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Abrar Hussain, Ahmed Alsanad, Kifayat Ullah, Zeeshan Ali, Muhammad Kamran Jamil, Mogeeb A. A. Mosleh, "Investigating the Short-Circuit Problem Using the Planarity Index of Complex q-Rung Orthopair Fuzzy Planar Graphs", Complexity, vol. 2021, Article ID 8295997, 22 pages, 2021. https://doi.org/10.1155/2021/8295997
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.
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 . In an IFS, the addition of the falsity degree enhanced the conception of the FS (introduced by Zadeh)  where only one aspect of the uncertain situation is expressed by a truth degree. This model of IFS is based on a certain restriction where the total of both falsity and truth degree lies in that restricts the allocation of the falsity and truth degrees. This barrier of allocation of the degrees of falsity and truth was relaxed by Yager  by introducing the conception of the Pythagorean FS (PyFS) where the total of the squares of both degrees allows to be in . The 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 . The 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 [5–10] for some recent accomplishment.
The earlier discussed fuzzy layouts including FS, IFS, PyFS, and QROFS take degrees of falsity and truth from the real unit interval . This was generalized by Ramot et al.  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). This was a new addition in the fuzzy environment and was followed by many researchers for practical usage. This notion of CFS was followed by Alkouri and Salleh , Ullah et al. , and Liu et al. [14, 15] who proposed the layouts of the complex IFS (CIFS), complex PyFS (CPyFS), and complex QROFS (CQROFS), respectively. These 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 and the CPyFS allows the total of the squares of the complex truth and falsity degrees in the range , while the CQROFS allows the total of the qth powers of the complex falsity and truth degrees in the range . In these investigations, if a complex truth or falsity degree is expressed in terms of polar form, say , then and 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 [16–24].
The theory of fuzzy graph (FG) was discovered by Rosenfeld  where the nodes and edges are defined in terms of the membership degrees. There 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  by associating the falsity degree with FG. The 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 . Moreover, Akram et al.  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 . Furthermore, Habib et al.  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 . The theory of QROFG was further studied and investigated by  where the notion of hypergraphs is introduced in the frame of QROFGs.
Motivated by the work of Ramot et al.  and Rosenfeld , the notion of complex FG (CFG) was introduced by Thirunavukarasu et al. . In CFGs, the degrees of truth are described using complex numbers instead of real values from . This notion of CFG cannot be applicable in some cases; for this, the theory of complex IFG (CIFG) was discovered by Yaqoob et al.  by discussing the complex-valued falsity degree in the environment of CFG. The theory of CIFG is an enhanced version of the CFG to manage awkward and complicated information in realistic issues. Moreover, Akram and Naz  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 . The 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 [36–40]. The concept of planarity is also investigated in several other frameworks including CPyFGs in [29, 41, 42]. There 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). The 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. The 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)The planarity of these outlines relies on the fact that the information is described in terms of a complex number instead of crisp values from . 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 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.
The 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.
The 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 , CPyFS , CPyFG , CPyFMS , QROFS , QROFG , and CQROFS  in this section. Throughout this paper, and denote the truth and falsity degrees of the information respectively, whose values are assigned from . Furthermore, denotes a universal set and be its arbitrary element.
Definition 1 (see ). A PyFS is of the form
Definition 2 (see ). A CPyFS is of the form
Definition 3 (see ). A CPyFG on is a pair, where is a CPyF vertex set on and is a CPyF relation on such thatwhere , for all and for all .
Definition 4 (see ). A CPyFMS is characterized in a way where falsity degree and truth degree functions give every object degrees of truth and falsity in given by and , that is, and and . The elements of , where (for amplitude terms) and (for phase terms). A similar pattern can be followed for the elements of such that and for all . For convenience, we have
Definition 5 (see ). A QROFS is of the form
Definition 6 (see ). A QROFG on a nonempty set is a pair where is a QROF vertex set on and is a QROF relation on such thatwhere and with .
Definition 7 (see ). A CQROFS is of the form
3. Complex q-Rung Orthopair Fuzzy Graphs
As discussed earlier, the notion of CQROFS  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.
Definition 8. A CQROFR is defined bywhere and .
Definition 9. A CQROFG is a pair , where is a CQROF vertex set on and is a QROF relation on such thatand , , for all .
Definition 10. A CQROFMS is characterized in a way where falsity degree and truth degree functions give to every object degrees of truth and falsity in by and that is, and and . The elements of , where (for amplitude terms) and (for phase terms). A similar pattern can be followed for the elements of such that and for all . For convenience, we have
Definition 11. Let be a CQROFS on , and let be a CQROFMS on such that.
Then, is called a CQROFMG.
Example 1. Let be an MG, where and . Suppose a CQROF vertex set and CQROF multiedge set defined on as , respectively.Then, the pair forms a CQROFMG and is portrayed in Figure 1.
Example 2. By using the information of Example 1, we have as a strong edge because and, on the other hand, .
Example 3. In Figure 1 is said to be an effective edge as in case of amplitude terms, and for phase terms, .
Example 4. Consider an MG , where and Suppose a CQROF vertex set and a CQROF edge set defined on and , respectively, as follows:
4. 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 . The 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 in a complex plane instead of crisp real values from .
First, we discuss the notion of strength of CQROF edges and support this concept by Example 5 as follows.
Definition 15. In a CQROFG, the strength of an edge is of the formIf , and . Then, the edge is known as strong and otherwise weak.
Example 5. Let be an MG, where and . Suppose a CQROF vertex set and CQROF multiedge set defined on and , respectively, as follows:Then, the strength of each edge is computed as follows:Since , edge is the only strong edge in the CQROFMG depicted in Figure 3.
Definition 16. Consider as a CQROFMG. Let us assume two edges in , i.e., and , intersect at point . The value of cut point is given bywhereIn the CQROFMG, the cut point and planarity are inversely proportional, so planarity decreases with the increase in the number of cut points.
Definition 17. Consider as a CQROFMG. Suppose to be the cut points of the edges. So, is called a complex QROF planar graph (CQROFPG) with CQROF PI of in the formIf there is no cut point in the CQROFPG, then its CQROF PI is assumed as 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.
Example 6. Suppose that is an MG in which and .
Let and be the CQROF vertex set and CQROF multiedge set defined on and , respectively, given as follows by a CQROFMG as depicted in Figure 4:Let and be two cut points in the CQROFMG, as shown in Figure 4. For the edges,The strength isThe first and second cut points are given as follows:The PI is given by
Theorem 1. Let be a CQROFMG whose intersecting edges are effective. Then, CQROF planarity index of is given aswhere and denote the number of crossing among the edges in .
Proof. Let be a CQROFMG having intersection edges effective, that is,Let be the cut points of the edges in , where is the positive integer. For each edge in ,Therefore, the cut point of the edges and isSo, for . The PI of the CQROFMG isHere, denotes the number of crossing among the edges in .
5. Faces and Dual Graphs of CQROFPGs
Faces in graphs are termed as those regions which are surrounded by edges. The 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 assures that there is no crossing between the CQQROF edges.
Definition 18. Let a CQROFPG with PI andA CQROFF is a region bordered by a CQROF edge set of of the CQROFPG defined as
Definition 19. A CQROFF is considered a strong CQROFF if , , , and and otherwise weak.
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 is a CQROFPG, as shown in Figure 5. Let , and be the CQROFFs given as follows:
Now, we see that is strong and the other two faces and are weak faces.
Remark 3. Every strong CQROFF has a strong CQROF edge.
The conception of dual graphs has been widely discussed in the literature, and it has a close connection with planar graphs. There 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.
Definition 20. Consider as a CQROFPG whereSuppose that the strong CQROFFs of are . Then, the CQROFDG of is a CQROFPG where denotes the nodes of for CQROFFs . Moreover, the degrees of truth and falsity of the vertices are given by mapping such thatThe edge is near to the strong CQROFF. Between every two CQROFFs and , there may exist more than one common edge. Hence, there may occur more than one edge between two nodes in CQROFDG . The degrees of the truth and falsity of CQROF edges of the CQROFDG can be computed aswhere denotes those edges that lie in the surrounding of strong CQROFFs and where . As there are no cut points in the CQROFDG of CQROFPG , its PI is .
Example 8. Consider a CQROFPG as portrayed in Figure 6 with . Calculation shows that three strong CQROFFs , and exist whose vertices are shown by little circles and the edges are displayed dotted. The nodes of CQROFDGs are computed as follows:We observed that comes out to be the only common edge between and in . Hence, there exists only one edge between the vertices and in CQROFDG of since , ,