Edge Domination and Incidence Domination in Vague Incidence Graphs and Its Application

Abstract

In this article, we present a novel framework for edge domination in vague incidence graphs (VIG). We introduce the notion of certain types of vague incidence graphs and extend the concepts of dominations into edge and incidence domination in VIG. In particular, we propose the idea of order, size in VIG, isolated vertex, and cardinalities related to a dominating set. Additionally, the strong and weak domination for VIG were obtained and discussed with some theorems to support the context. We also initiate some definitions of edge domination and incidence domination on VIG and propose a model for the application of edge and incidence domination on VIG.

1. Introduction

Atanassov [1] proposed the concept of an intuitionistic fuzzy graph. He also added a fuzzy graph component that determines the degree of non-membership. In IFG, Parvathi and Karunambigai [2] defined a unique instance. Then, in [3,4], Dinesh developed fuzzy incidence graphs, a crucial new approach to fuzzy graphs. Moderson [5,6,7,8,9,10] went on to research fuzzy incidence graphs, end nodes, and connectedness notions later on. Somasundaram [11] first proposed domination in a fuzzy graph. Later, Ramakrishna [12] proposed the vague graph, which is a generalized version of the fuzzy graph. Domination number was described in a VG by Parvathi and Thamizhendi [13].

References [14,15,16,17] discussed several sorts of domination, including edge domination and double domination, on IFG.

In IFG, Gani et al. [18] described several fuzzy dominant set features.

Borzooei and Rashmanlou [19,20,21,22,23] studied VG principles. Akram [24] explored the concept of a strong IFG. Talebi, in [25], introduced the concepts of domination sets in vague graphs.According to Bustince [26], Vague sets are intuitionistic fuzzy sets. Thus, we developed the novel concept and definitions of vague incidence graphs by combining the traits and concepts of an intuitionistic fuzzy graph with a fuzzy incidence graph. Some elementary theorems were also proven.

The goal of this work is to present the novel notion of edge domination in VIG and to discuss the concepts of valid degree and cardinalities as they relate to the degree order and size of domination. The strong and weak dominations for VIG were also determined and presented using several theorems. We suggested a model for using edge and incidence domination on VIG to reduce the frequency of traffic accidents on the road transportation network. We represent domination and edge domination, including membership and non-membership functions of VIG, which produce accurate outcomes of the desirable parameters.

2. Preliminaries

In this section, we set some basic definitions for analysis from [7,13,19].

Graph G = (V, E, I), an incidence graph, holds $E\subseteq V\times$ V and I $\subseteq V\times E$, where V is the set of vertices and E is the set of edges in G.

Definition 1.

G = (V, E) is said to be an IFG where the mapping

(i)

$\mathcal{V}=\left\{\mu \left(a\right),\gamma \left(a\right)|a\in V\right\}, \mu \left(a\right):\mathcal{V}\to \left[0,1\right], \gamma \left(a\right):\mathcal{V}\to \left[0,1\right]$

where  $0 \le \mu \left(a\right)+\gamma \left(a \right) \le 1$

(ii)

$ℰ \subseteq \mathcal{V}\times \mathcal{V}$ were $ℰ=\left\{\mu \left(ab\right),\gamma \left(ab\right)|ab\in E\right\}$

$\mu \left(ab\right):\mathcal{V}\times \mathcal{V} \to \left[0,1\right], \gamma \left(ab\right):\mathcal{V}\times \mathcal{V} \to \left[0,1\right]$ where $\mu \left(ab\right)\le \mu \left(a\right)\bigwedge \mu \left(b \right)$ and

$\gamma \left(ab\right)\ge \gamma \left(a\right) \bigvee \gamma \left(b \right)$ such that $0 \le \mu \left(ab\right)+\gamma \left(ab\right) \le 1$

where $\mathrm{a},\mathrm{b}\in \mathrm{V}$ ; also, $\mathsf{\mu }$ denotes the degree of membership and $\gamma$ denotes the degree of non-membership.

Definition 2.

Let $\psi$ be a fuzzy incidence of $G=\left(\sigma ,\mu \right)$, where $\sigma$ and $\mu$ are fuzzy subsets of $V and E$ respectively. Then $\tilde{G}=\left(\sigma ,\mu ,\psi \right)$ is said to be FIG if $\psi \left(v,e\right)\le \sigma \left(v\right)\bigwedge \mu \left(e\right)$ for all

$$v\in V ,e\in E.$$

Definition 3.

For $\tilde{G}=\left(\sigma ,\mu ,\psi \right)$, If $ab\in supp \left(\mu \right)$ and if $\left(a,ab\right),\left(b,ab\right) \in Supp \left(\psi \right),$ then $\left(a,ab\right)$ and $\left(b,ab\right)$ are said to be incidence pairs.

Definition 4.

$\tilde{G}$ is connected if any two vertices and edges are joined by a path.

Definition 5.

G = (V, E, I) is said to be VIG if for a 3-tuple of the form $\Im =\left(\mathcal{V},ℰ,ℐ\right)$, where

(i)

$\mathcal{V}=\left\{\mu \left(a\right),\gamma \left(a\right)|a\in V\right\}, \mu :\mathcal{V}\to \left[0,1\right], \gamma :\mathcal{V}\to \left[0,1\right]$

where $0 \le \mu \left(a\right)+\gamma \left(a \right) \le 1$

(ii)

$ℰ \subseteq \mathcal{V}\times \mathcal{V}$ where $ℰ=\left\{\mu \left(ab\right),\gamma \left(ab\right)|ab\in E\right\}$

$$\mu :\mathcal{V}\times \mathcal{V}\to \left[0,1\right], \gamma :\mathcal{V}\times \mathcal{V}\to \left[0,1\right]$$

where $\mu \left(ab\right)\le \mu \left(a\right)\bigwedge \mu \left(b\right)$ and $\gamma \left(ab\right)\ge \gamma \left(a\right) \bigvee \gamma \left(b\right)$

Such that $0 \le \mu \left(ab\right)+\gamma \left(ab\right) \le 1$

(iii)

$ℐ\subseteq \mathcal{V}\times ℰ$ where $ℐ=\left\{\mu \left(a,ab\right),\gamma \left(a,ab\right)|\left(a,ab\right)\in I\right\}$

$$\mu :\mathcal{V}\times ℰ\to \left[0,1\right], \gamma :\mathcal{V}\times ℰ\to \left[0,1\right]$$

where $\mu \left(a,ab\right)\le \mu \left(a\right)\bigwedge \mu \left(ab\right)$ and $\gamma \left(a,ab\right)\ge \gamma \left(a\right) \bigvee \gamma \left(ab\right)$

Such that, 0 ≤ μ(a,ab) + γ(a,ab) ≤ 1, for every $\left(a,ab\right)\in \mathcal{V}\times ℰ$, where $\mu$ denotes the degree of membership and $\gamma$ denotes the degree of non-membership.

Definition 6.

The cardinality of a VIG is defined as

$$|ℐ|=\left|{\displaystyle \sum }_{aϵv}\frac{1+\mu \left(a\right)-\gamma \left(a\right)}{2}+{\displaystyle \sum }_{abϵℰ}\frac{1+\mu \left(ab\right)-\gamma \left(ab\right)}{2}+{\displaystyle \sum }_{a,baϵℐ}\frac{1+\mu \left(a,ab\right)-\gamma \left(a,ab\right)}{2}\right|$$

(i)

The vertex cardinality of $\mathcal{J} is \left|\mathcal{V}\right|={\displaystyle \sum }_{aϵ\mathcal{V}}\left|\frac{1+\mu \left(a\right)-\gamma \left(b\right)}{2}\right|$

(ii)

The edge cardinality of $ℐ$ is $|ℰ|={\displaystyle \sum }_{abϵℰ}\left|\frac{1+\mu \left(ab\right)-\gamma \left(ab\right)}{2}\right|$

(iii)

The incidence cardinality of $ℐ is |ℐ|={\displaystyle \sum }_{a,abϵℐ}\left|\frac{1+\mu \left(a,ab\right)-\gamma \left(a,ab\right)}{2}\right|$

Definition 7.

An edge $\left(a,b\right)$ is said to be strong if $\mu \left(a,b\right)\ge {\mu }^{\infty }\left(a,b\right)$ and $\gamma \left(a,b\right)\ge {\gamma }^{\infty }\left(a,b\right).$

Definition 8.

In G, let $a,b\in V$, we say that a dominates b in G if there exists a strong edge between them.

Definition 9.

Let $\tilde{H}=\left(\tau , \upsilon ,\Omega \right)$ be a fuzzy incidence subgraph of $\tilde{G} if \tau \subseteq \sigma , \upsilon \subseteq \mu , and \Omega \subseteq \psi .$

Definition 10.

$\tilde{G}$ is called a cycle if $\left(Supp\left(\sigma \right), Supp \left(\mu \right), Supp \left(\psi \right)\right)$ is a cycle and fuzzy incidence cycle, if there is no unique $\left(a, ab\right) \in Supp\left(\psi \right)$ such that $\psi \left(a, ab\right)=\Lambda \left\{\psi \left(u,uv\right) | \left(u,uv\right) \in Supp \left(\psi \right)\right\}$.

Definition 11.

$\tilde{G}$ is complete if $\left(u,uv\right) \in E^{\left(i\right)}$ . Then an incidence cut pair $\left(u,uv\right)$ is ${\psi }^{\iota \infty }\left(a, ab\right)<{\psi }^{\infty }\left(a, ab\right)$ for some x, y ∈ V, where ${\psi }^{\iota } \left(u,uv\right)=0 and$ ${\psi }^{\iota }=\psi ,$ elsewhere; also, if $\left(u,uv\right) \in V\times E, \psi \left(u,uv\right)=\sigma \left(u\right) \Lambda \mu \left(uv\right).$

3. Main Results

In this section, we extend the fuzzy incidence graph and intuitionistic fuzzy graph to the concept of a vague incidence graph, $\Im$(VIG).

Definition 12.

A vague incidence relation is a subset $\mathcal{V}\times \mathcal{V}$ then the expression $ℛ$ is given by

$$ℛ=\left\{\langle \left(a,ab\right),\mu \left(a,ab\right) \gamma \left(a,ab\right)\rangle \right\} | a\in \mathcal{V}, ab\in ℰ, \left(a,ab\right)\in ℐ$$

where $\mu \left(a\right) : \mathcal{V} \to \left[0,1\right]$; $\gamma \left(a\right) : \mathcal{V} \to \left[0,1\right]$, $\mu \left(ab\right): \mathcal{V}\times \mathcal{V}\to \left[0,1\right]$;

$\gamma \left(ab\right) : \mathcal{V}\times \mathcal{V}\to \left[0,1\right]$ $\mu \left(a,ab\right) : \mathcal{V}\times ℰ\to \left[0,1\right] ;\gamma \left(a,ab\right): \mathcal{V}\times ℰ\to \left[0,1\right]$

Satisfying the conditions $0 \le \mu \left(ab\right)+\gamma \left(ab\right) \le 1$ and $0\le \mu \left(a,ab\right)+\gamma \left(a,ab\right) \le 1$.

Definition 13.

The vertices a and b are neighbors in VIG if any of the following conditions hold.

(i)

$\mu \left(a,b\right) =0, \gamma \left(a,b\right)>0$

(ii)

$\mu \left(a,b\right)>0,\gamma \left(a,b\right) =0$

(iii)

$\mu \left(aa,b\right) =0 ,\gamma \left(aa,b\right)>0$

(iv)

$\mu \left(a,ab\right)>0,\gamma \left(a,ab\right)=0,$ For some $a,b \in \mathcal{V}, \left(ab\right) \in , \left(a,ab\right)\in ℐ$

Definition 14.

For an incidence graph G = (V, E, I) and $\Im =\left(\mathcal{V},ℰ,ℐ\right)$, if VIG has distinct vertices, then it is said to be incidence paths in $\Im$.

Definition 15.

An edge $ab$ is strong if $\mu \left(ab\right)>{\mu }^{\infty }\left(ab\right)$ and $\gamma \left(ab\right)>{\gamma }^{\infty }\left(ab\right)$ where

$$\begin{gathered} \mu \left(ab\right)=\sup \left\{{\mu }^{\kappa }\left(ab\right) | K=1,2\dots n\right\} \\ \gamma \left(ab\right)=\inf \left\{{\gamma }^{\kappa }\left(ab\right) | K=1,2\dots n\right\} \end{gathered}$$

${\mu }^k\left(ab\right)$ = sup $\left\{\mu \left(a{b}_1\right)\bigwedge \mu \left(b_1{b}_2\right)\bigwedge \dots \dots \mu \left(b_{k-1}b\right)\right\}$ and

${\gamma }^k\left(ab\right)$ = inf $\left\{\gamma \left(a{b}_1\right)\bigvee \gamma \left(b_1{b}_2\right)\bigvee \dots \dots \dots \gamma \left(b_{k-1}b\right)| a,b_1,b_2,\dots b\in \mathcal{V}\right\}$.

Definition 16.

An incidence $\left(a,ab\right)$ is strong if $\mu \left(a,ab\right)\ge {\mu }^{\infty }\left(a,ab\right)and \gamma \left(a,ab\right)\ge {\gamma }^{\infty }\left(a,ab\right)$

where ${\mu }^{\infty }\left(a,ab\right)=\sup \left\{{\mu }^{\kappa }\left(a,ab\right) | K=1,2\dots n\right\}$

$${\gamma }^{\infty }\left(a,ab\right)=\inf \left\{{\gamma }^k\left(a,ab\right) | K=1,2\dots n\right\}$$

${\mu }^k\left(a,ab\right)$ = sup $\left\{\mu \left(a,a{b}_1\right)\bigwedge \mu \left(b_{1 },b_1{b}_2\right)\bigwedge \dots \dots \mu \left(b_{k-1},b_{k-1}b\right)\right\}$ and

${\gamma }^k\left(a,ab\right)$ = inf $\left\{\gamma \left(a,ab\right)\bigvee \gamma \left(b_1,b_1{b}_2\right)\bigvee \dots \dots \dots \gamma \left(b_{k-1},b_{k-1}b\right)| a,b_1,b_2,\dots b\in \mathcal{V}\right\}$.

Definition 17.

The strength of the incidence whose vertices a, b are connected is defined as $\left(min\left(u, uv\right), max\left(u, uv\right)\right),$ where $min\left(u, uv\right)$ is the $\mu$ γ-strength of the weakest incidence and $max\left(u, uv\right)$ the $\gamma$ γ-strength of the strongest incidence.

Definition 18.

Let $N\left(a\right)=\left\{b:b\in v \& \left(a,ab\right)is a strong incidence\right\}$, where a, b are the vertices. Then $N\left(a\right)$ is said to be the neighborhood of a.

Definition 19.

$a\in v$ is isolated if $\mu \left(a,ab\right)=0 \& \gamma \left(a,ab\right)=0 ⍱ b \in V$ . That is $N\left(a\right)=\Phi$.

Definition 20.

If $a\in \mathcal{V}-H$, there exists $a\in H$ that dominates B, then a subset $H of \mathcal{V}$ is a dominating set.

Definition 21.

The lower the number of $ℐ,$ indicated by $d(ℐ)$, the smaller the cardinality of all minimal dominating sets.

The higher the number of $ℐ,$ indicated by $\mathcal{D}(ℐ)$, the greater the cardinality of all maximal dominating sets.

Definition 22.

Let $\Im =\left(\mathcal{V},ℰ,ℐ\right)$ be a VIG. A set $\mathcal{D}$ is a total dominating set if for all $b ϵ \nu \& a ϵ s , a\ne b$ such that a dominates b.

Definition 23.

The Minimum cardinality of $\mathcal{D}$ is called the minimal total domination number of $ℐ$ denoted by $\mathcal{t}(ℐ)$.

The Maximum cardinality of $\mathcal{D}$ is called the maximal domination number of $ℐ,$ denoted by $T(ℐ)$

Definition 24.

The complement of a VIG is represented as $\overline{ℐ}$ = ( $\overline{\mathcal{V}}$ $\overline{\in }$ $\overline{ℐ})$ and is defined as

(i)

$\overline{\mu }$ $\left(a\right)$ = $\mu \left(a\right); \overline{\gamma }\left(a\right)$ = $\gamma \left(a\right)$

(ii)

$\overline{\mu }$ $\left(ab\right)$ = min $\left[\mu \left(a\right), \mu \left(b\right)\right]$ $-\mu \left(ab\right); \overline{\gamma }\left(ab\right)$ = max $\left[\gamma \left(a\right), \gamma \left(b\right)\right]$ $-\gamma \left(ab\right)$

(iii)

$\overline{\mu }$ $\left(a,ab\right)$ = min $\left[\max \left(\mu \left(a\right), \mu \left(ab\right)\right)-\mu \left(a,ab\right),\min \left(\mu \left(a\right), \mu \left(b\right)\right)-\mu \left(ab\right)\right];$

$\overline{\gamma }$ $\left(a,ab\right)$ = max $\left[\min \left(\gamma \left(a\right), \gamma \left(ab\right)\right)-\gamma \left(a,ab\right), \max \left(\gamma \left(a\right), \gamma \left(b\right)\right)-\gamma \left(ab\right)\right]$.

Definition 25.

The number of vertices in $ℐ$ is called the order of a VIG, denoted by $O(ℐ)={\displaystyle \sum }_{\begin{array}{c}a \ne b\\ a,b\in V\end{array}}\frac{1+\mu \left(a,ab\right)-\gamma \left(\bigwedge ,ab\right)}{2}$ for all, $\bigwedge \in \mathcal{V}$.

The number of edges in $ℐ$ is called the size of a VIG, denoted by $S(ℐ)={\displaystyle \sum }_{a,b\in ℰ}\frac{1+\mu \left(a,b\right)-\gamma \left(ab\right)}{2}$.

Theorem 1.

A dominating set $\mathcal{D}$ of a VIG is minimal if and only if for all $\mathcal{d} ϵ \mathcal{D}$, one of the following conditions holds:

(i)

$d$ is a weak neighbor of any vertex in $\mathcal{D}$

(ii)

$\mathcal{V} \in V-\mathcal{D}$ such that $\mathcal{N} \left(\mathcal{V}\right) {\displaystyle \cap } \mathcal{D}=d.$

Proof. 

Let $\mathcal{D}$ be a minimal dominating set; then, for all, $d\in \mathcal{D}$, $\mathcal{D}-d$ is not a dominating set. There exists $\mathcal{V} \in V-\left(\mathcal{D}-\left\{d\right\}\right),$ which is not dominated by any vertex in $\mathcal{D}-\left\{d\right\}$. If

$\mathcal{V}=d$ then $\mathcal{V}$ is a week neighbor of any vertex in $\mathcal{D}$

If $\mathcal{V} \ne d$ then $\mathcal{V}$ is dominated by $\mathcal{D}$.

Then $\mathcal{V}$ is strong only to $d$.

That is, $\mathcal{N} \left(\mathcal{V}\right) {\displaystyle \cap } \mathcal{D}=\left\{d\right\}$.

Conversely, assume $\mathcal{D}$ is the dominating set for $d\in \mathcal{D}$ and one of the conditions holds. Suppose $\mathcal{D}$ is not minimal then $d\in \mathcal{D}, \mathcal{D}-\left\{d\right\}$ is a dominating set. Hence, $d$ is strong to at least one neighbor in $\mathcal{D}-\left\{d\right\}$. If $\mathcal{D}-\left\{d\right\}$ is also the dominating set, then every vertex is $\mathcal{V}-\mathcal{D}$. So, both the conditions do not hold, which is contradictory. So, $\mathcal{D}$ is a minimal dominating set. □

Theorem 2.

Let a VIG has no isolated vertices and $\mathcal{D}$ be a minimal dominating set. Then $\mathcal{V}-\mathcal{D}$ is dominating set.

Proof. 

The vertex $\mathcal{V}$ of the minimal dominating set $\mathcal{D}$ must be dominated by at least a vertex $\mathcal{D}-\left\{\mathcal{V}\right\}$ because G has no isolated vertex and also $d \in \mathcal{N}\left(\mathcal{V}\right)$. By Theorem 3, $d\in \mathcal{V}-\mathcal{D}.$ Thus $\mathcal{V}-\mathcal{D}$ is a dominating set because every vertex in $\mathcal{D}$ is dominated by at least a vertex in $\mathcal{V}-\mathcal{D}$. □

Theorem 3.

In an IFII, $d (ℐ)+d\left(\overline{ℐ}\right) \le 2 O (ℐ),$ where it holds if and only if $0<\gamma \left(ab\right)<{\gamma }^{\infty }\left(a b\right)$ for all $a,b \in v$.

Proof. 

The result is trivial when $d (ℐ)+d\left(\overline{ℐ}\right) \le 2 O (ℐ)$. Also $d(ℐ)=O(ℐ)$ iff $\mu \left(ab\right)<{\mu }^{\infty }\left(ab\right)$ and $\mu \left(ab\right)<{\gamma }^{\infty }\left(ab\right)$ for all, $b \in v.$ $d\left(\overline{ℐ}\right) \le 2 O(ℐ)$ iff $\mu \left(ab\right)-\mu \left(a b\right)<{\mu }^{\infty }\left(ab\right)$ and $\gamma \left(ab\right)-\gamma \left(ab\right)<{\gamma }^{\infty }\left(a b\right)$ for all $a,b \in v$ gives $\mu \left(ab\right) and \gamma \left(ab\right)>0$.

Hence, $d (ℐ)+d\left(\overline{ℐ}\right) \le 2 O (ℐ)$. □

Corollary 1.

If a VII has no isolated vertex, then $d (ℐ)\le O(ℐ)/2$.

4. Strong and Weak Domination in VIG

In this section, we discuss strong and weak domination in VIG.

Definition 26.

Let a VIG $\Im =\left(\mathcal{V},ℰ,ℐ\right)$ and let a, b be the vertices of $ℐ$ . Then a strongly dominates b or b weakly dominates a, If

(i) 

$d_i\left(a\right)\ge d_i\left(b\right)$,

(ii) 

$ℐ \left(a,ab\right)=\mathcal{V}\left(b\right)\wedge ℰ \left(ab\right)$

Definition 27.

A set $ℛ \le \mathcal{V}$ is a strongly dominated set in VIG if every vertex in $\mathcal{V}-ℛ$ is strongly dominated by at least one vertex in $ℛ$.

Definition 28.

The lowest cardinality of a strong and weak dominating set is the strong and weak domination number of an IFIG respectively and is denoted by $\left|{\Im }_{SD}\right|$ and $\left|{\Im }_{WD}\right|$ respectively.

Definition 29.

If $\mathcal{V}\left(a\right)>0$ then a is in supp $\mathcal{V}.$ If $\epsilon \left(ab\right)>0$ then ab is in supp $ℰ.$ If $ℐ\left(a,ab\right)>0$ then $\left(a,ab\right)$ is in supp $ℐ.$

Definition 30.

A VIG is said to be complete if $ℐ \left(a,ab\right)=\mathcal{V}\left(b\right)\wedge ℰ \left(ab\right)$ for each $ℐ \left(a,ab\right)\in {ℐ}^{*}$ ; also, for each $ℐ \left(a,ab\right)=ℐ \left(b,ab\right) for each a,b\in {\mathcal{V}}^{*}$ and is denoted by ${\Im }_c$.

Theorem 4.

For a complete VIG with $ℐ \left(a,ab\right)=\mathcal{V}\left(b\right)\wedge ℰ \left(ab\right)$ for all $a\in \mathcal{V}, ab\in ℰ,$ the inequality holds ${\Im }_{CWD}\le {\Im }_{CSD}$.

Proof. 

Let $\Im =\left(\mathcal{V},ℰ,ℐ\right)$ be a VIG with $ℐ \left(a,ab\right)=\mathcal{V}\left(b\right)\wedge ℰ \left(ab\right)$. Assume $a_i\in \mathcal{V}$ and $\mathcal{V}\left(a_i\right)$ are all the same. Since VIG is complete, $ℰ\left(a_i{a}_j\right)=\mathcal{V}\left(a_i\right)\wedge \mathcal{V}\left(a_j\right)$ for all $a_i\in \mathcal{V}, a_i{a}_j\in ℰ$. Thus, for all, $a_i\in \mathcal{V}$ is both a strong and weak vague incidence dominating set. Thus,

${\Im }_{CWD}={\Im }_{CSD}$ (i)

Now, let $a_i\in \mathcal{V}$ and $\mathcal{V}\left(a_i\right)$ be not the same. Then, in a complete VIG with $d_i\left(a_i\right)\ge d_i\left(a_j\right)$, one of them strongly dominates the other. If it is small, then it is weak. That is, ${\Im }_{CWD}=\mathcal{V}\left(a_i\right)$ with $d_i\left(a_i\right)\le d_i\left(a_j\right)$ for all $a_i{a}_j\in \mathcal{V}$ and $ℐ\left(a_i,a_i{a}_j\right)=\mathcal{V}\left(a_i\right)\wedge ℰ\left(a_i{a}_j\right)$ for all $a_i\in \mathcal{V}, a_i{a}_j\in ℰ.$ It means a strong dominant set has other node sets. That is, ${\Im }_{CWD}<{\Im }_{CSD}$ (ii)

From Equations (i) and (ii), we get, ${\Im }_{CWD}\le {\Im }_{CSD}.$ □

5. Edge Domination and Incidence Domination

Definition 31.

Let $\Im =\left(\mathcal{V},ℰ,ℐ\right)$ be a VIG. Let a, b be two edges. Then, a dominates b if a is strong in $ℐ$ and adjacent to b.

Definition 32.

Let $\mathcal{D}$ be a minimal dominating set of VIG. For $a\in ℰ-\mathcal{D}$, it is such that a dominates b. Then $\mathcal{D}$ is an edge dominating set. The minimum intuitionistic fuzzy cardinality of all $\mathcal{D}$ is the edge domination number and denoted by $ℰ(\Im ).$

For example, consider Figure 1.

Here, $\mathcal{D}=\left\{{ℰ}_1,{ℰ}_6 \right\}$ is an edge dominating set and $ℰ-\mathcal{D}=\left\{{ℰ}_2,{ℰ}_3,{ℰ}_4,{ℰ}_5,{ℰ}_7\right\}$.

The edge domination number $ℰ(\Im )=0.35.$

Definition 33.

The strong neighborhood of an incidence ${ℐ}_u$ in VIG is

$\mathcal{N}\left({ℐ}_u\right)=\left\{{ℐ}_{v }\in \Im /{ℐ}_{v }is a strong incidence in \Im and adjacent to {ℐ}_u\right\}$.

Definition 34.

Let $\Im =\left(\mathcal{V},ℰ,ℐ\right)$ be a VIG. Let ${ℐ}_u$ and ${ℐ}_{v }$ be the two incidences of $\Im$ . We say that ${ℐ}_u$ dominates ${ℐ}_{v }$ if ${ℐ}_{u }$ is the strong incidence in ℑ and adjacent to ${ℐ}_{v }$.

Definition 35.

Let $\Im$ be a VIG and $\mathcal{D}$ be an edge dominating set. Then the incidence cover of $\mathcal{D}$ in $\Im$ is defined as the set of all incidence to each edge in $\mathcal{D}$.

6. Application of Edge Domination of VIG in Road Transport Network

In this section, we consider a VIG model to illustrate a road transportation network. Since there are labeling data for vertices such as location, importance, and so on, and edges such as length, traffic, quality, etc., the best way to depict the road transportation network is to use VIG, with vertices and edges representing locations and routes.

Consider the traffic systems of various cities to determine the most common cause of accidents. To reduce the number of road accidents, several substantial measures should be done. We provide a model to solve the problem here.

Roads with a large flow of vehicles become a source of the most serious traffic accidents. Government can take measures by speed breakers, speed bumps, and deploying more traffic police to minimize road accidents. Here we apply VIG to the traffic systems of different cities.

Consider the network of VIG in Figure 2.

Consider the network of VIG consisting of 5 vertices indicating different cities $\left\{{\nu }_1, {\nu }_{2,}, {\nu }_{3,}{\nu }_4, {\nu }_{5,}\right\}$ and the edges $\left\{{\epsilon }_1, {\epsilon }_2,{\epsilon }_3,{\epsilon }_{4,}{\epsilon }_5, {\epsilon }_6,{\epsilon }_7\right\}$ are the roads connecting the cities.

The flow of traffic from one city to another indicates the incidence pairs. For example, $\left({\nu }_1, {\nu }_1{\nu }_2\right)$ is the flow of traffic from ${\nu }_1, to {\nu }_2 \mathrm{and} \left({\nu }_2, {\nu }_1{\nu }_2\right)$ is from ${\nu }_2,to {\nu }_1$. The membership value of the edges shows light motor vehicles (LMV). Non-membership values of the edges show other vehicles flowing through the roads among the different cities.

The edge dominating sets for Figure 2 are as follows:

$$\begin{gathered} {\mathcal{D}}_1=\left\{{\epsilon }_1,{\mathcal{e}}_4\right\} \\ {\mathcal{D}}_2=\left\{{\epsilon }_1,{\mathcal{e}}_3\right\} \\ {\mathcal{D}}_3=\left\{{\mathcal{e}}_3,{\mathcal{e}}_5\right\} \\ {\mathcal{D}}_4=\left\{{\mathcal{e}}_3,{\mathcal{e}}_{6,}{\mathcal{e}}_7\right\} \\ {\mathcal{D}}_5=\left\{{\epsilon }_1,{\mathcal{e}}_{3,}{\mathcal{e}}_5\right\} \\ {\mathcal{D}}_6=\left\{{\mathcal{e}}_3,{\mathcal{e}}_{4,}{\mathcal{e}}_6\right\} \\ {\mathcal{D}}_7=\left\{{\epsilon }_1,{\mathcal{e}}_{3,}{\mathcal{e}}_{4,}{\mathcal{e}}_7\right\} \\ {\mathcal{D}}_8=\left\{{\mathcal{e}}_3,{\mathcal{e}}_{4,}{\mathcal{e}}_{6,}{\mathcal{e}}_7\right\} \end{gathered}$$

The edge cardinalities of the dominating edge sets are

$$\begin{gathered} \left|{\mathcal{D}}_1\right|=0.9 \\ \left|{\mathcal{D}}_2\right|=0.85 \\ \left|{\mathcal{D}}_3\right|=0.95 \\ \left|{\mathcal{D}}_4\right|=1.25 \\ \left|{\mathcal{D}}_5\right|=1.3 \\ \left|{\mathcal{D}}_6\right|=1.45 \\ \left|{\mathcal{D}}_7\right|=1.75 \\ \left|{\mathcal{D}}_8\right|=1.8 \end{gathered}$$

The dominating edge set ${\mathcal{D}}_2$ has the smallest cardinality and ${\mathcal{D}}_8$ has the largest cardinality among other dominating sets. Therefore, we conclude more LMV flows through the roads ${\epsilon }_3, {\epsilon }_4, {\epsilon }_6, {\epsilon }_7$, which corresponds to the dominating set ${\mathcal{D}}_8$, which gives the highest percentage of road accidents.

Furthermore, the edge cardinalities are

$$\begin{gathered} \left|{\epsilon }_1\right|=0.35, \\ \left|{\epsilon }_2\right|=0.4, \\ \left|{\epsilon }_3\right|=0.5, \\ \left|{\epsilon }_4\right|=0.55, \\ \left|{\epsilon }_5\right|=0.45, \\ \left|{\epsilon }_6\right|=0.4, \\ \left|{\epsilon }_7\right|=0.35. \end{gathered}$$

The highest edge cardinality $\left|{\epsilon }_4\right|=0.55.$ So, we can take measures like strictly avoiding the following:

(i)

Over speeding;

(ii)

Drunken drive;

(iii)

Driving in the wrong lane.

The above are the major causes of the accidents. Moreover, if these measures were taken particularly for the membership LMV, it will play a major role in minimizing road accidents.

We arrived at a similar result for non-membership also. We can apply this to a large number of inputs. Only in VIG we can get the desired values of both incidences, membership and non-membership.

7. Conclusions

The concept of IFG has a large variety of applications in computational intelligence and computer science domains. In this study, as an extension of IFG and FIG, we introduced some concepts of edge domination and incidence domination of VIG and illustrated them with some definitions and theorems. The strong and weak domination of VIG also were examined, illustrating a model for the application of edge domination and incidence domination of VIG to minimize road accidents. In the future we intend to extend our research to the energy of VIG, double domination, the strength of domination, and the coloring of VIG that can be found using domination of VIG.