Strong and superstrong vertices in intuitionistic fuzzy graphs

1 Introduction

Fuzzy set theory invented by L.A. Zadeh [15] in 1965, generalises 0 and 1 membership values of a crisp set to membership funtion of a fuzzy set [16, 17]. Krassimir T Atanassov [3, 4] introduced the concept of intuitionistic fuzzy sets (IFSs), which is a generalisation of a fuzzy set and IFS can present the degrees of both membership and non-membership along with a degree of hesitancy. Rosenfeld [12] considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs. The theory of intuitionistic fuzzy graphs (IFGs) was introduced by Krassimir T Atanassov in [14]. In [6], Karunambigai and Parvathi introduced intuitionistic fuzzy graph as a special case of Atanassov’s IFG. In [11], Parvathi, et al. studied operations on intuitionistic fuzzy graphs. Akram and Dawaz [1] discussed strong intuitionistic fuzzy graphs. Akram and Dudek [2] studied intuitionistic fuzzy hypergraphs and their applications. Parvathi, et al. [10] obtained intuitionistic fuzzy shortest hypergraph in a network. Bhutani and Rosenfeld [5] discussed strong arcs in fuzzy graphs. In [7], the edges were classifed into α-strong, β-strong and δ-weak depending on the strength of connectedness between the two vertices. Strong and superstrong vertices in a fuzzy graph are defined by Ramakrishnan, et al. [13] and by Nagoorgani, et al. [9]. In this way, the authors got motivated to introduce strong and superstrong vertices in IFGs. It is verified that a vertex v _i is strong if every edge incident at v _i has same membership and non-membership values and also proved that a vertex v _i is strong and superstrong if every edge incident at v _i has the same membership and non-membership values and membership is minimum and non-membership is maximum among all the edges in an IFG. Further, it is shown that if an IFG has more than one superstrong vertex, then the strength of connectedness between all the superstrong vertices are same.

The paper is organised as follows: In Section 2, preliminaries required for this research work are given. Section 3 deals with the definitions and theoretical concepts of strong and superstrong vertices in an IFG by using the strength of connectedness. Also, the necessary and sufficient conditions for a vertex to be strong and superstrong in IFGs are analysed. In Section 4, it is proved that every edge is a strong edge in a strong IFG and some results on the strong and superstrong vertices in a strong IFG has been proved. In Section 5, Intuitionistic fuzzy steady path algorithm is discussed.

2 Preliminaries

In this section, some basic definitions and theorems which are useful in constructing the properties relating to this study, are given.

Definition 2.1. [6] Minmax intuitionistic fuzzy graph (IFG) is of the form G = (V, E), where

(i) V = {v₁, v₂, … v _n } such that μ _i  : V → [0, 1] and ν _i  : V → [0, 1] denote the degrees of membership and non - membership of the element v _i  ∈ V respectively and 0 ≤ μ _i  + ν _i  ≤ 1, for every v _i  ∈ V  (i = 1, 2, …, n).

(ii) E ⊂ V × V where μ _ij  : V × V → [0, 1] and ν _ij  : V × V → [0, 1] are such that μ _ij  ≤ min (μ _i , μ _j )

ν _ij  ≤ max (ν _i , ν _j )

and 0 ≤ μ _ij  + ν _ij  ≤ 1 for every e _ij  ∈ E .

Here the triple (v _i , μ _i , ν _i ) denotes the degree of membership and degree of non - membership of the vertex v _i . The triple (e _ij , μ _ij , ν _ij ) denotes the degree of membership and degree of non - membership of the edge e _ij between v _i and v _j on V × V.

For an IFG G, the degree of hesitancy (hesitation degree) of the vertex v _i  ∈ V in G is Π _i  = 1 - μ _i  - ν _i and the degree of hesitancy (hesitation degree) of an edge e _ij  ∈ E in G is Π _ij  = 1 - μ _ij  - ν _ij .

Notation: Here after an IFG, G = (V, E) means a minmax IFG G = (V, E).

Note 1. When μ _ij  = ν _ij  = 0 for some i and j, then there is no edge between v _i and v _j . Otherwise, there exists an edge between v _i and v _j .

Definition 2.2. [6] An IFG, is said to be strong IFG if μ _ij  = min (μ _i , μ _j ) and ν _ij  = max (ν _i , ν _j ) for all e _ij  ∈ E.

Definition 2.3. [6] An IFG, is said to be complete IFG if μ _ij  = min (μ _i , μ _j ) and ν _ij  = max (ν _i , ν _j ) for every v _i , v _j  ∈ V.

Definition 2.4. [6] A path P in an IFG is a sequence of distinct vertices v₁, v₂, . . . . , v _n for all i, j = 1, 2, . . . , n such that either one of the following conditions is satisfied.

μ _ij  > 0 and ν _ij  = 0 for some i and j.

Definition 2.5. [7] The μ - strength of a path P = v₁, v₂, . . . . , v _n is defined as min {μ _ij } for all i, j = 1, 2, . . . n and it is denoted by S _μ  (P).

Definition 2.6. [7] The ν - strength of a path P = v₁, v₂, . . . . , v _n is defined as max {ν _ij } for all i, j = 1, 2, . . . n and it is denoted by S _ν  (P).

Definition 2.7. [7] If v _i , v _j  ∈ V ⊆ G, the μ - strength of connectedness between two vertices v _i and v _j is CONN_μ(G) (v _i , v _j ) = max{ S _μ  (P _n ) } and ν - strength of connectedness between two vertices v _i and v _j is CONN_ν(G) (v _i , v _j ) = min{ S _ν  (P _n ) } of all possible paths P₁, P₂, . . . . . . P _n between v _i and v _j .

Notation: The strength of connectedness between two vertices v _i and v _j are denoted by (CONN_μ(G) (v _i , v _j ) , CONN_ν(G) (v _i , v _j )).

Note 2. [7] (CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) , CONN_{ν(G)-(v i ,v j )} (v _i , v _j )) is the strength of connectedness between v _i and v _j in the IFG obtained from G by deleting the edge (v _i , v _j ).

Definition 2.8. [7] An edge e _ij is said to be strong edge if μ _ij  ≥ CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) and ν _ij  ≤ CONN_{ν(G)-(v i ,v j )} (v _i , v _j ) for every v _i , v _j  ∈ V.

Definition 2.9. [7] An edge e _ij is said to be weak edge if μ _ij  < CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) and ν _ij  > CONN_{ν(G)-(v i ,v j )} (v _i , v _j ) for every v _i , v _j  ∈ V.

Definition 2.10. [7] In an IFG G = (V, E), a path P between any two vertices is called the strongest path if its strength equals the strength of connectedness (CONN_μ(G) (v _i , v _j ) , CONN_ν(G) (v _i , v _j )) and both the values lie in the same edge of P.

Definition 2.11. [7] A v _i v _j path P in an IFG G = (V, E) is called a strong path if P contains only strong edges.

Definition 2.12. A vertex of a graph is said to be pendant if it is of degree one.

Definition 2.13. Two vertices v _i and v _j in an IFG G are said to be neighbors or adjacent in G if e _ij  ∈ E.

3 Strong and superstrong vertices of intuitionistic fuzzy graphs

Definition 3.1. Let G = (V, E) be an IFG. A vertex v _i  ∈ V is said to be strong if e _ij is a strong edge for all v _j incident with v _i .

Definition 3.2. Let G = (V, E) be an intuitionistic fuzzy graph. A vertex v _i  ∈ V is said to be superstrong if CONN_μ(G) (v _i , v _j ) = k₁ and CONN_ν(G) (v _i , v _j ) = k₂ for every v _j  (≠ v _i ) ∈ V and for some k₁, k₂ ∈ (0, 1].

Example 3.1. In Fig. 1, e₁₃, e₃₄, e₂₄, e₂₅ are strong edges and v₃ is a strong vertex, because all the edges incident at v₃ are strong. Also, the vertex v₁ is a superstrong vertex since CONN_μ(G)(v₁, v₂) = CONN_μ(G) (v₁, v₃) = CONN_μ(G) (v₁, v₄)= CONN_μ(G) (v₁, v₅) =0.3 and CONN_ν(G) (v₁, v₂) = CONN_ν(G) (v₁, v₃)   =  CONN_ν(G) (v₁, v₄)   =  CONN_ν(G)(v₁, v₅) =0.5.

Lemma 3.1. Let G = (V, E) be a connected and finite IFG G. Let v _i  ∈ V is a superstrong vertex in G. Then, for every v _j  ∈ V, CONN_μ(G) (v _i , v _j ) ≥ μ _ik and CONN_ν(G) (v _i , v _j ) ≤ ν _ik for all v _k adjacent to v _i .

Proof. Let G = (V, E) be a connected and finite IFG G. Let v _i  ∈ V is a superstrong vertex in G. By definition of superstrong vertex,CONN μ ( G ) ( v i , v j ) = CONN μ ( G ) ( v i , v k )(1)andCONN ν ( G ) ( v i , v j ) = CONN ν ( G ) ( v i , v k )(2)for every v _j  ∈ V. By Theorem 3.2.2 in [8], we have,CONN μ ( G ) ( v i , v j ) ≥ μ ik(3)andCONN ν ( G ) ( v i , v j ) ≤ ν ik(4)for all v _k adjacent to v _i and for every v _j  ∈ V. Therefore, Equations.(1) and (2) are true which implies v _i is a superstrong in IFG G.□

Example 3.2. In Fig. 2, v₄ is a superstrong vertex.

Corollary 3.2. If a vertex v _i  ∈ V is a superstrong in G, then CONN_μ(G) (v _i , v _j ) = min{ μ _ij  } and CONN_ν(G) (v _i , v _j ) = max{ ν _ij  } of any path P connecting v _i to v _j in G.

Theorem 3.3. Let G = (V, E) be a connected IFG. If every edge incident at v _i has the same membership and non-membership value, then v _i is a strong vertex in G.

Proof. Assume that every edge incident at v _i has same membership and non-membership value. To prove v _i is strong, we have to show that every edge incident at v _i is strong. If an edge incident at v _i is strong, then, μ _ij  ≥ CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) and ν _ij  ≤ CONN_{ν(G)-(v i ,v j )} (v _i , v _j ).

Case (i): If there is only one v _j adjacent to v _i , then the case is trivial.

Case (ii): There are more than one vertex which are adjacent to v _i . Since all the possible paths between v _i and v _j containing one of the e _ij and it is given that all the incident edges having same membership and non-membership values, we have, μ _ij  ≥ CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) and ν _ij  ≤ CONN_{ν(G)-(v i ,v j )} (v _i , v _j )

which implies that e _ij is a strong edge for every v _j adjacent to v _i . Hence, v _i is a strong vertex.□

Remark 3.1. The converse of Theorem 3.3 need not be true. In Fig. 3, v₆ is a strong vertex, but μ₆₂ ≠ μ₆₄ and ν₆₂ ≠ ν₆₄.

Example 3.3. In Fig. 3, v₁ is a strong vertex since the incident edges are strong.

Theorem 3.4. Let G = (V, E) be a connected IFG. Every edges incident with v _i  ∈ V have the same membership and non-membership value and μ _ij is minimum membership and ν _ij is maximum non-membership among all edges of E in G iff v _i is strong and superstrong vertex.

Proof. Let v _i  ∈ V be a vertex in G. Every edge incident at v _i has same membership and non-membership values for all v _j adjacent to v _i . From Theorem 3.3, v _i  ∈ V is a strong vertex.

Now we have to prove that v _i  ∈ V is a superstrong vertex of G. Since μ _ij is the minimum and ν _ij is the maximum among all edges of E in G, then the μ-strength of all paths from the vertex v _i to v _j  (≠ v _i ) ∈ V is μ _ij and the ν-strength of all paths from the vertex v _i to v _j  (≠ v _i ) ∈ V is ν _ij . Therefore,CONN μ ( G ) ( v i , v j ) = μ ij and CONN ν ( G ) ( v i , v j ) = ν ij(5)for all v _j adjacent to v _i . Let v _k be non-adjacent vertex to v _i . From the given, it is obvious that,CONN μ ( G ) ( v i , v k ) = μ ij and CONN ν ( G ) ( v i , v k ) = ν ij(6)for all v _k non-adjacent to v _i . From Equations.(5) and (6), it implies that v _i is a superstrong vertex of G.

Conversely, if v _i is a strong and superstrong vertex, then we have to show that every edge incident at v _i has same membership and non-membership value say μ _ij and ν _ij for all v _j adjacent to v _i . By definition, if v _i is strong, then all the incident edges at v _i are strong. By Theorem 3.9, if an edge is strong, then CONN_μ(G) (v _i , v _j ) = μ _ij and CONN_ν(G) (v _i , v _j ) = ν _ij . By definition, if v _i is a superstrong, then CONN_μ(G) (v _i , v _j ) = k₁ and CONN_ν(G) (v _i , v _j ) = k₂ for all v _j  (≠ v _i ) ∈ V. From the above observation, k₁ = CONN_μ(G) (v _i , v _j ) = μ _ij and k₂ = CONN_ν(G) (v _i , v _j ) = ν _ij for all v _j adjacent to v _i . Therefore all the edges e _ij incident with v _i have same membership and non-membership value.□

Theorem 3.5. Let G = (V, E) be a connected IFG. If the edges, say e _ij , incident with v _i  ∈ V have the same membership and non-membership value for all v _j adjacent to v _i and μ _ij is minimum and ν _ij is maximum among all edges of E in G, then v _i is a unique superstrong vertex in G.

Proof. Given that, the edges say e _ij , incident withv _i  ∈ V have the same membership and non-membership value for all v _j adjacent to v _i . It is also given that μ _ij = minimum and ν _ij = maximum among all edges of E in G. Therefore by Theorem 3.4, v _i  ∈ V is a superstrong vertex of G = (V, E) which implies that CONN_μ(G) (v _i , v _j ) = μ _ij and CONN_ν(G) (v _i , v _j ) = ν _ij for all v _j  (≠ v _i ). Now we have to prove the uniqueness of v _i . Let v _k  ∈ V is another superstrong vertex of G. Then, CONN μ ( G ) ( v i , v k ) = μ ij = CONN μ ( G ) ( v k , v i ) and CONN ν ( G ) ( v i , v k ) = ν ij = CONN ν ( G ) ( v k , v i ) .

Now, consider a vertex v _l  ∈ V in which v _i ≠v _l  ≠ v _k  ∈ V, then

CONN_μ(G) (v _k , v _i ) = μ _ij  ≠ CONN_μ(G) (v _k , v _l ) (since μ _ij is minimum and μ _ij  < μ _kl )CONN μ ( G ) ( v k , v i ) ≠ CONN μ ( G ) ( v k , v l )(7)and

CONN_ν(G) (v _k , v _i ) = ν _ij  ≠ CONN_ν(G) (v _k , v _l ) (since ν _ij is maximum and ν _ij  > ν _kl )CONN ν ( G ) ( v k , v i ) ≠ CONN ν ( G ) ( v k , v l )(8)Equations.(7) and (8) gives a contradiction to the fact v _k is superstrong. Hence v _i is a unique superstrong vertex in G.□

Lemma 3.6. Let G = (V, E) be a connected IFG. If G has more than one superstorng vertex, then the strength of connectedness between the superstrong vertices are the same.

Proof. Let G be a connected IFG. Let v _i , v _k  ∈ V be two superstrong vertices in G. Assume (c₁, c₂) are the strength of connectedness between v _i and v _k and (d₁, d₂) are the strength of connectedness between v _k and v _i . Then, CONN μ ( G ) ( v i , v k ) = c 1 and CONN ν ( G ) ( v i , v k ) = c 2 CONN μ ( G ) ( v k , v i ) = d 1 and CONN ν ( G ) ( v k , v i ) = d 2 .

Therefore, CONN μ ( G ) ( v i , v k ) = c 1 = CONN μ ( G ) ( v k , v i ) = d 1 and CONN ν ( G ) ( v i , v k ) = c 2 = CONN ν ( G ) ( v k , v i ) = d 2 .

which is a contradiction. Hence the lemma.□

Corollary 3.7. Let G = (V, E) be an IFG. If μ _ij and ν _ij are constant for every e _ij  ∈ E, then every vertex of G is both strong and superstrong.

Lemma 3.8. Let G = (V, E) be a connected IFG. Let v _i  ∈ V be a pendant vertex and e _ij is the only one edge incident with v _i and degree of membership value μ _ij of e _ij is minimum and degree of non-membership value ν _ij of e _ij is maximum among all e _jk  ∈ E where e _jk are non-incident edges with v _i , then v _i is both strong and superstrong vertex.

Proof. Let G = (V, E) be a connected IFG. Letv _i  ∈ V be a pendant vertex in G. Given that the edge e _ij  ∈ E incident at v _i has μ _ij = minimum andν _ij = maximum among all e _jk  ∈ E where e _jk are non-incident edges with v _i . There exists only one edge between v _i and v _j such that CONN_μ(G) (v _i , v _j ) = μ _ij and CONN_ν(G) (v _i , v _j ) = ν _ij . By Theorem 3.9, if an edge e _ij is strong, thenCONN μ ( G ) ( v i , v j ) = μ ij and CONN ν ( G ) ( v i , v j ) = ν ij(9)

Thus, the pendant vertex v _i  ∈ V is a strong vertex.

Now, we have to prove that v _i  ∈ V is a superstrong. Let v _k  ∈ V be a vertex in G. From the given, for all v _k  ∈ V, the strength of connectedness between v _i and v _k isCONN μ ( G ) ( v i , v k ) = μ ij and CONN ν ( G ) ( v i , v k ) = ν ij(10)

From (9) and (10), CONN_μ(G) (v _i , v _j ) = CONN_μ(G) (v _i , v _k ),CONN_ν(G) (v _i , v _j ) = CONN_ν(G) (v _i , v _k ). Hence the lemma.□

Theorem 3.9. Let G = (V, E) be a connected IFG. An edge e _ij  ∈ G is strong iff

CONN μ ( G ) ( v i , v j ) = μ ij and CONN ν ( G ) ( v i , v j ) = ν ij .

Proof. Let G = (V, E) be a connected IFG. Let us assume that e _ij  ∈ E is a strong edge. Then, μ _ij  ≥ CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) and ν _ij  ≤ CONN_{ν(G)-(v i ,v j )} (v _i , v _j ). But, CONN_μ(G) (v _i , v _j ) = max { S _μ  } = max { min { μ _ij  }} = max { CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) , μ _ij  } = μ _ij (since μ _ij is strong, then μ _ij  ≥ CONN_{μ(G)-(v i ,v j )} (v _i , v _j )) and CONN_ν(G) (v _i , v _j ) = min { S _ν  } = min { max { ν _ij  }} = min { CONN_{ν(G)-(v i ,v j )} (v _i , v _j ) , ν _ij  } = ν _ij (since ν _ij is strong, then ν _ij  ≤ CONN_{ν(G)-(v i ,v j )} (v _i , v _j ))

Conversely, given that CONN_μ(G) (v _i , v _j ) = μ _ij and CONN_ν(G) (v _i , v _j ) = ν _ij . It is obvious that we must have, μ _ij  ≥ CONN_{μ(G)-(v i ,v j )} (v _i , v _j ) and ν _ij  ≤ CONN_{ν(G)-(v i ,v j )} (v _i , v _j ). Hence the theorem.□

Theorem 3.10. Let G = (V, E) be a connected IFG. For any strong v _i  ∈ V, if every edge incident with v _i has distinct membership and non-membership values, then v _i is not superstrong.

Proof. Let v _i be a strong vertex in G. Then, the edges incident with v _i are strong edges. Therefore, CONN_μ(G) (v _i , v _j ) = μ _ij and CONN_ν(G) (v _i , v _j ) = ν _ij by Theorem 3.9. Given that the edges incident with v _i has distinct membership and non-membership values. Thus, μ _ij  ≠ μ _ik and ν _ij  ≠ ν _ik for some v _j , v _k adjacent to v _i where v _i , v _j , v _k  ∈ V. From the definition of superstrong vertex, it is clear that if v _i  ∈ V is not a superstrong vertex, then CONN_μ(G) (v _i , v _j ) ≠ k₁ and CONN_ν(G) (v _i , v _j ) ≠ k₂ for some v _j  (≠ v _i ) ∈ V and for some k₁, k₂ ∈ (0, 1]. Since the edges which are incident with v _i are strong with distinct membership and non-membership values, we have μ _ij  ≠ μ _ik and ν _ij  ≠ ν _ik . That is, CONN_μ(G) (v _i , v _j ) = μ _ij  ≠ μ _ik  = CONN_μ(G) (v _i , v _k ) ⇒CONN_μ(G) (v _i , v _j ) ≠ CONN_μ(G) (v _i , v _k ) and CONN_ν(G) (v _i , v _j ) = ν _ij  ≠ ν _ik  = CONN_ν(G) (v _i , v _k ) ⇒CONN_ν(G) (v _i , v _j ) ≠ CONN_ν(G) (v _i , v _k ). Therefore, v _i is not superstrong.□

Remark 3.2. Converse of Theorem 3.10 need not be true. In Fig. 4, v₅ is neither superstrong nor strong.

Theorem 3.11. Let G = (V, E) be an IFG which itself is a cycle C in crisp sense. If the edges e _ij  ∈ E have membership values (μ _ij ) in ascending order and non-membership values (ν _ij ) in descending order for any vertex v _j  ∈ V, then v _i is superstrong and whose incident edges are having successive membership and non-membership values.

Proof. Since G is a cycle, there are only two edges incident with any vertex in V and hence with the source vertex v _i  ∈ V. Therefore, there exist atmost two possible paths P₁ and P₂ between any pair of vertices of G = C. Let C be v _i v _l v₁v₂ . . . . . . . v_j-1v _j v _k v _i . Assume e _li , e _ik , e _kj , e_j(j-1), . . . . , e₂₁, e_1l are successive edges in which the membership values are in ascending order and the non-membership values are in descending order. Letμ li < μ ik < μ kj < μ j ( j - 1 ) < . . . . . . < μ 21 < μ 1 l(11)andν li > ν ik > ν kj > ν j ( j - 1 ) > . . . . . . > ν 21 > ν 1 l(12)

Now, to claim v _i is superstrong, we have to prove that CONN_μ(G) (v _i , v _l ) = CONN_μ(G) (v _i , v _k ) = CONN_μ(G) (v _i , v _j ) and CONN_ν(G) (v _i , v _l ) = CONN_ν(G) (v _i , v _k ) = CONN_ν(G) (v _i , v _j ). That is,

CONN_μ(G) (v _i , v _l ) = max{ S _μ  (P _n ) } = max{ S _μ  (P₁) , S _μ  (P₂) } = max{ min { μ _il  } , min { μ _ik , μ _kj , μ_j(j-1), . . . . , μ₂₁, μ_1l }} = max{ μ _il , μ _ik  }, by (11) = μ _ik , since μ _ik  > μ _il .

CONN_μ(G) (v _i , v _k ) = max{ S _μ  (P _n ) } = max{ S _μ  (P₁) , S _μ  (P₂) } = max{ min { μ _ik  } , min { μ _il , μ_l1, μ₁₂ . . . . μ_(j-1)j, μ _jk  }} = max{ μ _ik , μ _il  }, by (11) = μ _ik , since μ _ik  > μ _il and, CONN_μ(G) (v _i , v _j ) = max{ S _μ  (P _n ) } = max{ S _μ  (P₁) , S _μ  (P₂) } = max{ min { μ _il , μ_l1, μ₁₂, . . . . , μ_(j-1)j } , min { μ _ik , μ _kj  }} = max{ μ _il , μ _ik  }, by (11) = μ _ik , since μ _ik  > μ _il .

Similarly,

CONN_ν(G) (v _i , v _l ) = min{ S _ν  (P _n ) } = min{ S _ν  (P₁) , S _ν  (P₂) } = min{ max { ν _il  } , max { ν _ik , ν _kj , ν_j(j-1), . . . . , ν₂₁, ν_1l }}= min{ ν _il , ν _ik  }, by (12) = ν _ik , since ν _ik  < ν _il .

CONN_ν(G) (v _i , v _k ) = min{ S _ν  (P _n ) } = min{ S _ν  (P₁) , S _ν  (P₂) } = min{ max { ν _ik  } , max { ν _il , ν_l1, ν₁₂ . . . . ν_(j-1)j, ν _jk  }} = min{ ν _ik , ν _il  }, by (12) = ν _ik , since ν _ik  < ν _il and, CONN_ν(G) (v _i , v _j ) = min{ S _ν  (P _n ) } = min{ S _ν  (P₁) , S _ν  (P₂) } = min{ max { ν _il , ν_l1, ν₁₂, . . . . , ν_(j-1)j } , max { ν _ik , ν _kj  }}= min{ ν _il , ν _ik  }, by (12) = ν _ik , since ν _ik  < ν _il . Hence, v _i is a superstrong in G.□

Example 3.4. In Fig. 5, v₆ is a superstrong vertex in IFG G which is a cycle.

Remark 3.3. In general, if there exists a superstrong vertex v _i  ∈ V in a connected IFG G = (V, E), then CONN_μ(G) (v _i , v _j )≤ min { CONN_μ(G) (v _j , v _k ) } and CONN_ν(G) (v _i , v _j )≥ max { CONN_ν(G) (v _j , v _k ) } for all v _j , v _k  ∈ V.

Theorem 4.1. Let G = (V, E) be a connected and strong IFG. Then every edge e _ij  ∈ E in G is strong.

Proof. By Theorem 3.9, an edge is strong iff CONN_μ(G) (v _i , v _j ) = μ _ij and CONN_ν(G) (v _i , v _j ) = ν _ij . Since G is strong IFG, μ _ij  = min (μ _i , μ _j ) and ν _ij  = max (ν _i , ν _j ) for all e _ij  ∈ E. Then the connectedness between v _i and v _j is CONN_μ(G) (v _i , v _j ) = μ _ij  = min (μ _i , μ _j ) and CONN_ν(G) (v _i , v _j ) = ν _ij  = max (ν _i , ν _j ) for everye _ij  ∈ E. Therefore e _ij is strong. Hence every edge in strong IFG is strong.□

Theorem 4.2. Let G = (V, E) be a connected and strong IFG. If μ _i = minimum and ν _i = maximum among all v _i  ∈ V, then v _i is strong and superstrong.

Proof. Let G = (V, E) be a connected and strong IFG. By the Theorem 4.1, if G is a strong IFG, then every edge e _ij  ∈ E in G is strong. Hence every vertex v _i  ∈ V is strong. To prove v _i  ∈ V is superstrong. Consider,μ _i = minimum and ν _i = maximum among all v _i  ∈ V. By the definition of strong IFG, μ _ij  = min (μ _i , μ _j ) and ν _ij  = max (ν _i , ν _j ). Then there exists v _i , v _j  ∈ V such that μ _ij  = min (μ _i , μ _j ) = μ _i since μ _i is minimum and ν _ij  = max (ν _i , ν _j ) = ν _i since ν _i is maximum among all v _i  ∈ V. From this, μ _ij = minimum and ν _ij = maximum for the edges incident with v _i among all edges in G. By Theorem 3.5, it is clear that v _i  ∈ V is a superstrong vertex.□

Corollary 4.3. Let G = (V, E) be a connected and complete IFG. If μ _i = minimum and ν _i = maximum among all v _i  ∈ V, then v _i is strong and superstrong.

5 Intuitionistic fuzzy steady path in an IFG

Definition 5.1. A v _i v _j path P in IFG G is said to be intuitionistic fuzzy steady path, if all the vertices in P are strong vertices for an arbitrary source vertex v _i .

Note 3. In Definition 5.1, the source vertex v _i need not be strong.

6 Conclusion

In this paper, an attempt has been made to introduce the concepts of strong and superstrong vertices in IFGs. The existence and non-existence of strong and superstrong vertices in IFGs by imposing certain conditions are studied. These concepts have also been extended to strong IFG. An algorithm for the identification of a intuitionistic fuzzy steady path between two vertices in IFGs is also designed. Further, the authors propose to extend these results to all other IFGs, in future. The concepts of strong and superstrong vertices alongwith distance and centre of IFGs may find applications in clustering.