Graph theory has significantly grown in importance as a field of study due to its widespread applications. The significance of graph theory has escalated, owing to its versatile uses. For instance, it plays a crucial role in calculating the shortest path between two nodes in services like Google Maps. Graph theory finds applications in various domains, such as computer networks, image processing, electric circuits, and road networks. For applications of graph theory in chemistry, refer to [1–4].

In certain graph networks, intersections between edges give rise to problems, such as planning issues for electrical circuits, metros, and utility corridors. The elimination of crossings between electric wires might be feasible, but avoiding such crossings in a road network is often more complex. In road networks, nodes represent specific locations and edges symbolize highways. Hence, it becomes critical to determine the necessity of road crossings. The solution of this problem may involve the construction of underpasses or flyovers, but this may result in higher costs. However, this approach effectively resolves numerous issues, including road accidents and traffic jams that arise in congested areas. It is important to note that the term “congested” is a linguistic expression without precise boundaries. Various degrees of congestion, such as “very highly congested,” “very congested,” “congested,” “low congested,” and “very low congested,” can be employed. Additionally, these linguistic terms are associated with membership values, allowing for a more nuanced representation of congestion levels.

Strong and weak routes are distinguished by their congestion levels, with strong routes being congested and weak routes remaining non-congested. When developing a city road network, intersections between two weak routes hold significance, while intersections between two strong routes can be inconvenient. These problems can be effectively addressed using a fuzzy planar graph. In 1964, Zadeh [5] recognized the lack of clarity and ambiguity in real-world problems, leading to the introduction of fuzzy sets. This advancement has transformed the landscape of science and technology, especially when dealing with partial information or inaccessible data, leading to the emergence of fuzzy theories.

The concept of fuzzy graph theory was introduced by Rosenfeld [6], marking the beginning of fuzzy research. Kaufmann [7] introduced the theory of fuzzy graphs (FGs) based on fuzzy relations. Mordeson and Nair [8] introduced the FG complement and its associated operations. Shannon and Atanassov [9] introduced the intuitionistic fuzzy relation (IFR) and intuitionistic fuzzy graph (IFG) as its foundation. Parvathi et al. [10] introduced various operations on IFGs, such as union and join. The study of intuitionistic fuzzy cycles related to IFGs was conducted in [11]. Jabbar et al. [12] introduced the concept of fuzzy dual graphs, while Samanta and Pal [13] introduced fuzzy planar graphs. Alshehri and Akram [14], and Akram et al. [15] introduced intuitionistic and Pythagorean fuzzy planar graphs. Nirmal and Dhanabal [16] discussed specific fuzzy planar graphs in non-deterministic polynomial time. Pal et al. [17] examined planarity in FGs and proposed an alternative approach without restricting edge crossings. For various applications of fuzzy graphs, refer to [18–35].

Zadeh [36] presented interval-valued fuzzy sets (IVFS) as extension of fuzzy sets. Akram and Dudek [37] described the basic concepts of the interval-valued fuzzy graph (IVFG). These IVFGs were further studied in [38]. The notion of the IVFG is perhaps more generalized and flexible than a fuzzy graph because the membership degrees of vertices and edges are in the form of intervals. The interval-valued intuitionistic fuzzy graphs were discussed in [39]. In [40], interval-valued intuitionistic fuzzy competition graphs are investigated. Pramanik et al. [41] proposed interval-valued planar graphs. On the other hand, the cubic set proposed by Jun et al. [42] is a blending of ideas of IVFS and a fuzzy set. During recent 5 years, a lot of research work has been performed in the domain of the cubic set. Rashid et al. [43] presented the basic theory of cubic graphs. Khan et al. [44] investigated cubic intuitionistic FGs. The cubic planar graphs have been studied by Muhiuddin et al. [45]. They also applied the planarity of cubic graphs in a road network problem. In [46], the cubic Pythagorean FGs were investigated. The vertex regularity for cubic fuzzy graph structures is investigated in [47].

Definition 2.1

[5, 7]. Consider a non-empty set S and the mapping J : S → [ 0,1 ] . Subsequently, a fuzzy set can be defined as Q = f , J f : f ∈ S . In addition, with a mapping H : S × S → [ 0,1 ] , a fuzzy relation is defined as W = ⟨ f , t , H f , t ⟩ : f , t ∈ S . A pair G = ( Q , W ) is a fuzzy graph where a fuzzy set is denoted by Q and a fuzzy relation is denoted by W on S such that H f , t ≤ min J f , J t , for all f , t ∈ S , where H ( f , t ) and J ( f ) indicate the edge (f, t) membership value and the vertex f membership value, respectively. If a graph’s vertex f does not have any loops, then H ( f , f ) = 0 and a graph with a loop at the vertex f, then H ( f , f ) ≠ 0 .

Definition 2.2

[48]. Consider a non-empty set S and the mapping J : S → [ 0,1 ] . Subsequently, a fuzzy multi-set is defined over S as R = { f , J p ( f ) : f ∈ S } , for all p = 0, 1, 2, … , q _f , where q f = max { p : J p ≠ 0 } .Taking a non-empty set S along with the mapping J : S → [ 0,1 ] , H : S × S → [ 0,1 ] . A fuzzy multi graph is defined as G = ⟨ f , t , H p f , t , p = 1,2 , … , q f t : f , t ∈ S × S ⟩ , such that H p ( f , t ) ≤ min { J ( f ) , J ( t ) } , where q f t = max { ( f , t ) : H p ( f , t ) ≠ 0 } and H p ( f , t ) and J ( f ) indicate, respectively, the membership value of the fuzzy multi-edge and fuzzy vertex.

Definition 2.3

[49]. Consider a mapping J + : S → [ 0,1 ] and J − : S → [ 0,1 ] . Following that, an interval-valued fuzzy set (IVFS) is defined as T ′ = J − f , J + f : f ∈ S , where 0 ≤ J − ≤ J + ≤ 1 . The IVFG is described by G = ( T ′ , Y ′ ) such that T ′ is an IVFS on S and Y ′ is on S × S , satisfying H − f , t ≤ min J − f , J − t , H + f , t ≤ min J + f , J + t , for all f , t ∈ S .Let J p + : S → [ 0,1 ] and J p − : S → [ 0,1 ] be the mappings such that J p − ≤ J p + ∀ f ∈ S and p = 0,1 , … , q ̇ f , where q ̇ f = max { p , J ( f ) ≠ 0 } . IVFMS, or the interval-valued fuzzy multiset, on S can be defined as T = S , J p − , J p + = f , J p − , J p + | f ∈ S , p = 0,1 , … , q ̇ f . (1) Consider the mappings H v − : S × S → [ 0,1 ] and H v + : S × S → [ 0,1 ] . Subsequently, the interval-valued fuzzy multi graph (IVFMG) on S × S is described by ( T , U ) such that T provided in (1) and U are given as follows: U = S × S , H v − , H v + , v = 0,1 , … , q ̇ f , such that H v − ( f , t ) ≤ min { J p − ( f ) , J p − ( t ) } and H v + ( f , t ) ≤ min { J p + ( f ) , J p + ( t ) } , where p = 0, 1, 2, … , q _f , v = 0, 1, 2, … , q _f , q f = max { p : J p ≠ 0 } , and q f t = max { ( f , t ) : H v ( f , t ) ≠ 0 } .

Definition 2.4

A cubic multi-set is a combination of IVFMS and FMS expressed as O = ⟨ J p − , J p + , J p * ⟩ | p = 0,1 , … , q ̇ f , where q ̇ f = max { p , J ( f ) ≠ 0 } .

Definition 2.5

[50]. Take the mappings J : S → [ 0,1 ] and A : S → [ 0,1 ] to define the intuitionistic fuzzy set (IFS) as B = f , J f , A f | f ∈ S , where J ( f ) and A ( f ) are the membership value and non-membership value, respectively, satisfying J ( f ) + A ( f ) ≤ 1 . In the universe of discourse, an intuitionistic fuzzy relation (IFR) S × S along with the mappings H : S × S → [ 0,1 ] and L : S × S → [ 0,1 ] is also described as an IFS, which is P = f , t , H f , t , L f , t | f , t ∈ S × S , along with H ( f , t ) + L ( f , t ) ≤ 1 . Consider a graph G with a pair of IFS and IFR, i.e., G = ( B , P ) . Then, G is known as intuitionistic FG if the following conditions hold: H f , t ≤ min J t , J t and L f , t ≤ max A t , A t , such that 0 ≤ H ( f , t ) + L ( f , t ) ≤ 1 .

Definition 2.6

A count membership function and a count non-membership function define an intuitionistic fuzzy multi-set (IFMS) and is described as C M B : S → Q and C N B : S → Q , where B is a function and a crisp multiset chosen from [0,1] is Q . If f ∈ S then C M B ( t ) and C N B ( t ) are the crisp multisets. Additionally, the sequence of the membership is specified in the descending order, although the non-membership sequence may not be in that order. The IFMS is described as I ′ = t , J a t , A a t | t ∈ S , where a = 1, 2, … , q _t and q t = max { a : J a ≠ 0 } .

Definition 2.7

[51]. Consider the mappings J − : S → [ 0,1 ] , J + : S → [ 0,1 ] , A + : S → [ 0,1 ] , and A − : S → [ 0,1 ] . A definition of the interval-valued intuitionistic fuzzy set (IVIFS) is given as V = f , t , J − , J + , A − , A + : f , t ∈ S such that 0 ≤ [ J − , J + ] + [ A − , A + ] ≤ 1 , which is observable as 0 ≤ J − + A − ≤ 1 and 0 ≤ J + + A + ≤ 1 . Similarly, the interval-valued intuitionistic fuzzy relation (IVIFR) is expressed as D = f , t , H − , H + , L − , L + : f , t ∈ S × S along with the mappings H − : S × S → [ 0,1 ] , H + : S × S → [ 0,1 ] , L + : S × S → [ 0,1 ] , and L − : S × S → [ 0,1 ] , fulfilling the requirements 0 ≤ [ H − , H + ] + [ L − , L + ] ≤ 1 . Take a graph G with pairs of IVIFS and IVIFR such as G = ( V , D ) , then G is referred to as an intuitionistic interval-valued fuzzy graph (IIVFG) if it met the following criteria: H − f , t ≤ min J − f , J − t , H + f , t ≤ min J + f , J + t , L − f , t ≤ max A − f , A − t , L + f , t ≤ max A + f , A + t , for all f , t ∈ S .

Definition 2.8

A count membership function C M H : S → Q and count non-membership function C N H : S → Q defined an intuitionistic interval-valued fuzzy multi-set (IIVFMS), and Q denotes the crisp  multiset obtained from [0,1]. If ( f , t ) ∈ S , then CM _H (f, t) and CN _H (f, t) are the crisp multisets. Additionally, the non-membership sequence is not necessarily in the decreasing order, while the membership sequence  is defined in the decreasing order. The IVIFMS can also be described as H = f , t , J a f , t , A a f , t | f , t ∈ S , where a = 1, 2, … , q _p and q p = max { a : J a ≠ 0 } .

Definition 3.1

[44]. A graph that contains both IFS and IVIFS, as defined by the cubic intuitionistic fuzzy set (CIFS): C = f , V , B : f ∈ S , where IVIFS is V and IFS is B .Now a cubic intuitionistic fuzzy graph is a pair G = ( V , B ) of the crisp graph G * = ( S , E ) , where V = { f , ⟨ ( [ J − , J + ] , [ A − , A + ] ) , ( J * , A * ) ⟩ : f ∈ S } is a cubic intuitionistic fuzzy set and B = f , t , ⟨ H − , H + , L − , L + , H * , L * ⟩ : f , t ∈ S × S is a cubic intuitionistic fuzzy relation on S × S , satisfying the following conditions: H − f , t ≤ min J − f , J − t , H + f , t ≤ min J + f , J + t , L − f , t ≤ max A − f , A − t , L + f , t ≤ max A + f , A + t , H * f , t ≤ min J * t , J * t , L * f , t ≤ max A * t , A * t , and 0 ≤ [ J − , J + ] + [ A − , A + ] ≤ 1 , 0 ≤ [ H − , H + ] + [ L − , L + ] ≤ 1 , 0 ≤ J * + A * ≤ 1 , and 0 ≤ H * + L * ≤ 1 along with the mappings H − : S × S → [ 0,1 ] , H + : S × S → [ 0,1 ] , L + : S × S → [ 0,1 ] , L − : S × S → [ 0,1 ] , J − : S → [ 0,1 ] , J + : S → [ 0,1 ] , A + : S → [ 0,1 ] , A − : S → [ 0,1 ] , J * : S → [ 0,1 ] , and A * : S → [ 0,1 ] .

Definition 3.2

IVIFMS ( H ) and IFMS ( I ′ ) as a pair forms the intuitionistic cubic multiset C that is C B = ( H , I ′ ) . Similarly, an intuitionistic cubic multi graph is G = ( V , B ) on the crisp graph G * = ( S * , E * ) , where V = { f , ⟨ ( [ J p − , J p + ] , [ A p − , A p + ] ) ( J p * , A p * ) ⟩ : f ∈ S } is a cubic intuitionistic fuzzy set and B = { ( f , t ) , ⟨ ( [ H v − , H v + ] , [ L v − , L v + ] ) , ( H v * , L v * ) ⟩ : ( f , t ) ∈ S × S } is a cubic intuitionistic fuzzy relation on S × S , satisfying the following conditions: H v − f , t ≤ min J z − f , J z − t , H v + f , t ≤ min J z + f , J z + t , L v − f , t ≤ max A z − f , A z − t , L v + f , t ≤ max A z + f , A z + t , H v * f , t ≤ min J z * t , J z * t , L v * f , t ≤ max A z * t , A z * t , and 0 ≤ [ J p − , J p + ] + [ A p − , A p + ] ≤ 1 , 0 ≤ [ H v − , H v + ] + [ L v − , L v + ] ≤ 1 , 0 ≤ J p * + A p * ≤ 1 , and 0 ≤ H v * + A D v D v * ≤ 1 along with the mappings H v − : S × S → [ 0,1 ] , H v + : S × S → [ 0,1 ] , L v + : S × S → [ 0,1 ] , L v − : S × S → [ 0,1 ] , J p − : S → [ 0,1 ] , J p + : S → [ 0,1 ] , A p + : S → [ 0,1 ] , A p − : S → [ 0,1 ] , J p * : S → [ 0,1 ] , and A p * : S → [ 0,1 ] , where p = 0, 1, 2, … , q _o , v = 0, 1, 2, … , q _f , q f = max { p : J p ≠ 0 } , and q f t = max { ( f , t ) : H v ( f , t ) ≠ 0 } .

Example 3.3

Let S = { a , b , c } and J − ( a ) = 0.2 , J + ( a ) = 0.4 , J * ( a ) = 0.5 , A − ( a ) = 0.3 , A + ( a ) = 0.4 , A * ( a ) = 0.2 . Next, for the vertex b, J − ( b ) = 0.1 , J + ( b ) = 0.5 , J * ( b ) = 0.3 , A − ( b ) = 0.4 , A + ( a ) = 0.5 , A * ( b ) = 0.4 , and for the vertex c, J − ( c ) = 0.3 , J + ( c ) = 0.4 , J * ( c ) = 0.7 , A − ( c ) = 0.4 , A + ( c ) = 0.6 , A * ( c ) = 0.3 . The vertices a and b have two edges between them, indicating that the graph is an ICMG. The values of the several edges are shown in Figure 1.

Definition 3.4

The strength of an edge for an ICMG G = ( V , B ) can be determined by using I o p = M 𝔬 𝔭 , N 𝔬 𝔭 = 〈 M 𝔬 𝔭 − , M 𝔬 𝔭 + , N 𝔬 𝔭 , N 𝔬 𝔭 + , M 𝔬 𝔭 , N 𝔬 𝔭 * 〉 , = 〈 H o p − min J o + , J o + , H o p + min J o + , J o + , L o p − max A o + , A o + , L o p + max A o + , A o + , H o p * min J o * , J o * , L o p * max A o * , A o * 〉 , where the edge strength for the membership value is represented by M 𝔬 𝔭 , and in the same manner, the edge strength of the non-membership value is represented by N 𝔬 𝔭 . Intuitionistic strong cubic edges are described as such if M 𝔬 𝔭 ≥ 0.5 and N 𝔬 𝔭 ≤ 0.5 . If not, it is obviously a weak edge.

When using the ICMG, the intersection of two edges provides a specific value. This amount is known as a intuitionistic cubic-valued number. If two edges (l, k) and (e, r) cut each other at the point P , then the intuitionistic cubic number at P is determined as I P = M P , N P = M l k + M e r 2 , N l k + N e r 2 . A graph containing no cutting points between the edges is a planar graph. Planarity is affected by the number of cutting edges. If the number of cutting edges is greater, planarity is reduced, and vice versa. This motivates to introduce the concept of an intuitionistic cubic planar graph.

Definition 3.5

Suppose G = ( V , B ) is an ICMG along the cutting points P 1 , P 2 , … , P k and a specific graphical orientation, then the degree of planarity of planar graph G can be calculated as F = F M , F N = ⟨ F M − , F M + F N − , F N + F M * , F N * ⟩ , where F M − = 1 1 + M P 1 + , M P 2 + , … , M P k + , F M + = 1 1 + M P 1 − , M P 2 − , … , M P k − , F M * = 1 1 + M P 1 * , M P 2 * , … , M P k * , F N − = 1 1 + N P 1 + , N P 2 + , … , N P k + , F N + = 1 1 + N P 1 − , N P 2 − , … , N P k − , F N * = 1 1 + N P 1 * , N P 2 * , … , N P k * . Here, we can observe 0 ≤ F M ≤ 1 and 0 ≤ F N ≤ 1 .

A graph without any cutting point have a planarity value ⟨ [18][1, 1](1, 1)⟩. The cutting point between edges increases if F N decreases and F M increases and vice versa. Each ICMG having a certain planarity value is an intuitionistic cubic planar graph.

Example 3.6

Take G = ( V , B ) as a ICMG with a crisp graph G * = ( S , E ) , where S = { l , k , e , r } and E = { l k , k e , e r , r l } . The vertex and edge membership and non-membership are represented in Figure 2 and Table 1. Consider the cutting edge lr. Then, we compute M l r − = 0.2 min 0.4 , 0.4 = 0.2 0.4 = 0.5 , M l r + = 0.3 min 0.4 , 0.4 = 0.3 0.4 = 0.75 , M l r * = 0.1 min 0.5 , 0.1 = 0.1 0.1 = 1 , N l r − = 0.4 max 0.6 , 0.5 = 0.4 0.6 = 0.66 , N l r + = 0.6 max 0.6 , 0.5 = 0.6 0.6 = 1 , N l r * = 0.4 max 0.4 , 0.3 = 0.4 0.4 = 1 . So, I _lr = ⟨[0.5, 0.75][0.66, 1](1, 1)⟩. By repeating the same technique for the edge (k, e), we obtain I _ke = ⟨[0.5, 1][0.5, 1](1, 1)⟩. M P = 0.5 + 0.5 2 , 1 + 0.75 2 , 1 + 1 2 = 0.5 , 0.87 , 1 . Similarly, N P = 0.66 + 0.5 2 , 1 + 1 2 , 1 + 1 2 = 0.58 , 1 , 1 , I P = ⟨ 0.5 , 0.87 , 0.58 , 1 , 1,1 ⟩ . Next, we evaluate the valueofplanarity as F M − = 1 1 + 0.87 = 0.53 , F M + = 1 1 + 0.5 = 0.66 , F M * = 1 1 + 1 = 0.5 , F N − = 1 1 + 1 = 0.5 , F N + = 1 1 + 0.58 = 0.63 ,and F N * = 1 1 + 1 = 0.5 . F = ⟨ [ 0.53 , 0.66 ] , [ 0.5 , 0.63 ] , ( 0.5 , 0.5 ) ⟩ .

Theorem 3.7

The value of planarity is F M = 1 1 + n P , F N = 1 1 + n P , and F M + F N ≤ 1 for a complete CIG, where the total number of cutting points between the edges is n P .

Strong cubic intuitionistic planar graphs (SCIPGs) are graphs that have a cubic planarity value of at least 0.5 for their membership parts, and the non-membership component is less than or equal to 0.5. Alternatively, we have F = ( F M , F N ) , where F M ≥ 0.5 and F N ≤ 0.5 .

Theorem 3.8

The number of cutting points between the strong edges for SCIPG is at most 1.

Proof

Consider two cutting points P 1 and P 2 for cubic intuitionistic strong edges (o, p) and (f, t) along M 𝔬 𝔭 ≥ 0.5 and N 𝔬 𝔭 ≤ 0.5 cubic intuitionistic planarity value. Then, we obtain, N 𝔬 𝔭 + N o p 2 ≤ 0.5 and M 𝔬 𝔭 + M o p 2 ≥ 0.5 , which implies that 1 + N P 1 + N P 2 ≤ 2 and 1 + M P 1 + M P 2 ≥ 2 , demonstrating that F N ≥ 0.5 and F M ≤ 0.5 . This implies that G is a SCIPG. Therefore, there can only be one cutting point among strong edges. Planarity decreases as the amount or number of cutting places between the strong cubic intuitionistic edges increases. The degree of planarity possessed by the SCIPG is F N ≤ 0.5 and F M ≥ 0.5 . The graph will be a planar graph if there is exactly one cutting point between any two edges and if there are no crossings between any two edges. Therefore, there can be precisely one cutting point between any two edges.

Theorem 3.9

Consider a CIPG G with the planarity value F = ( F M , F N ) , where F N ≤ ⟨ [ 0.33 , 0.33 ] , 0.33 ⟩ and F M ≥ ⟨ [ 0.67 , 0.67 ] , 0.67 ⟩ . Therefore, none of the strong edges in G have a cutting point.

Proof

Now, we take a CIPG G with a planarity value F = ⟨ [ 0.67 , 0.67 ] , [ 0.33 , 0.33 ] , ( 0.67 , 0.33 ) ⟩ , and two strong cubic intuitionistic edges (l, k) (e, r) cut each other at Q . As these are strong edges, so M 𝔩 𝔨 ≥ 0.5 , N 𝔩 𝔨 ≤ 0.5 , M _er ≥ 0.5, and N _er ≤ 0.5. Now, by Theorem 3.7, the given value of F is F N = 1 1 + 0.5 ≥ 0.33 , and F M = 1 1 + 0.5 ≤ 0.67 which are in opposition to the degree of planarity of strong edges because F N ≥ 0.33 and F M ≥ 0.67 . Therefore, the cubic intuitionistic strong edges do not meet at a cutting point.

Definition 3.10

Consider a rational number 0 ≤ c ≤ 0.5 for a CIPG. Then, an edge (l, k) is an considerable edge if ⟨ [ H l k − min ( J l + , J l + ) , H l k + min ( J l + , J l + ) ] , H l k * min ( J l * , J l * ) ⟩ ≥ c , ⟨ [ L l k − max ( A l + , A l + ) , L l k + max ( A l + , A l + ) ] , L l k * max ( A l * , A l * ) ⟩ ≤ c ; a non-considerable edge is a term used to describe an edge that does not meet the criteria listed previously. A multi-edge lk for the CIMPG is considered significant if N_lk ≤ c and M l k ≥ c ∀ l , k ∈ G .

Remark: The number c is known as the considerable number if every edge is considerable. c assumes a specific pre-assigned value depending on the problem or an application. This pre-assigned number may or may not be unique.

Theorem 3.11

For a CIPG G having a considerable number c , the number of cutting points between considerable edges is at most [ 1 c ] .

Proof

Consider a strong CIPG G such that 0 ≤ c ≤ 0.5 F = ( F M , F N ) . If (l, k) is a considerable edge, then ⟨ [ L l k − max ( A l + , A l + ) , L l k + max ( A l + , A l + ) ] , L l k * max ( A l * , A l * ) ⟩ ≤ c , ⟨ [ H l k − min ( J l + , J l + ) , H l k + min ( J l + , J l + ) ] , H l k * min ( J l * , J l * ) ⟩ ≥ c , that is N 𝔩 𝔨 ≤ c and M 𝔩 𝔨 ≥ c . Here, P 1 , P 2 , … , P n are the n cutting points of (l, k) and (f, t). Subsequently, N P N 𝔩 𝔨 + N l k 2 ≤ c and M P = M 𝔩 𝔨 + M l k 2 ≥ c . Then, ∑ k i = 1 M P i ≥ n c , ∑ k i = 1 N P i ≤ n c . Hence, F N ≥ 1 1 + n c and F M ≤ 1 1 + n c . However, we know that a strong CIPG G has planarity values ( [ 0.33 , 0.33 ] , 0.33 ) ≥ F N ≥ 1 1 + n c and ( [ 0.67 , 0.67 ] , 0.67 ) ≤ F M ≤ 1 1 + n c , which shows that 0.5 ≥ F N ≥ 1 1 + n c and 0.5 ≤ F M ≤ 1 1 + n c . So, clearly n ≤ 1 c or 0.5 ≤ 1 1 + n c . One thing we can observe is that n = 1 c − 1 if  1 c ∈ Z , 1 c if  1 c ∉ Z . Here, the set of integers is denoted by Z.

The area covered by the edges is known as the faces of the graph. These faces of the graph can be categorized into two types: one is the outer face and the other is the inner face. The area surrounded by some specific edges and limited region is known as the inner face. The unlimited zone surrounded by the side edges of the graph is known as the outer face. A graph with the planarity value [18], [0, 0](1, 0) is known as a crisp planar graph. By assigning the degree of planarity F = [ 0,0 ] , [ 1,1 ] , ( 0,1 ) to an edge of a CIPG or by removing an edge, the faces of the graph automatically decreased. Here, we can observe that there is a direct relation among the faces and edges of the CIPG. By increasing the number of edges among vertices, the number of faces will be increased automatically.

Definition 3.12

Let G be a CIPG with F = ⟨ [ 1,1 ] , [ 0,0 ] , ( 1,0 ) ⟩ , which is the degree of planarity. Then, G possesses a cubic intuitionistic face (CIF) f that is formed by cubic intuitionistic edges. The non-membership and membership values of the face for the multi-edge B = ( f , t ) , 〈 [ H v − , H v + ] , [ L v − , L v + ] , ( H v * , L v * ) 〉 : ( f , t ) ∈ S × S can be computed as f N = { max { N f t v } , v = 1,2 , … , k : f t ∈ S × S } and f M = { min { M f t v } , v = 1,2 , … , k : f t ∈ S × S } .The face is referred as a strong face if it is enclosed by edges (face) with the membership value f M > ( [ 0.5 , 0.5 ] , 0.5 ) and a non-membership value f N < ( [ 0.5 , 0.5 ] , 0.5 ) . On the other hand, it is referred as a weak face if f M ≤ ( [ 0.5 , 0.5 ] , 0.5 ) and f N ≥ ( [ 0.5 , 0.5 ] , 0.5 ) .

Example 3.13

A graph displayed in Figure 3 is a CIPG. It contains faces f 1 , f 2 , f 3 . The edges {(l, e), (e, r), and (r, l)} surround the face f 1 . Similarly, the area enclosed by the edges {(l, k), (k, e), and (e, l)} is f 2 and the outer face is f 3 , which is surrounded by the set of edges {(l, k), (k, e), (e, r), and (l, r)}. s t r e n g t h o f m e m b e r s h i p o f f a c e 1 = min M l e − , M e r − , M 𝔩 𝔨 − , min M l e + , M e r + , M 𝔩 𝔨 + , min M l e * , M e r * , M 𝔩 𝔨 * = min 0.33 , 0.33 , 0.75 , min 0.66 , 0.66 , 1 , min 0.5 , 0.33 , 1 = 0.33 , 0.6 , 0.33 . s t r e n g t h o f n o n − m e m b e r s h i p o f f a c e 1 = max N l e − , N e r − , N 𝔩 𝔨 − , max N l e + , N e r + , N 𝔩 𝔨 + , max N l e * , N e r * , N 𝔩 𝔨 * = max 0.5 , 0.66 , 0.8 , max 0.66 , 1,1 , max 0.5 , 1,1 = 0.8 , 1 , 1 .

This implies that the strength of f 1 is [0.33, 0.6][0.8, 1](0.33, 1). The strengths of faces 2 and 3 by continuing the same procedure are evaluated as f 2 = [ 0.5 , 1 ] [ 1,1 ] ( 0.5 , 1 ) and f 3 = [ 0.33 , 0.66 ] [ 1,1 ] ( 0.5 , 1 ) . Additionally, it is evident that every face is weak because its non-membership strength is > 0.5 .

In this section, the notion of a cubic intuitionistic dual graph (CIDG) is presented. We can draw a CIDG only if the graph is either planar or have planarity measure f N ≤ ⟨ [ 0.33 , 0.33 ] , 0.33 ⟩ and f M ≥ ⟨ [ 0.67 , 0.67 ] , 0.67 ⟩ . The edges of the CIPG correspond to the edges of the CIDG, while faces of the CIPG correspond to the vertices of the CIDG.

Definition 4.1

Let G = ( V , B ) be a CIPG such that B = { ( f , t ) , ⟨ [ H v − , H v + ] , [ L v − , L v + ] , ( H v * , L v * ) ⟩ , v = 1,2 , … , k : f t ∈ S × S ) and f 1 , f 2 , … , f n be the faces of G surrounded by edges. So, G ′ = ( I ′ , S ′ ) is the dual graph of the CIPG, where the vertex set is denoted by I ′ and the edge set is denoted by S ′ . The following expression is used to calculate the membership and non-membership values of vertices: J y v = max H v f , t , v = 1,2 , … , k : f t is one of the edge along the cubic intuitionistic faces , A y v = min L v f , t , v = 1,2 , … , k : f t  is  one  of  the  edge along  the  cubic  intuitionistic  faces . It is noted that two faces may share more than one common edge. As a result, vertices of the CIDG may have several edges crossing them. If L v ( z i , z j ) and A v ( z i , z j ) indicate the non-membership and membership values, respectively, of an edge (z _i , z _j ), then for the membership and non-membership values of the cubic intuitionistic edge of the CIDG, we have L z i , z j v = L v f , t p a n d A z i , z j v = A v f , t p , where v = 1, 2, … , k and k is the number of common edges. The pendant vertex in the CIPG is associated with a loop in the CIDG with the degrees of membership and non-membership being the same as that of the pendant vertex. According to the value of planarity [18][0, 0](1, 0), the graph CIDG is always planar. By taking the vertices of the CIDG as the faces and the edges as the edges of the CIPG, we may create the CIPG from the CIDG.