an intuitionistic set of edges. Here, $U=\langle V\times V,\mu ,\nu \rangle $ is an intuitionistic fuzzy relation that satisfies the following condition

## 3. Dominating Vertex Set (DVS)

**Definition**

**7**

**.**The subgraph of a graph $G$ that is created by all of the vertices around a vertex $v$ is known as the neighbourhood of vertex $v$ denoted by $N\left(v\right)$.

**Definition**

**8**

**.**Consider $G=\left(V,E\right)$ be a crisp graph where $V$ represents the collection of vertices and $E$ represents the collection of edges. A set $D\subseteq V$ is a DVS of $G$ if $\forall v\in V\backslash D,N\left(v\right){\displaystyle \cap}D\ne \varnothing .$

#### DVS for IFG

**Definition**

**9**

**.**The set $X$ is called an intuitionistic DVS for vertex $y$ with the intuitionistic degree of domination ${p}_{X}\left(y\right).$

**Definition**

**10**

**.**The set $X$ is an intuitionistic DVS for graph $\tilde{G}=\left(V,U\right)$ with the intuitionistic degree of domination:

**Example**

**1.**

**Definition**

**11**

**.**An IFS

## 4. Existing Algorithm for Finding DVS for an IF Graph

- ${x}_{i}\in {X}_{\beta};$
- If ${x}_{i}\notin {X}_{\beta}$ then there exists a vertex ${x}_{j}$ so that it corresponds to set ${X}_{\beta}$ while the vertex ${x}_{j}$ is adjacent to vertex ${x}_{i}$ with the degree $\left(\mu \left({x}_{j},{x}_{i}\right),\nu \left({x}_{j},{x}_{i}\right)\right)\ge \beta .$

Algorithm 1. Minimization of expression (7) | ||

Input: An IFG $\tilde{G}=\left(V,U\right)$ with intuitionistic DVS ${X}_{\beta}$; $\beta =\left({\mu}_{\beta},{\nu}_{\beta}\right)$ being degree of domination. | ||

Output: Minimized expression (7). | ||

1 | Begin | |

2 | Construct the expression: ${\Phi}_{D}=\underset{i=1,n}{\&}({p}_{i}\vee \underset{j=1,n}{\bigvee}\left({p}_{j}{\xi}_{ji}\right))$.// ${\xi}_{ji}$ represents | |

3 | $\left(\mu \left({x}_{j},{x}_{i}\right),\nu \left({x}_{j},{x}_{i}\right)\right)\ge \left({\mu}_{\beta},{\nu}_{\beta}\right)$ and ${p}_{i}=1$ if ${x}_{i}\in {X}_{\beta}$ and $0$ otherwise | |

4 | Construct the expression: ${\Phi}_{D}=\underset{i=1,n}{\&}(\underset{j=1,n}{\bigvee}\left({p}_{j}{\xi}_{ji}\right))$. | |

5 | Compute the expression: ${\Phi}_{D}={\bigvee}_{i=1,i}\left({p}_{1i}\&{p}_{2i}\&\dots \&{p}_{ki}\&{\beta}_{i}\right).$//According to the rules in (9) | |

6 | end |

**Example**

**2.**

## 5. Modified Algorithm for Finding DVS

- (a)
- ${x}_{i}\in {X}_{\beta};$
- (b)
- ${x}_{i}\notin {X}_{\beta}$,

**Step 1:**To find DVSs of degree 1 (if they exist), find ${x}_{j}$, such that

**Step 2:**For deriving DVSs of order 2 or more, there exists ${X}_{\beta}{}^{*}\subseteq {X}_{\beta}$, such that

Algorithm 2. Minimization of expression (11) | ||

Input: An IFG $\tilde{G}=\left(V,U\right)$ with intuitionistic DVS ${X}_{\beta}$; $\beta =\left({\mu}_{\beta},{\nu}_{\beta}\right)$ being degree of domination. | ||

Output: Minimized expression (11). | ||

1 | Begin | |

2 | Construct ${\beta}_{1}^{0}=\underset{j}{\bigvee}{}_{{x}_{r}\in V-\left\{{x}_{j}\right\}}{}^{\&}p({x}_{j},{x}_{r})$.//calculated for all ${x}_{j}$ such that $p\left({x}_{j},{x}_{r}\right)\ne \left(0,1\right)$ for all ${x}_{r}\in V-\left\{{x}_{j}\right\}$ | |

2 | Construct the expression: ${\Phi}_{D}={}_{i=1,n}{}^{}({p}_{i}\vee \underset{j=1,n}{\bigvee}\left({p}_{j}{\xi}_{jik}\right))$.// ${\xi}_{jik}$ represents | |

3 | $\mu \left({x}_{i},{x}_{j}\right)\ge {\mu}_{\beta}\bigwedge \nu \left({x}_{i},{x}_{k}\right)\le {\nu}_{\beta}$ and ${p}_{i}=1$ if ${x}_{i}\in {X}_{\beta}$ and $0$ otherwise | |

4 | Construct the expression: ${\Phi}_{D}={}_{i=1,n}{}^{}(\underset{j=1,n}{\bigvee}\left({p}_{j}{\xi}_{jik}\right))$. | |

5 | Compute the expression: ${\Phi}_{D}={\bigvee}_{i=1,i}\left({p}_{1i}\&{p}_{2i}\&\dots \&{p}_{ki}\&{\beta}_{i}\right).$//According to the rules in (9) | |

6 | end |

**Step 1:**

**Step 2:**

## 6. Comparative Analysis

**Example**

**4.**

#### 6.1. Solution of Water Flow Network through Algorithm 1

#### 6.2. Some Observations

- For the IFG in Figure 4, taking

#### 6.3. Solution of Water Flow Network through Algorithm 2

**Step 1:**The set of all vertices is $X=\left\{{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}\right\}$. Here ${x}_{1}$ and ${x}_{2}$ are two vertices whose $p\left({x}_{1},{x}_{r}\right)\ne \left(0,1\right)$ s.t ${x}_{r}\in X-\left\{{x}_{1}\right\}$ and $p\left({x}_{2},{x}_{j}\right)\ne \left(0,1\right)$ s.t ${x}_{j}\in X-\left\{{x}_{2}\right\}$.

**Step 2:**Using the adjacency matrix for the graph in Figure 4 (calculated in Section 6.1), ${\phi}_{D}$ will be as below

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Abbreviation | Meaning |

IF | Intuitionistic fuzzy |

IFS | Intuitionistic fuzzy set |

IFG | Intuitionistic fuzzy graph |

DVS | Dominating vertex set |

FS | Fuzzy set |

FG | Fuzzy graph |

MF | Membership function |

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**MDPI and ACS Style**

Nazir, N.; Shaheen, T.; Jin, L.; Senapati, T.
An Improved Algorithm for Identification of Dominating Vertex Set in Intuitionistic Fuzzy Graphs. *Axioms* **2023**, *12*, 289.
https://doi.org/10.3390/axioms12030289

**AMA Style**

Nazir N, Shaheen T, Jin L, Senapati T.
An Improved Algorithm for Identification of Dominating Vertex Set in Intuitionistic Fuzzy Graphs. *Axioms*. 2023; 12(3):289.
https://doi.org/10.3390/axioms12030289

**Chicago/Turabian Style**

Nazir, Nazia, Tanzeela Shaheen, LeSheng Jin, and Tapan Senapati.
2023. "An Improved Algorithm for Identification of Dominating Vertex Set in Intuitionistic Fuzzy Graphs" *Axioms* 12, no. 3: 289.
https://doi.org/10.3390/axioms12030289