Investigating the Short-Circuit Problem Using the Planarity Index of Complex q-Rung Orthopair Fuzzy Planar Graphs

Planar graphs play an effective role in many practical applications where the crossing of edges becomes problematic. This paper aims to investigate the complex q-rung orthopair fuzzy (CQROF) planar graphs (CQROFPGs). In a CQROFPG, the nodes and edges are based on complex QROF information that represents the uncertain knowledge in the range of unit circles in terms of complex numbers. The motivation in discussing such a topic is the wide flexibility of QROF information in the expression of uncertain knowledge compared to intuitionistic and Pythagorean fuzzy settings. We discussed the complex QROF graphs (CQROFGs), complex QROF multigraphs (CQROFMGs), and related terms followed by examples. Furthermore, the notion of strength and planarity index (PI) of the CQROFPGs is defined and exemplified followed by a study of strong and weak edges. We further defined the notion of complex QROF face (CQROFF) and complex QROF dual graph (CQROFDG) and exemplified these concepts. A study of isomorphism, coweak and weak isomorphism, is set up, and some results relating to the CQROFPG and isomorphisms are explored using examples. Furthermore, the problem of short circuits that results due to crossing is discussed because of the proposed study where an algorithm based on complex QROF (CQROF) information is presented for reducing the crossing in networks. Some advantages of the projected study over the previous study are observed, and some future study is predicted.


Introduction
Multiattribute decision making (MADM), pattern recognition, clustering, shortest path problems, and networking problems are some famous areas of real life where uncertainties are involved. In these situations, fuzzy set and its extended frameworks can be very beneficial, and many such problems are discussed in different fuzzy frameworks. In short, where there is uncertainty, there is room for a fuzzy set and its extensions to model such problems. To discuss several real-life problems involving complex information, the conception of the intuitionistic fuzzy set (IFS) based on falsity degree endowed with a truth degree was introduced by Atanassov [1]. In an IFS, the addition of the falsity degree enhanced the conception of the FS (introduced by Zadeh) [2] where only one aspect of the uncertain situation is expressed by a truth degree. is model of IFS is based on a certain restriction where the total of both falsity and truth degree lies in [0, 1] that restricts the allocation of the falsity and truth degrees. is barrier of allocation of the degrees of falsity and truth was relaxed by Yager [3] by introducing the conception of the Pythagorean FS (PyFS) where the total of the squares of both degrees allows to be in [0,1]. e PyFS is a significant improvement in fuzzy settings, but still, allocation of the degrees of falsity and truth needed more flexibility which was granted by the conception of the QROF set (QROFS) by Yager [4]. e layout of the QROFS removes any kind of barrier once and for all whenever information needs to be portrayed by duplets, i.e., falsity and truth degrees. A significant amount of research is performed in the frame of these duplets, and one is referred to [5][6][7][8][9][10] for some recent accomplishment. e earlier discussed fuzzy layouts including FS, IFS, PyFS, and QROFS take degrees of falsity and truth from the real unit interval [0, 1]. is was generalized by Ramot et al. [11] where complex numbers are used to portray uncertain information from a unit circle by the notion of complex FS (CFS). In the settings of CFS, truth degrees of the information are expressed by those complex numbers that lie in a unit circle (complex numbers with magnitude less than or equal to 1). is was a new addition in the fuzzy environment and was followed by many researchers for practical usage. is notion of CFS was followed by Alkouri and Salleh [12], Ullah et al. [13], and Liu et al. [14,15] who proposed the layouts of the complex IFS (CIFS), complex PyFS (CPyFS), and complex QROFS (CQROFS), respectively. ese investigations are quite familiar with the work carried out in [1,3,4] with the rules that the CIFS allows the total of complex truth and falsity degrees in the range [0, 1] and the CPyFS allows the total of the squares of the complex truth and falsity degrees in the range [0, 1], while the CQROFS allows the total of the q th powers of the complex falsity and truth degrees in the range [0, 1]. In these investigations, if a complex truth or falsity degree is expressed in terms of polar form, say (r.e iθ ), then r and θ denote the amplitude and phase terms, respectively. Although these conceptions are very recently introduced, still a great amount of attraction is given to these notions and some remarkable work has been carried out in [16][17][18][19][20][21][22][23][24]. e theory of fuzzy graph (FG) was discovered by Rosenfeld [25] where the nodes and edges are defined in terms of the membership degrees. ere were some cases where the falsity degree needs to be employed along with truth degree and the theory of FG was not able to cope with such issues; for this, the theory of intuitionistic fuzzy graph (IFG) was discovered by Parvathi and Karunambigai [26] by associating the falsity degree with FG. e theory of IFGs is an advanced and elastic version than FGs to manage awkward and complicated information in realistic issues. A deep study about the investigations of the IFGs is carried out in [27]. Moreover, Akram et al. [28] modified the theory of IFG by discussing the theory of Pythagorean FG (PyFG) where the restraint imposed on the IFG was improved by allowing the total of the squares of the truth and falsity degrees in the range [0, 1]. Furthermore, Habib et al. [29] modified the theory of PyFG to discover the theory of QROF graphs (QROFGs) by improving the restriction on duplets of information by allowing the sum of the q-powers of the truth and falsity grades in the range [0, 1]. e theory of QROFG was further studied and investigated by [30] where the notion of hypergraphs is introduced in the frame of QROFGs.
Motivated by the work of Ramot et al. [11] and Rosenfeld [25], the notion of complex FG (CFG) was introduced by irunavukarasu et al. [31]. In CFGs, the degrees of truth are described using complex numbers instead of real values from [0, 1]. is notion of CFG cannot be applicable in some cases; for this, the theory of complex IFG (CIFG) was discovered by Yaqoob et al. [32] by discussing the complexvalued falsity degree in the environment of CFG. e theory of CIFG is an enhanced version of the CFG to manage awkward and complicated information in realistic issues. Moreover, Akram and Naz [33] modified the theory of CIFG to discover the theory of complex PyFG (CPyFG) by improving the condition of CIFG, i.e., the sum of the squares of the complex-valued truth and complex-valued falsity grades ranges in [0, 1]. e theory of CPyFG is more flexible than CIFG to manage awkward and complicated information in realistic issues. A comprehensive study is established on the frame of CPyFG in [34,35].
PI is one of the widely discussed characteristics in graph theory which has some potential application in electric circuit theory. A planar graph is a graph that has no crossings of its edges, and this concept is very beneficial in reducing short circuits in electric networks. In the framework of FGs, IFGs, and PyFGs, the PI is discussed comprehensively with its applications, and for details, one can see [36][37][38][39][40]. e concept of planarity is also investigated in several other frameworks including CPyFGs in [29,41,42].
ere are some issues that cannot be resolved by using the CPyFG, for instance, if a decision maker provides information that cannot be handled by the CPyFG due to its constraint that the sum of the squares of both grades cannot exceed from unit interval. For managing such types of issues, we aim to enhance the theory on the PI of the FG, IFG, PyFS, CIFG, and CPyFG by investigating the theory of the PI of the complex QROFG (CQROFG). e theory of PI in the environment of the CQROFG has two main advantages over previous work as it discusses two aspects of opinion by falsity and truth degree. Secondly, it represents uncertain information with the help of complex truth and falsity degrees with greater flexibility. e summary of the proposed work is discussed as follows: (1) CQROFPG is an enhanced version of the QROFPG to cope with awkward and complicated information in realistic decision issues. In this investigation work, the idea of CQROFPG and its related ideas are discovered and verified with the help of examples. (2) e planarity of these outlines relies on the fact that the information is described in terms of a complex number instead of crisp values from [0, 1]. Here, the contemplations of CQROFMGs, CQROFPGs, and some perceived pieces of these graphs are presented by examining the complex QROF planarity using weak and strong edges. (3) A close-by association is set up among CQROFPGs and dual graphs. Discussion about the nonplanarity 2 Complexity of graphs and the thoughts of coweak isomorphism, isomorphism moreover, weak isomorphism for CQROFPGs is in like manner added. (4) By using numerical examples, the applications of the discovered theories are also utilized. To find the reliability and effectiveness of the presented work, we discussed the comparative study, advantages, and geometrical representations of the explored approaches.
e manuscript is organized as follows: In Section 2, we recall the ideas of the QROFS, QROFG, and CQROFSs and some other basic terms. In Section 3, the conception of the complex q-rung orthopair fuzzy relation (CQROFR), CQROFG, complex QROF multiset (CQROFMS), and CQROFMG and the ideas of strong and effective edges are elaborated. Section 4 is based on notions of strong edges in the CQROFGs, CQROFPG, and PI where examples are provided in support of these notions. In Section 5, the concepts related to CQROFF and CQROFDGs are discussed along with related results supported by examples. In Section 6, the isomorphism between two CQROFPGs is established where the coweak and weak isomorphic relations are investigated with the help of examples. In Section 7, we present an application of the PI of CQROFPGs in short-circuit problems with the help of illustrative examples. In Section 8, we comparatively examine the advantages of the CQROFPGs over the previous study followed by some conclusive remarks in Section 9.

Preliminaries
e purpose of this section is to recall some basic terms and notions based on which we proposed the new concepts and results. We discussed some basic definitions of the PyFS [2], CPyFS [13], CPyFG [36], CPyFMS [36], QROFS [4], QROFG [30], and CQROFS [14] in this section. roughout this paper, μ, α and ], β denote the truth and falsity degrees of the information respectively, whose values are assigned from [0, 1]. Furthermore, X denotes a universal set and m be its arbitrary element.
Definition 4 (see [36]). A CPyFMS T is characterized in a way where falsity degree F � D T (m) and truth degree Ţ � D T (m) functions give every object m ∈ X degrees of truth and falsity in C given by Ţ � D T (m): X ⟶ Q and F � D T (m): T (m) (for amplitude terms) and α 1 T (m) ≥ α 2 T (m) ≥ · · · ≥ α s T (m) (for phase terms). A similar pattern can be followed for the elements of F � D T (m) such that 0 ≤ (μ p T (m)) 2 + (] p T (m)) 2 ≤ 1 and 0 ≤ (α p T (m)) 2 + (β p T (m)) 2 ≤ 1 for all p � 1, 2, . . . , s. For convenience, we have Definition 5 (see [4]). A QROFS is of the form Definition 6 (see [30]). A QROFG on a nonempty set X is a pair G � (T, S) where T is a QROF vertex set on X and S is a QROF relation on E ⊆ X × X such that Definition 7 (see [14]). A CQROFS is of the form

Complex q-Rung Orthopair Fuzzy Graphs
As discussed earlier, the notion of CQROFS [14] flexibly described the falsity and truth degrees under uncertainty with the help of complex numbers. In comparison with the CIFS, the CPyFS gives the same results by allowing us wide ground for the selection of degrees of falsity and truth. In this section, we aim to present the notion of CQROFR followed by the conception of CQROFGs and related ideas.
en, G � (T, S) is called a CQROFMG. 14) , en, the pair (A, B) forms a CQROFMG and is portrayed in Figure 1.
To illustrate Definition 5, we consider an example based on Figure 1 as follows.
Example 2. By using the information of Example 1, we have An edge mn of G is called effective if the following axioms are satisfied for some fixed k: Example 3 supports Definition 13.

Definition 14. Let G � (T, S) be a CQROFMG and
be a CQROF multiedge set. en, a CQROFMG G is known as complete if To support Definition 14, we consider Figure 2 followed by Example 4.

Example 4. Consider an MG
Suppose a CQROF vertex set T and a CQROF edge set S defined on V and V × V, respectively, as follows:

Complex q-Rung Orthopair Fuzzy Planar Graphs
Planar graphs are those where there is no crossing of edges. Planarity is discussed under uncertainty on a wide range [36]. e aim of this section is to discuss the planarity in the environment of CQROFGs where the truth and falsity degrees of information are flexibly described with a parameter q in a complex plane instead of crisp real values from [0, 1].
First, we discuss the notion of strength of CQROF edges and support this concept by Example 5 as follows.
Suppose a CQROF vertex set T and CQROF multiedge set S defined on V and V × V, respectively, as follows: en, the strength of each edge is computed as follows: Since Figure 3.  Figure 2: Complete complex q-rung orthopair fuzzy multigraph. Complexity where In the CQROFMG, the cut point O and planarity are inversely proportional, so planarity decreases with the increase in the number of cut points.
Definition 17. Consider G as a CQROFMG. Suppose O 1 , O 2 , . . . , O p to be the cut points of the edges. So, G is called a complex QROF planar graph (CQROFPG) with CQROF PI of P in the form If there is no cut point in the CQROFPG, then its CQROF PI is assumed as (1, 1) and the corresponding crisp graph is a planar graph.

Remark 1. Every CQROFMG with some precise PI is a CQROFPG.
To support the abovementioned remark, we present Example 6. Figure 3: Complex q-rung orthopair fuzzy multigraph.

Complexity
Let T and S be the CQROF vertex set and CQROF multiedge set defined on V and V × V, respectively, given as follows by a CQROFMG as depicted in Figure 4: Let O 1 and O 2 be two cut points in the CQROFMG, as shown in Figure 4. For the edges, 10) . (26) e first and second cut points are given as follows: e PI is given by Theorem 1. Let G be a CQROFMG whose intersecting edges are effective. en, CQROF planarity index P � (P M , P N ) of G is given as where 0 ≤ (P M ) q + (P N ) q ≤ 1 and m O denote the number of crossing among the edges in G.
Proof. Let G be a CQROFMG having intersection edges effective, that is, Let O 1 , O 2 , . . . , O p be the cut points of the edges in G, where p is the positive integer. For each edge mn in G, erefore, the cut point O 1 of the edges mn and xy is Complexity 9 Here, O � 1 + 1 + · · · + 1 { } denotes the number of crossing among the edges in G.

Faces and Dual Graphs of CQROFPGs
Faces in graphs are termed as those regions which are surrounded by edges. e presence of the CQROFF is due to the presence of the edges in CQROFGs. Even the slightest strength CQROF edges guaranteed the existence of CQROFFs. A PI of (1e i(2π) , 1e i(2π) ) assures that there is no crossing between the CQQROF edges.

Remark 2.
An outer CQROFF is said to possess a limitless region of the CQROFPG; otherwise, it is considered as an inner CQROFF.
Example 7. Suppose that G is a CQROFPG, as shown in Figure 5. Let f 1 , f 2 , and f 3 be the CQROFFs given as follows: Now, we see that f 1 is strong and the other two faces f 2 and f 3 are weak faces.
Remark 3. Every strong CQROFF has a strong CQROF edge.
e conception of dual graphs has been widely discussed in the literature, and it has a close connection with planar graphs. ere exist some results that relate planarity with duality, i.e., the graphs for which dual graphs exist are known as planar graphs. Due to this reason, we aim to discuss the CQROFDG in our next study and establish its relationship with CQROFPGs.
Suppose that the strong CQROFFs of G are f 1 , f 2 , f 3 , . . . , f s . en, the CQROFDG of G is a CQROFPG G ′ � (Z ′ , T ′ , S ′ ), where Z ′ � ş k , k � 1, 2, 3, . . . , p} denotes the nodes of G for CQROFFs f k . Moreover, the degrees of truth and falsity of the vertices are given by mapping e edge ts is near to the strong CQROFF. Between every two CQROFFs f k and f h , there may exist more than one common edge. Hence, there may occur more than one edge ş k ş h between two nodes in CQROFDG G ′ . e degrees of the truth and falsity of CQROF edges of the CQROFDG can be computed as where (ts) r denotes those edges that lie in the surrounding of strong CQROFFs f h and f k where r � 1, 2, 3, . . . , s. As there are no cut points in the CQROFDG G ′ of CQROFPG G, its PI is (1, 1). Figure 6 with Z � v 1 , v 2 , v 3 , v 4 . Calculation shows that three strong CQROFFs f 1 , f 2 , and f 3 exist whose vertices are shown by little circles and the edges are displayed dotted. e nodes of CQROFDGs z 1 , z 2 , z 3 are computed as follows:

Example 8. Consider a CQROFPG G � (Z, T, S) as portrayed in
We observed that v 1 v 3 comes out to be the only common edge between f 1 and f 2 in G. Hence, there exists only one edge between the vertices z 1 and z 2 e edge set of the CQROFDG is given by 26) , 0.29e i2π(0.14) , z 2 z 3 , 0.31e i2π(0. 26) , 0.29e i2π(0. 14) .
Hence, G ′ � (Z ′ , T ′ , S ′ ) is a CQROFDG of G � (Z, T, S). e dotted line shows a CQROFDG of the CQROFPG which is represented by the bold edges in Figure 6.   .

absence of a weak edge in a CQROFDG G ′ of a CQROFPG G assures that the degrees of truth and falsity of CQROF edges of G ′ and G are equivalent.
Proof. Suppose that G is a CQROFPG having no weak edges. e CQROFDG of G is the family of G ′ , and it contained no crossing between any edge. Consider that f 1 , f 2 , f 3 , . . . , f s are strong CQROFFs of G. By using the definition of CQROFDG, the degrees of truth and falsity of CQROF edges of the CQROFDG are given as is represents (ts) r is an edge bordered by strong CQROFFs f h and f k . e number of common edges in the surrounding between f h and f k is r, where r � 1, 2, . . . , s. e two CQROFGs G and G ′ have the same number of CQROF edges, as there are no weak edges in G. Hence, there is a CQROF edge in G ′ , for every CQROF edge of G, with the same degree of truth and falsity.

Isomorphism between Two Complex QROF Planar Graphs
In this section, we study the isomorphism in CQROFPGs and establish some results to discuss the isomorphic, weak, and coweak isomorphic relations. Each concept is supported by an example.

Definition 21.
Let us consider two QROFGs G 1 � (T 1 , S 1 ) and G 2 � (T 2 , S 2 ). A mapping g: G 1 ⟶ G 2 is called homomorphism defined by g: V 1 ⟶ V 2 such that (i) μ T 1 (y) ≤ μ T 2 (g(y)), ] T 1 (y) ≥ ] T 2 (g(y)) (ii) μ S 1 (yz) ≤ μ S 2 (g(y)g(z)), ] S 1 (yz) ≥ ] S 2 (g(y)g(z)), ∀y ∈ V 1 , yz ∈ E 1 An isomorphic relation between two graphs tells us about the nature of the graph in case we know the properties of one graph. An isomorphism can be established between two CQROFPGs if they have equal division. In our next discussion, we aim to study isomorphisms and their relevant notions in CQROFPGs.

Definition 22.
An isomorphism is a bijective mapping H: Z 1 ⟶ Z 2 of two CQROFPGs G 1 and G 2 defined by H: G 1 ⟶ G 2 given as To illustrate the isomorphism of CQROFPGs, we consider Example 9.
Theorem 5. If a CQROPG G is isomorphic to a CQROFG K by an isomorphism H: G ⟶ K, then K has the same PI as G.
Proof. Consider a CQROPG G with no weak edge in it. Suppose that G 1 is a CQROFDG of G and G 2 is a CQROFDG of G 1 . Let F 1 denote the strong CQROFFs of G 1 and v 2 and v denote the number of vertices of G 2 and G. en, F 1 � v 2 and F 1 � v imply that v 2 � v. Furthermore, by the definition of duality, the truth and falsity degrees of an edge in CQROFDG and CQROFPG are equal. So, a coweak isomorphism between G and G 2 exists.
Example 12 is given to support the result in eorem 7. □ Example 12. In this example, the CQROFDGs G, G 1 , and G 2 are depicted as in respectively. ese CQROFGs shown verify the result discussed in eorem 7 and establish a coweak isomorphism between G 2 and G.

Application
In this section, we present an application of the CQROFPGs in electric networks to reduce the short circuits. We used the idea of CQROFPGs to analyze the short-circuit problem based on the planarity of the graph network. We assumed a problem of electricity supply from grid stations using CQROFPGs given below.
Grid stations are the main resources of power supply to our houses that use several lines (known as power lines) that are present throughout our city and streets.
e supply lines carry a huge amount of voltage, and to convert it to low voltage, the transformers are used. e transformers consist of various units which further consist of small, tiny wires responsible for the carriage of current. During current flow, sometimes a crossing occurs between the small wires while connecting various units. In a very limited time duration, often the voltage becomes so high that leads to short circuits that are very harmful to the electrical appliances in our houses. e short circuits can be reduced by arranging the wires in a way where the number of crossing needs to be less enough. We use the idea of CQROFPGs to deal with such problems for reducing the number of crossing. A brief algorithm for reducing the edge crossing in the setting of CQROFPGs followed by practical Example 13 is given below.

Algorithm.
is algorithm is to reduce the edge-crossing between nodes. e algorithms involve information in the form of CQROFPGs and use to the concept of strength, values of cut points, and PI in a network. e steps of the algorithm are as follows: where p � 1, 2, 3, . . . , s and E m , E n ∈ E (iv) evaluate the planarity value using P � (P M , P N ) � (1/(1 + M L 1 + M L 2 + · · · + M L p ), 1/(1 + N L 1 + N L 2 + · · · + N L p } )) (v) If E m and E n give no crossing, then we keep the formation of such edges as it is; otherwise, we alter the representation of such edges until the crossing is reduced (vi) By changing the layout of the E m and E n , the intersection becomes minimum and the PI gets maximized 16 Complexity Example 13. Let us assume a layout of units in the transformer, as depicted in Figure 13. e nodes represent units . . V 7 , and the connections between units are represented by edges, as shown in Figure 13. At the unit V 1 , we assume that the chances for a low spark are 61% and those for a high spark are 69%. Similarly, in the case of phase terms, suppose that, in a specific time, the chances for a low spark are 41% and those for a high spark are 49%.
is concludes that, for lower destruction that can occur at any time, an electrical model based on CQROF information can be adapted.

Comparative Study
In this segment, we illustrate a comparison of the results discussed in this paper with those already established. A brief comparative analysis of the current work with previous studies is discussed pointwise as follows: (1) e work in [36,38] discusses only one aspect of the vague information with the help of truth degree which leads to the loss of information as the falsity degree is ignored and results might not be effective. (2) e concept of planarity discussed in [39] illustrates PI in an uncertain environment with the help of truth and falsity degrees of an IFS which have certain restraints on the falsity and truth degrees of the objects and, hence, cannot be used smoothly. (3) e PI and its relevant theories and application in the setting of PyFG are discussed in [40] where a larger space for assigning the falsity and truth degrees is provided, but still, some have limitations when it comes to defining nodes and edges under some specific restraints of PyFGs. (4) Complex falsity and truth degrees are adapted to analyze the theory and results based on PI of CPyFGs. Application of the PI of the CPyFG is also analyzed in [41]. e work in [41] is considered to be more general than the work in [36,38,39]. In the setting of CPyFGs, one has an edge by describing the uncertain information with the help of two complex membership grades, but it has restraint on the grades due to which it becomes inapplicable in some cases. (5) e current study involves complex truth and falsity degrees associated with a variable parameter q that only produces flexibility in taking the complex truth and falsity degrees but can describe the opinion with the help of two kinds of grades that enhance the representation of the uncertain information.

Conclusions
In this script, we presented a theory related to the CQROFPG keeping in mind its usefulness. We discussed planarity in the setting of CQROFGs where an example is given to support each concept. An algorithm to study a short-circuit problem is developed. A summary of our research is given as follows: (1) We developed the notions of CQROFR, which leads to the concept of CQROFG. We also introduced the concept of CQROFMS which leads to the concept of CQROFMG. Some examples are given to support the theory. (4) We further establish a comprehensive study on isomorphism in the context of CQROFPGs and discussed the notions of coweek and weak isomorphisms. A few results on the isomorphisms of the CQROFPGs are also investigated. (5) An application of the CQROFPGs in solving a shortcircuit problem is investigated to see the effectiveness of the proposed work. A practical example is taken to illustrate the applicability of the newly established results in short-circuit problems. (6) A comparative investigation of the current work is set up with previously established results to show the superiority of the current work. (7) In the near future, we aim to extend our study of CQROFGs by defining several other relevant terms and notions and to study the applicability of the study in shortest path problems, social network problems, etc. is study when extended to the frame of T-spherical fuzzy graphs [43,44] can give us flexible and more accurate results.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest about the publication of this research article.