Some properties of vague graphs with application

1 Introduction

The origin of graph theory started with the problem of Konigsberg bridge, in 1735. This problem led to the concept of Eulerian graph. Euler studied the problem of Konigsberg bridge and constructed a structure that solved the problem called Eulerian graph. In 1840, Mobious gave the idea of complete graph and bipartite graph and Kuratowski proved that they were planar by means of recreational problems. At present, graph theoretical concepts are highly utilized by computer science applications. Especially in research areas of computer science including datamining, image segmentation, clustering, and networking. The introduction of fuzzy sets by Zadeh [12] in 1965 changed the face of science and technology to a great extent. The most important feature of a fuzzy set is that a fuzzy set A is a class of objects that satisfies a certain property. Each object x in A has a membership degree of A, denoted as μ _A  (x). In 1993, Gau and Buehrer [3] proposed the concept of vague sets by replacing the value of an element in a set with a subinterval of [0, 1]. Namely, a true-membership function t _v  (x) and a false membership function f _v  (x) are used to describe the bounderies of the membership degree. The first definition of fuzzy graphs was proposed by Kaufman [5] in 1973, from Zadeh’s fuzzy relations [12, 13]. But Rosenfeld [11] introduced other elaborated definitions including fuzzy vertex, fuzzy edges, and several fuzzy analogs of graph theoretic concepts such as paths, cycles, and connectedness. Jun [4] investigated intuitionisticfuzzy graphs. Ramakrishna [7] introduced the concept of vague graphs and studied some of their properties. Akram et al. [1] studied certain types of vague graphs. Rashmanlou et al..[8, 10] defined complete interval-valued fuzzy graphs, and new concepts of bipolar fuzzy graphs. Borzooei and Rashmanlou [2] investigated domination in vague graphs. In this paper, busy vertices, free vertices, strong arc,isolated vertex, fully free vertex and partially free vertex in vague graphs are defined. Also, some interesting properties of isomorphism and weak isomorphism on them are introduced. Finally, we give an application of vague digraph in vulnerability assessment of gas pipeline networks.

2 Preliminaries

Let X be a set, and II = {[b, c] | 0 ≤ b ≤ c ≤ 1} (i.e. the set of all closed intervals in [0, 1]). Every mapping A : X → II is called a vague set on X. The value A (x) is written as [t _A , f _A ]. For the sake of simplicity we write A = 〈t _A , f _A 〉. Obviously, t _A and f _A are mapping from X to [0, 1] satisfying 0 ≤ t _A  (x) + f _A  (x) ≤1, for all x ∈ X. Note that t _A  (x) is considered as the lower bound for degree of membership of x in A and f _A  (x) is the lower bound for negative of membership of x in A. So, the degree of membership of x in the vague set A is characterized by interval [t _A  (x) , 1 - f _A  (x)]. Hence, a vague set is a special case of interval-valued sets studiedby many mathematicians and applied in many branches of mathematics. It is worth to mention here that interval-valued fuzzy sets are not vague sets. In interval-valued fuzzy sets, an interval-valued membership value is assigned to each element of the universe considering the evidence for x only, without considering evidence against x. In vague sets both memberships are independently proposed by the decision maker. This makes a major difference in the judgment about the grade of membership. For a given set V, define an equivalence relation ∼ on V × V - {(x, x) | x ∈ V} as follows: ( x 1 , y 1 ) ∼ ( x 2 , y 2 ) ⇔ { x 1 , y 1 } = { x 2 , y 2 } .

The quotient set obtained in this way is denoted as V 2 ˜, and the equivalent class that contains the element (x, y) is denoted as [(x, y)], xy or yx (See [3]).

Definition 2.1. [1, 7] A vague graph is a pair G = (A, B), where A and B are vague sets on V and V 2 ˜ respectively which satisfy t _B  (xy) ≤ t _A  (x) ∧ t _A  (y) and f _B  (xy) ≥ f _A  (x) ∨ f _A  (y), for all ( x , y ) ∈ V 2 ˜ and t _B  (xy) = f _B  (xy) =0, for all ( x , y ) ∈ V 2 ˜ - E (E is the edge set).

Here, t _B  (xy) and f _B  (xy) represent the membership and non-membership values of the edge (x, y). Also, t _A  (x) and f _A  (x) represent the membership and non-membership values of the vertex x.A vague graph G is said to be complete if t _B  (v _i v _j ) = min {t _A  (v _i ) , t _A  (v _j )} and f _B  (v _i v _j ) = max {f _A  (v _i ) , f _A  (v _j )}, for all v _i , v _j  ∈ V. The complement of a vague graph G = (A, B) is a vague graph G ¯ = ( A ¯ , B ¯ ), where A ¯ = A = ( t A , f A ) and B ¯ = ( t B ¯ , f B ¯ ) is defined by:

t B ¯ ( xy ) = min ( t A ( x ) , t A ( y ) ) - t B ( xy ) , f B ¯ = f B ( xy ) - max ( f A ( x ) , f A ( y ) ) .

Definition 2.2. [2] Let G 1 * = ( V 1 , E 1 ) (resp., G 2 * = ( V 2 , E 2 )) be a given graph, and G₁ = (A₁, B₁) (resp., G₂ = (A₂, B₂)) be a vague graph of G 1 * = ( V 1 , E 1 ) (resp., G 2 * = ( V 2 , E 2 )).

A homomorphism from G₁ to G₂ is a mapping g : V₁ → V₂ that satisfies the following two conditions:

t _{A 1}  (x₁) ≤ t _{A 2}  (g (x₁)) , f _{A 1}  (x₁) f _{A 2}  (g (x₁)) , (∀ x₁ ∈ V₁),

3 Busy vertices and free vertices in vaguegraphs

In this section, we define strong arc, isolated vertex, busy vertices and free vertices in vague graphs and study their image under an isomorphism.

Definition 3.1.Let G = (A, B) be a vague graph on G^∗. Then the order of G is denoted by |V| (or O (G)), and is defined by O ( G ) = | V | = ∑ x ∈ V 1 + t A ( x ) + f A ( x ) 2 . The size of G is denoted by |E| (or S (G)), and is defined by S ( G ) = | E | = ∑ xy ∈ E 1 + t B ( xy ) + f B ( xy ) 2 . The neighborhood degree of a vertex x is defined as deg(x) = (d _t  (x) , d _f  (x)), where d t ( x ) = ∑ y ∈ N ( x ) t A ( y ) and d f ( x ) = ∑ y ∈ N ( x ) f A ( y ) . A path in G is a sequence of distinct vertices v₁, v₂, ⋯ , v_n+1 such that

t _B  (xy) >0 and f _B  (xy) =0 for some x, y∈{v₁, v₂, ⋯ , v_n+1}.

A vertex u of G is said to be an isolated vertex if f _B  (uv) =0, for all v ∈ V. Finally G is called connected if t _B  (xy) >0 and f _B  (xy) >0, for all xy ∈ E.

Example 3.2. Consider a vague graph G such that V = {x, y, z}, E = {xy, yz, xz} (see Fig. 1).

By routine computations, it is easy to show that O (G) =1.55, S (G) =1.6, d _t  (x) =0.3, d _t  (y) =0.4, d _t  (z) =0.5, d _f  (x) =1.1, d _f  (y) =1, d _f  (z) =0.9.

Notations. In a vague graph G = (A, B), we consider t B k , f B k , t B ∞ and f B ∞ as follows; t B k ( uv ) = sup { t B ( uv 1 ) ∧ t B ( v 1 v 2 ) ∧ t B ( v 2 v 3 ) ∧ ⋯ ∧ t B ( v k - 1 v ) ∣ u , v 1 , v 2 , ⋯ , v k - 1 , v ∈ V } f B k ( uv ) = sup { f B ( uv 1 ) ∧ f B ( v 1 v 2 ) ∧ f B ( v 2 v 3 ) ∧ ⋯ ∧ f B ( v k - 1 v ) ∣ u , v 1 , v 2 , ⋯ , v k - 1 , v ∈ V } t B ∞ ( uv ) = sup { t B k ( uv ) ∣ k = 1 , 2 , 3 , ⋯ } , f B ∞ ( uv ) = sup { f B k ( uv ) ∣ k = 1 , 2 , 3 , ⋯ }

Proposition 3.3.LetG₁andG₂be two vague graphs andhbe an isomorphism fromG₁ to G₂. Then, for everyx, y ∈ V₁we have t B 1 ∞ ( xy ) = t B 2 ∞ ( h ( x ) h ( y ) ) , f B 1 ∞ ( xy ) = f B 2 ∞ ( h ( x ) h ( y ) ) .

Proof. By definition, for every x, y ∈ V₁, we have t _{A 1}  (x) = t _{A 2}  (h (x)), f _{A 1}  (x) = f _{A 2}  (h (x)), for all x ∈ V₁ and t _{B 1}  (xy) = t _{B 2}  (h (x) h (y)), f _{B 1}  (xy) = f _{B 2}  (h (x) h (y)), for all xy ∈ E₁. Hence, f B 1 ∞ ( xy ) = sup { f B 1 k ( xy ) ∣ k = 1 , 2 , 3 , ⋯ } = sup { ⋀ i = 1 k f B 1 ( x i - 1 x i ) ∣ x = x 0 , x 1 , ⋯ , x k = y ∈ V 1 , k = 1 , 2 , ⋯ } = sup { ⋀ i = 1 k f B 2 ( h ( x i - 1 ) h ( x i ) ) ∣ x = x 0 , x 1 , ⋯ , x k = y ∈ V 1 , k = 1 , 2 , ⋯ } = sup { f B 2 k ( h ( x ) h ( y ) ) ∣ k = 1 , 2 , ⋯ } = f B 2 ∞ ( h ( x ) h ( y ) ) . Thus, f B 1 ∞ ( xy ) = f B 2 ∞ ( h ( x ) h ( y ) ), for all x, y ∈ V₁. Similarly, we can prove that t B 1 ∞ ( xy ) = t B 2 ∞ ( h ( x ) h ( y ) ) for all x, y ∈ V₁. □

Definition 3.4. In vague graph G = (A, B), the arc (u, v) is said to be strong, if t B ( uv ) ≥ t B ∞ ( uv ) and f B ( uv ) ≥ f B ∞ ( uv ).

Example 3.5. In this example, it is obvious that (x, p) and (y, z) are strong arcs (see Fig. 2).

Theorem 3.6.LetG₁andG₂be two vague graphs andG₁ ≅ G₂. Then the arc inG₁is strong if and only if the corresponding arc inG₂is strong, too.

Proof. Let h be an isomorphism from G₁ to G₂, and (x, y) in G₁ be a strong arc so,

t B 1 ( xy ) ≥ t B 1 ∞ ( xy ) .(1) Since h is an isomorphism between G₁ and G₂,t _{B 1}  (xy) = t _{B 2}  (h (x) h (y)) and f _{B 1}  (xy) = f _{B 2}  (h (x)h (y)). By (3.1) and Proposition 3.3, t B 2 ( h ( x ) h ( y ) ) ≥ t B 1 ∞ ( xy ) = t B 2 ∞ ( h ( x ) h ( y ) ). Similarly, we have f B 2 ( h ( x ) h ( y ) ) ≥ f B 2 ∞ ( h ( x ) h ( y ) ). Hence, (h (x) h (y)) is a strong arc in G₂. Conversely, by bijectivity and isomorphism property of h, strong arc in G₂ implies its preimage in G₁ is also strong. □

Theorem 3.7.LetG₁andG₂be two vague graphs andG₁be isomorphic toG₂. ThenG₁is connected if and only ifG₂is connected.

Proof. Let G₁ be isomorphic to G₂. Then, there exists an isomorphism h from G₁ to G₂, such that for every x, y ∈ V, t _{A 1}  (x) = t _{A 2}  (h (x)), t _{B 1}  (xy) = t _{B 2}  (h (x) h (y)) and f _{A 1}  (x) = f _{A 2}  (h (x)), f _{B 1}  (xy) = f _{B 2}  (h (x) h (y)). Now we have G₁ is connected if and only if t B 1 ∞ ( xy ) > 0 and f B 1 ∞ ( xy ) > 0, for all x, y ∈ V₁ if and only if t B 2 ∞ ( h ( x ) h ( y ) ) > 0 and f B 2 ∞ ( h ( x ) h ( y ) ) > 0, for all x, y ∈ V₁ (By Proposition 3.3) if and only if G₂ is connected. □

Definition 3.8. The busy value of a vertex v of a vague graph G = (A, B) is defined to be D (v) = (D _t  (v) , D _f  (v)) where D _t  (v) = ∑ _i t _A  (v) ∧ t _A  (v _i ) and D _f  (v) = ∑ _i f _A  (v) ∨ f _A  (v _i ) which v _i are neighbors of v and the busy value of a vague graph G is defined to be the sum of the busy values of all vertices of G, i.e., D (G) = ∑ _i D (v _i ) where v _i are vertices of G.

By routine computations we have D t ( x ) = 1.4 , D t ( y ) = 1.5 , D t ( z ) = 0.8 , D t ( w ) = 1.1 D f ( x ) = 1.6 , D f ( y ) = 1.4 , D f ( z ) = 1.2 , D f ( w ) = 0.8 D ( x ) = ( 1.4 , 1.2 ) , D ( y ) = ( 1.5 , 1.2 ) , D ( z ) = ( 0.8 , 1 ) , D ( w ) = ( 1.1 , 0.8 )

Definition 3.10.The vertex v in a vague graph G = (A, B) is said to be

t-busy vertex if t _A  (v) ≤ d _t  (v) otherwise it is called a t-free vertex.

Example 3.11. Consider vague graphs G and G ¯ as follows (see Fig. 4).

Here, x is t-partial and f-partial busy vertex since it is t-busy vertex and f-busy vertex in both G and G ¯. y is t-fully busy vertex since it is t-busy vertex in G but it is t-free vertex in G ¯. z is f-partial busy vertex since it is f-busy vertex in both G and G ¯.

Definition 3.12.An edge uv of vague graph G = (A, B) is said to be an effective edge if t B ( uv ) = t A ( u ) ∧ t A ( v ) and f B ( uv ) = f A ( u ) ∨ f A ( v ) .

Example 3.13. For the graph of Fig. 1, xy and yz are effective edges.

Theorem 3.14.If a vague graphGhas effective edges, then it has at least onet-busy vertex andf-busy vertex.

Proof. Let G = (A, B) be a vague graph which has some effective edges. Let (v₁, v₂) be an effective edge of G. Without loss of generality, let us assume that t _A  (v₁) ≤ t _A  (v₂) and f _A  (v₁) ≤ f _A  (v₂). Then clearly d _t  (v₁) ≥ t _A  (v₁) and d _f  (v₁) ≥ f _A  (v₁) which means that v₁ is a t-busy vertex and f-busy vertex. □

Theorem 3.15.If vague graphG = (A, B) withn > 1 is complete, then it has at least (n - 1) t-fully busy vertex.

Proof. Let G = (A, B) be a vague graph with n vertex. Let us label the vertex as a₁, a₂, ⋯ , a _n such that t _A  (a₁) ≥ t _A  (a₂) ≥ ⋯ ≥ t _A  (a _n ). Then,

d t ( a 1 ) = t A ( a 2 ) + t A ( a 3 ) + ⋯ + t A ( a n ) d t ( a 2 ) = t A ( a 2 ) + t A ( a 3 ) + ⋯ + t A ( a n ) d t ( a 3 ) = 2 t A ( a 3 ) + t A ( a 4 ) + ⋯ + t A ( a n ) d t ( a 4 ) = 3 t A ( a 4 ) + t A ( a 5 ) + ⋯ + t A ( a n ) ⋮ d t ( a n - 1 ) = ( n - 2 ) t A ( a n - 1 ) + t A ( a n ) d t ( a n ) = ( n - 1 ) t A ( a n ) . From the above n equations it follows that d _t  (a _i ) ≥ t _A  (a _i ), for i = 1, 2, ⋯ , n. Hence, there are at least (n - 1) t-busy vertex. Since G ¯ contains only isolated vertices, it has at least (n - 1) t-free vertices.Therefore, in a complete vague graph, there are at least (n - 1) t-fully busy vertices. □

Remark 3.16. From the definition of a complete vague graph, it follows that for each vertex v, (d _t  (v) , d _f  (v)) = (D _t  (v) , D _f  (v)).

Theorem 3.17.LetG = (A, B) be a vague graph. Then a vertexvis a

t-fully free vertex andf-fully busy vertex if and only if ( d t ( v ) , d f ( v ) ) < ( t A ( v ) , f A ( v ) ) ≤ ( d ¯ t ( v ) , d ¯ f ( v ) )

Proof. (i) and (ii) follows from the definition. (iii) v is a t-partial free vertex and f-partial busy vertex if and only if (d _t  (v) , d _f  (v)) < (t _A  (v) , f _A  (v)) and (D _t  (v) , D _f  (v)) - (d _t  (v) , d _f  (v)) < (t _A  (v) , f _A  (v)) ifand only if (t _A  (v) + d _t  (v) , f _A  (v) + d _f  (v)) <2 (t _A  (v) , f _A  (v)) and (D _t  (v) , D _f  (v)) < (t _A  (v) + d _t  (v) , f _A  (v) + d _f  (v)) if and only if (D _t  (v) , D _f  (v)) < (t _A  (v) + d _t  (v) , f _A  (v) + d _f  (v)) <2 (t _A  (v) , f _A  (v)). (iv) v is a t-partial busy vertex and f-partial free vertex if and only if (d _t  (v) , d _f  (v)) ≥ (t _A  (v) , f _A  (v)) and (D _t  (v) , D _f  (v)) - (d _t  (v) , d _f  (v)) ≥ (t _A  (v) , f _A  (v)) ifand only if (t _A  (v) + d _t  (v) , f _A  (v) + d _f  (v)) ≥2 (t _A  (v) , f _A  (v)) and (D _t  (v) , D _f  (v)) ≥ (t _A  (v) + d _t  (v) , f _A  (v) + d _f  (v)) if and only if (D _t  (v) , D _f  (v)) ≥ (t _A  (v) + d _t  (v) , f _A  (v) + d _f  (v)). □

Lemma 3.18.LetG₁ ≅ G₂andhbe an isomorphism fromG₁ to G₂. Then, deg(x) = deg(h (x)), for allx ∈ V.

Proof. Since G₁ ≅ G₂, we have t _{A 1}  (x₁) = t _{A 2}  (h (x₁)), f _{A 1}  (x₁) = f _{A 2}  (h (x₁)), for all x₁ ∈ V₁. Hence d t ( x ) = ∑ y ∈ V 1 t A 1 ( y ) = ∑ y ∈ V 1 t A 2 ( h ( y ) ) = d t ( h ( x ) ) d f ( x ) = ∑ y ∈ V 1 f A 1 ( y ) = ∑ y ∈ V 1 f A 2 ( h ( y ) ) = d f ( h ( x ) )

Also, we know that deg(x) = (d _t  (x) , d _f  (x)), for all x ∈ V. Thus, deg(x) = deg(h (x)), for all x ∈ V. □

Theorem 3.19.IfG₁ ≅ G₂andxis a busy vertex inG₁, then it is a busy vertex inG₂, too.

Proof. Let g : V₁ → V₂ be an isomorphic from G₁ to G₂. Then t _{A 1}  (x) = t _{A 2}  (g (x)), f _{A 1}  (x) = f _{A 2}  (g (x)), for all x ∈ V₁ and t _{B 1}  (xy) = t _{B 2}  (g (x) g (y)), f _{B 1}  (xy) = f _{B 2}  (g (x) g (y)), for all xy ∈ E₁. Also g preserves the degree of vertices, by Lemma 3.18., i.e. d _t  (x) = d _t  (g (x)), d _f  (x) = d _f  (g (x)), for all x ∈ V. If x is a busy vertex in G₁, then t _{A 1}  (x) ≤ d _t  (x) and f _{A 1}  (x) ≤ d _f  (x). So, t _{A 2}  (g (x)) ≤ d _t  (g (x)), f _{A 2}  (g (x)) ≤ d _f  (g (x)). Hence, g (x) is a busy vertex in G₂. □

Theorem 3.20.LetG₁be a vague graph which is weak isomorphism toG₂. Ifu ∈ V₁is a busy vertex inG₁, then its image under a weak isomorphism inG₂is busy, too.

Proof. Let g : V₁ → V₂ be a weak isomorphism between G₁ and G₂. Then, for all x, y ∈ V₁t A 1 ( x ) = t A 2 ( g ( x ) ) , f A 1 ( x ) = f A 2 ( g ( x ) )(2) and t B 1 ( xy ) ≤ t B 2 ( g ( x ) g ( y ) ) , f B 1 ( xy ) ≥ f B 2 ( g ( x ) g ( y ) ) .(3) Let u ∈ V₁ be a busy vertex. Then, t A 1 ( u ) ≤ d t ( u ) , f A 1 ( u ) ≤ d f ( u ) .(4) From (3.2) and (3.4) we have t A 2 ( g ( u ) ) = t A 1 ( u ) ≤ d t ( u ) = ∑ v ∈ V 1 t A 1 ( v ) = ∑ v ∈ V 1 t A 2 ( g ( v ) ) = d t ( g ( u ) ) . Hence, t _{A 2}  (g (u)) ≤ d _t  (g (u)). Also f A 2 ( g ( u ) ) = f A 1 ( u ) ≤ d f ( u ) = ∑ v ∈ V 1 f A 1 ( v ) = ∑ v ∈ V 1 f A 2 ( g ( v ) ) = d f ( g ( u ) ) . Therefore, t _{A 2}  (g (u)) ≤ d _f  (g (u)). Hence, g (u) is a busy vertex in G₂. □

The vague digraph G = (F, P) of the gas pipeline network, shown in Fig. 5, is represented by the following adjacency matrix:

G = [ (0.0,1.0) (0.1,0.8) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.2,0.7) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.3,0.6) (0.2,0.7) (0.1,0.8) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.2,0.8) (0.1,0.7) (0.1,0.7) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) (0.0,1.0) ] The final weighted digraph WG that can be used for different kind of vulnerabilities can be calculated by finding the ranks of edges S _i  : = f_P(i) - t_P(i) * π_P(i). The final adjacency matrix and weighted digraph, shown in Fig. 6, are developed based on these weights:

WG = [ 0 0.73 0 0 0 0 0 0 0.6 0 0 0 0 0 0 0 0.51 0.6 0.73 0 0 0 0.68 0.64 0.64 0 0 0 0 0 0 0 0 0 0 0 ]

The overall algorithm is explained in Algorithm 1. It takes a vague set of pipeline fittings as an input. Lines 3-6 calculate the degrees of membership and non-membership for edges, and Line 7 assigns them to vague set of edges and adjacency matrix is prepared in Line 8. Finally, a weighted adjacency matrix is calculated in Lines 9-12 using rank techniques based on the degrees of membership and non-membership. This weighted matrix is printed in Line 13 and is used for calculating vulnerability in Line 14.

Algorithm 1. Vague digraph in vulnerability assessment of gas pipeline networks

(1) void fuzzy Pipeline Vulnerability (){

(2)  F=Vague set of Pipeline fitting;

(3)  Count Fitt = count(F);

(4)  P=Empty vague set;

(5)  for (int x = 0 ; x< Count Fitt ; x ++){

(6)    for (int y = 0 ; y< Count Fitt ; y ++){

(7)      if (F(x) is adjacent to F(y)){

(8)        t_P(xy) = min(t_F(x), t_F(y));

(9)        f_P(xy) = max(f_F(x), f_F(y));

(10)      }

(11)    }

(12)  }

(13)  P=Vague set of edges;

(14)  G=Vague relation (Adjacency matrix of F × F);

(15)  WG= Weighted relation (Adjacency matrix of F × F);

(16)  no Of Edges = count (P);

(17)  for (int i = 0; i < no Of Edges; i++){

(18)    S _i  = f _{P i}  - t _{P i}  ∗ π _{P i}

(19)    x=Adjacent from Node of P _i ;

(20)    y=Adjacent to Node of P _i ;

(21)    WG _xy  = S _i ;

(22)}

(23)  print WG;

(24)  Calculated Vulnerability using WG;

(25)  }

5 Conclusion

Graph theory has several interesting applications in system analysis, operations research, computer applications, and economics. Since most of the time the aspects of graph problems are uncertain, it is nice to deal with these aspects via the methods of fuzzy systems. It is known that fuzzy graph theory has numerous applications in modern science and engineering, especially in the field of information theory, expert systems, medical diagnosis, network routing, and control theory. In this paper, busy vertices, free vertices, strong arc, isolated vertex, fully free vertex and partial free vertex in vague graphs are defined. Likewise, some interesting properties of isomorphism and weak isomorphism on them are introduced.