Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 261 https://internationalpubls.com Multi-Bipolar Fuzzy Planar Graphs K. Rama Kishore a, b, Ch. Ramprasad b,*, P. L. N. Varma c a, c Department of Mathematics and Statistics, School of Applied Sciences, VFSTR (Deemed to be University), Vadlamudi, 522 213, India a, b, * Department of Mathematics, Vasireddy Venkatadri Institute of Technology, Namburu, 522 508, India Email ids: aitskrk987@gmail.com, b,*ramprasadchegu1984@gmail.com cplnvarma@gmail.com Article History: Received: 01-06-2024 Revised: 03-07-2024 Accepted: 29-07-2024 Abstract: Generalization of a bipolar fuzzy set helps to arrive at an m-BPF set that can be extended to understand important properties of planar and multigraph. The definition of m-BPFMGs, m- BPFPGs and m-BPFDGs are introduced, and some of their intriguing aspects are studied. An m-BPFPGs level of planarity is measured in this case using the term "degree of planarity." On the subject of planarity, some theorems have been proved. Additionally, we investigate isomorphism across m-BPFPGs. Keywords: Graphs, Multigraph, Planar Graph, FuzzyGraphs. Nomenclature m-bipolar fuzzy m-BPF m-bipolar fuzzy graph m-BPFG m-bipolar fuzzy multigraph m-BPFMG m- bipolar fuzzy multi line m-BPFML m-bipolar fuzzy planar graph m-BPFPG m-bipolar fuzzy complete multi graph m-BPFCMG m-bipolar fuzzy dual graph m-BPFDG 1. Introduction Ease of use and flexibility has made it possible to apply graph theory in real life situations requiring further research in this field. Ideally, problems related to traffic and power line management provide suitable areas of applications of graph theory. However, the complex nature of space organization and management through cost effective procedures is a dominant impediment to find a tangible solution, even to this date. For example, the problems of over lapping and cross-crossing power lines have traditional solutions of separating them in space and height. In printed circuit boards, this problem is solved by using layers of separation. We can try to minimize the height parameters and decrease the layers using the concept of planar graphs. When it comes to traffic management, routes with various degrees of congestion pose a common problem in several developed cities. With the help of the graph theory an application for an information system that informs road users and drivers about levels of congestion on roads can be developed. mailto:itskrk987@gmail.com mailto:ramprasadchegu1984@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 262 https://internationalpubls.com Electrical circuits can be represented using a graph model. In this system, minimizing no overlapping circuits is the primary goal. Rail lines, transmission lines, underground tunnels, etc. are all crucial components of city development. Accidents can happen when people are crossing. Despite the preference for routes without crossings, space constraints sometimes make them necessary. It is advisable to cross between congested and less congested routes rather than two congested ones. The definition of "congested" is ambiguous. We frequently use terms like "congested,""extremely congested,""very congested," etc. Linguistic terms are what they are, and they have certain relationship values. Strong routes are those that are congested, whereas weak routes are those that are less congested. Thus, in a city plan, crossing across strong and weak pathways may be allowed with a certain level of safety. Strong line and weak line of a fuzzy planar graph are denoted by the words "strong route" and "weak route," respectively, and the idea of fuzzy planar graph is denoted by the ability to traverse between strong and weak lines [1, 19, 23]. Zhang [32] originally put forward the concept of bipolar fuzzy sets in 1994 as an extension of fuzzy sets [30]. Fuzzy sets with a connection degree range of [-1, 1] are extended in bipolar fuzzy sets [30]. In a bipolar fuzzy set, an element's connection degree of 0 indicates that it has no bearing on the consequent property, its relationship degree of (0, 1] that it somewhat fulfills the property, and its relationship degree of [-1, 0) that it slightly satisfies the implied counter-property. Despite having a similar appearance, bipolar fuzzy sets and intuitionistic fuzzy sets are fundamentally distinct sets. It's crucial to have the ability to handle bipolar information in various fields. Positive information is known to reflect what is seen as possible, even as negative in order is known to stand for what is thought to be unfeasible. Recent fresh studies have been inspired by this area in various different areas. Bipolarity also occurs because we work with spatial information in applications such as image processing and spatial reasoning; as a result, fuzzy and possibility formalisms for bipolar information have been created [12]. For instance, positive information may be stated as a set of potential places and negative in order may be expressed as a set of unfeasible places when evaluating the position of an item in a space. The modeling of real-time systems where the underlying amount of information fluctuates with different degrees of precision is increasingly being done using fuzzy graph theory. Fuzzy models are beginning to show promise because they aim to reduce the differences between the symbolic models used in expert systems and the conventional numerical models used in research and engineering. Kaufmann proposed the first notion of a fuzzy graph [15] based on Zadeh's fuzzy relations [31]. Cut- nodes, roadways, connectivity, trees, and cycles are only a few of the fundamental graph-theoretic concepts that Rosenfeld [22] proposed as having a fuzzy equivalent. Sunitha and Vijaya kumar [27] described fuzzy trees, while Bhattacharya [10] made some comments on fuzzy graphs. Research in artificial intelligence and quantum computing makes it necessary to introduce concepts related to logical causality and decision-making that are integral to technical applications. This can be to some extent captured by the m-BPFG as this theory provides possibility for a ternary classification in the place at conventional binary classification. We know that we have positive (+) and negative (-) as well-established bipolar coordinates for traditional graph analysis. However, complex developments in the field of science tell us that there are categories which are both positive and negative at the given point of time [2-6]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 263 https://internationalpubls.com This can be demonstrated with the example of opinion making behavior of a country like India. Though India has a democratically elected government at the center, each state has its own democratically elected state governments. While opinions regarding many central government policies are expressed as agreements or oppositions by state governments, there are some central policies where there is a degree of both agreement and opposition. Let us explain an example from latest farm law proposed by the government of India with all 29 states taken as an m-block. In principle, this law was directly accepted by 20 states while six states opposed it. There are three states which expressed modifications in the policy as it evolved a mixed response. This is important as it influences the political system and patterns of voting in India. However, in each of the states that have accepted the policy some sections of the farmers opposed the policy; and in the states which opposed the policy there was a call for agreement from some sections of the farmers. This provides a suitable a set for applying the concept of m-BPFGs. m-BPFMGs, m- BPFPGs, and m-BPFDGs are introduced, and some of their interesting aspects are studied. Additionally, we discuss isomorphism across m-BPFPGs [20, 33-35]. In this work, we have only utilized accepted terminology and definitions. The readers are directed to [7, 8, 11, 13, 14, 16-18, 21, 24-26, 28, 29] extra symbols, jargon, and uses not addressed in the research. 2. Preliminaries The terms "m-BPFS," "m-BPFG," are defined in this section. For the m-BPFG to be generalized, an equivalence criterion was established. Create a relationship of equivalence from the given set V, ↔ on 𝑉 Γ— 𝑉 βˆ’ {(πœ„, πœ„): πœ„ ∈ 𝑉} as follows: (πœ„1,πœ…1) ↔ (πœ„2,πœ…2) if and only if either (πœ„1,πœ…1) = (πœ„2,πœ…2) or πœ„1, = πœ…2 , πœ…1 = πœ„2,. The equivalence class containing the components (πœ„, πœ…) is represented by πœ„πœ… or πœ…πœ„, while the quotient set is denoted by 𝑉2 ⃑ . Definition 2.1: An m-BPFG of a graph πΊβˆ— = (𝑉, 𝐸) is a pair 𝐺 = (𝑉, 𝑄, 𝑅) where 𝑄 = 〈[π‘ƒβ„ŽoΞ¨Q +, π‘ƒβ„ŽoΞ¨Q βˆ’] h=1 m βŒͺ , π‘ƒβ„ŽoΞ¨Q +: V β†’ [0, 1] and π‘ƒβ„ŽoΞ¨Q βˆ’: V β†’ [βˆ’1, 0] an m-BPFS is an m-BPFS on 𝑉 and 𝑅 = 〈[π‘ƒβ„ŽoΞ¨R +, π‘ƒβ„ŽoΞ¨R βˆ’]h=1 m βŒͺ , π‘ƒβ„ŽoΞ¨R +: 𝑉2 ⃑ β†’ [0, 1] and π‘ƒβ„ŽoΞ¨Q βˆ’: 𝑉2 ⃑ β†’ [βˆ’1, 0] in an m-BPFS in 𝑉2 ⃑ such that π‘ƒβ„ŽoΞ¨R +(πœ„πœ…) ≀ min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)}, π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…) β‰₯ max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} for all πœ„πœ… ∈ 𝑉2 ⃑ , β„Ž = 1, 2, β‹― , π‘š and π‘ƒβ„ŽoΞ¨R +(πœ„πœ…) = π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…) = 0 for all πœ„πœ… ∈ 𝑉2 ⃑ βˆ’ 𝐸. Definition 2.2: Let 𝑍 β‰  πœ™. Two functions, "count positive relationship" of 𝐻(𝐢𝐻 +)and "count negative relationship" of 𝐻(𝐢𝐻 βˆ’),are used to describe an m-bipolar fuzzy multiset (m-BPFMS) H generated from Z. Here𝐢𝐻 +: 𝑍 β†’ 𝑅1and 𝐢𝐻 βˆ’: 𝑍 β†’ 𝑅2, [0,1] and [-1,0] are the intervals from which the collections of every crisp multisite, 𝑅1 and 𝑅2are chosen. The definition of the positive sponsorship sequence is as follows: βŒ©π‘ƒ1oΞ¨H +(πœ„), 𝑃2oΞ¨H +(πœ„), β‹― , π‘ƒπ‘šoΞ¨H +(πœ„)βŒͺand βŒ©π‘ƒ1oΞ¨H βˆ’(πœ„), 𝑃2oΞ¨H βˆ’(πœ„), β‹― , π‘ƒπ‘šoΞ¨H βˆ’(πœ„)βŒͺwill be used to represent the negative relationship sequence. The following symbols stand for an m-BPFMS: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 264 https://internationalpubls.com {πœ„: 〈[𝑃1oΞ¨H +(πœ„), 𝑃1oΞ¨H βˆ’(πœ„)], [𝑃2oΞ¨H +(πœ„), 𝑃2oΞ¨H βˆ’(πœ„)], β‹― , [π‘ƒπ‘šoΞ¨H +(πœ„), π‘ƒπ‘šoΞ¨H βˆ’(πœ„)]βŒͺ: πœ„ ∈ 𝑍}. 3. m-Bipolar Fuzzy Planar Graphs (m-BPFPGs) Using the idea of an m-BPFMS, we first propose the idea of an m-BPFPG. Definition 3.1: Let 𝑉 β‰  πœ™ and 𝑄 = 〈[π‘ƒβ„ŽoΞ¨Q +, π‘ƒβ„ŽoΞ¨Q βˆ’] h=1 m βŒͺ be an m-BPFMS on 𝑉 . Let 𝑅 = { πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝑉 Γ— 𝑉} be an m-BPFMS of 𝑉 Γ— 𝑉 such that π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ≀ min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)}, π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒 β‰₯ max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} for all πœ„, πœ… ∈ 𝑉, 𝑒 = 1,2, β‹― , 𝑑 and β„Ž = 1, 2, β‹― , π‘š.Then𝐺is called an m-BPFMG. Remember that there could be a number of lines linking the nodes πœ„ and πœ… . [π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒, π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒]β„Ž=1 π‘š represent, respectively, the positive and negative relationship values of the line πœ„πœ… in 𝐺. 𝑒 stands for the quantity of lines connecting the nodes. 𝑅 is referred to as being m- BPFML set in m-BPFMG 𝐺 . π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 = 0 = π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒 for all πœ„πœ… ∈ 𝑉 Γ— 𝑉 βˆ’ 𝐸, 0 ≀ π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ≀ 1, βˆ’1 ≀ π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒 ≀ 0 for β„Ž = 1, 2, β‹― , π‘š . Example 1: Take a multigraph of πΊβˆ— = (𝑉, 𝐸) such that 𝑉 = {πœ„, πœ…, 𝜐, 𝜏}, 𝐸 = {πœ„πœ…, , πœ…πœ, 𝜐𝜏, πœ„πœ}. Let 𝑄 = 〈[π‘ƒβ„ŽoΞ¨Q +, π‘ƒβ„ŽoΞ¨Q βˆ’] h=1 m βŒͺ be an m-BPFS on 𝑉 and let 𝑅 = { πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝑉 Γ— 𝑉}be an m-BPFML set of 𝑉 Γ— 𝑉 defined by Fig. 1 m-Bipolar Fuzzy Multigraph It is clear from Fig. 1 through simple calculations that it is an m-BPFMG. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 265 https://internationalpubls.com Here 𝑄 = { πœ„ 〈[0.8,βˆ’0.3],[0.6,βˆ’0.8]βŒͺ , πœ… 〈[0.6,βˆ’0.7],[0.8,βˆ’0.9]βŒͺ , 𝜐 〈[0.9,βˆ’0.3],[0.8,βˆ’0.4]βŒͺ , 𝜏 〈[0.5,βˆ’0.8],[0.7,βˆ’0.3]βŒͺ } , 𝑅 = { πœ„πœ… 〈[0.4,βˆ’0.2],[0.5,βˆ’0.2]βŒͺ , πœ„πœ… 〈[0.3,βˆ’0.1],[0.3,βˆ’0.4]βŒͺ , πœ…πœ 〈[0.3,βˆ’0.2],[0.6,βˆ’0.1]βŒͺ , πœ…πœ 〈[0.2,βˆ’0.1],[0.4,βˆ’0.2]βŒͺ , 𝜐𝜏 〈[0.4,βˆ’0.2],[0.5,βˆ’0.1]βŒͺ , πœ„πœ 〈[0.1,βˆ’0.2],[0.5,βˆ’0.2]βŒͺ }. Definition 3.2: Let 𝑅 = { πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝑉 Γ— 𝑉}be an m-BPFME set in an m-BPFMG 𝐺. A multiline πœ„πœ… of 𝐺is strong if 1 2 min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)} ≀ π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒, 1 2 max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} ≀ π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒 for all 𝑒 = 1,2, β‹― , 𝑑 and β„Ž = 1, 2, β‹― , π‘š. Definition 3.3: Let 𝑅 = { πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 , π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝑉 Γ— 𝑉}be an m-BPFME set in an m-BPFMG 𝐺 . An m-BPFMG 𝐺 is complete if π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 = min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)}, π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒 = max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} for all πœ„, πœ… ∈ 𝑉, 𝑒 = 1,2, β‹― , 𝑑 and β„Ž = 1, 2, β‹― , π‘š. Example 2: Consider an m-BPFMG 𝐺 = (𝑉, 𝑄, 𝑅) as exposed in Fig. 2. It is obvious from Fig. 2 by simple calculations that it is an m-BPFCMG. Fig. 2. 2-Bipolar Fuzzy Complete Multigraph Here𝑄 = { 𝜐 〈[0.9,βˆ’0.8],[0.6,βˆ’0.5]βŒͺ , πœ„ 〈[0.7,βˆ’0.5],[0.5,βˆ’0.7]βŒͺ , 𝜏 〈[0.9,βˆ’0.5],[0.7,βˆ’0.9]βŒͺ }, 𝑅 = { πœπœ„ 〈[0.7,βˆ’0.5],[0.5,βˆ’0.5]βŒͺ , πœπœ„ 〈[0.7,βˆ’0.5],[0.5,βˆ’0.5]βŒͺ , 𝜐𝜏 〈[0.9,βˆ’0.5],[0.6,βˆ’0.5]βŒͺ , 𝜐𝜏 〈[0.9,βˆ’0.5],[0.6,βˆ’0.5]βŒͺ , πœ„πœ 〈[0.7,βˆ’0.5],[0.5,βˆ’0.7]βŒͺ }. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 266 https://internationalpubls.com Definition 3.4: Energy of an m-BPFE πœ„πœ… can be calculated by the value πΈπœ„πœ… = 〈[π‘ƒβ„ŽoπΈπœ„πœ… + , π‘ƒβ„ŽoπΈπœ„πœ… βˆ’ ]β„Ž=1 π‘š βŒͺ = 〈[ π‘ƒβ„ŽoΨ𝑅 +(πœ„πœ…)𝑒 min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)} , π‘ƒβ„ŽoΨ𝑅 βˆ’(πœ„πœ…)𝑒 max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} ] β„Ž=1 π‘š βŒͺ. Definition 3.5: Let 𝐺 be an m-BPFMG. A line πœ„πœ… should be an m-BPF strong if π‘ƒβ„ŽoπΈπœ„πœ… + β‰₯ 0.5 or π‘ƒβ„ŽoπΈπœ„πœ… βˆ’ ≀ 0.5, for β„Ž = 1, 2, β‹― , π‘š otherwise weak. Definition 3.6: Let 𝐺 be an m-BPFMG and let 𝑅 includes two lines πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ and 𝜐𝜏 〈[π‘ƒβ„ŽoΞ¨R +(𝜐𝜏)𝑠 ,π‘ƒβ„ŽoΞ¨R βˆ’(𝜐𝜏)𝑠] h=1 m βŒͺ which are intersect at a location π‘Œ, here 𝑒 and 𝑠 are permanent integers. The intersecting value at π‘Œ is defined as Ο‡π‘Œ = 〈[π‘ƒβ„ŽoΟ‡π‘Œ +, π‘ƒβ„ŽoΟ‡π‘Œ βˆ’]β„Ž=1 π‘š βŒͺ, where π‘ƒβ„ŽoΟ‡π‘Œ + = π‘ƒβ„ŽoπΈπœ„πœ… + +π‘ƒβ„Žo𝐸𝜐𝜏 + 2 , π‘ƒβ„ŽoΟ‡π‘Œ βˆ’ = π‘ƒβ„ŽoπΈπœ„πœ… βˆ’ +π‘ƒβ„Žo𝐸𝜐𝜏 βˆ’ 2 for β„Ž = 1, 2, β‹― , π‘š. Planarity falls as the amount of points of junction in an m-BPFMG rises. With this in consideration, we now presented the notion of an m-BPFPG below. Definition 3.7: Let 𝐺 be an m-BPFMG for a particular geometrical representation and defineπ‘Œ1 , π‘Œ2, β‹― , π‘Œπ‘Ÿ as the points where the lines are intersect. Consequently, 𝐺 is described as an m-BPFPG with a m-BPF planarity value Ξ“ = 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺ = 〈[ 1 1+{π‘ƒβ„ŽoΟ‡π‘Œ1 + +π‘ƒβ„ŽoΟ‡π‘Œ2 + +β‹―+π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ + } , βˆ’1 1+{π‘ƒβ„ŽoΟ‡π‘Œ1 βˆ’ +π‘ƒβ„ŽoΟ‡π‘Œ2 βˆ’ +β‹―+π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ βˆ’ } ] β„Ž=1 π‘š βŒͺ. It is obvious that Ξ“ is bounded, since 0 < π‘ƒβ„ŽoΞ“+ ≀ 1 and βˆ’1 ≀ π‘ƒβ„ŽoΞ“βˆ’ ≀ 0 for all β„Ž = 1, 2, β‹― , π‘š. The m-BPF planarity value of a given geometrical representation of an m-BPFPG is 〈[βˆ’1, 1]β„Ž=1 π‘š βŒͺ if there is no point of junction for such representation. In this instance, the crisp planar graph serves as the underlying crisp graph of this m-BPFG. Example 3: Take a multigraph πΊβˆ— = (𝑉, 𝐸) of 𝐺 = (𝑉, 𝑄, 𝑅) such that 𝑉 = {πœ„, πœ…, 𝜐, 𝜏} , 𝐸 = {πœ„πœ…, πœ„πœ, 𝜐𝜏, πœπœ…}. Fig. 3. 2-Bipolar Fuzzy Planar Graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 267 https://internationalpubls.com Let 𝑄 = { πœ„ 〈[0.9,βˆ’0.5],[0.7,βˆ’0.9]βŒͺ , πœ… 〈[0.5,βˆ’0.3],[0.3,βˆ’0.7]βŒͺ , 𝜐 〈[0.8,βˆ’0.4],[0.5,βˆ’0.7]βŒͺ , 𝜏 〈[0.9,βˆ’0.7],[0.5,βˆ’0.6]βŒͺ }be an m-BPF node set of 𝑉 and 𝑅 = { πœ„πœ… 〈[0.4,βˆ’0.2],[0.2,βˆ’0.6]βŒͺ , πœ„πœ… 〈[0.3,βˆ’0.1],[0.1,βˆ’0.5]βŒͺ , 𝜐𝜏 〈[0.7,βˆ’0.3],[0.3,βˆ’0.5]βŒͺ , πœ„πœ 〈[0.8,βˆ’0.4],[0.4,βˆ’0.5]βŒͺ , πœπœ… 〈[0.4,βˆ’0.2],[0.2,βˆ’0.5]βŒͺ }. be a m-BPFML set of 𝑉 Γ— 𝑉 is defined by an m-BPFMG as exposed in Fig. 3 has 2 point of junctions π‘Œ1 and π‘Œ2 . π‘Œ1 is a point between the lines πœ„πœ… 〈[0.4,βˆ’0.2],[0.2,βˆ’0.6]βŒͺ and 𝜐𝜏 〈[0.7,βˆ’0.3],[0.3,βˆ’0.5]βŒͺ and π‘Œ2 is between πœ„πœ… 〈[0.3,βˆ’0.1],[0.1,βˆ’0.5]βŒͺ and 𝜐𝜏 〈[0.7,βˆ’0.3],[0.3,βˆ’0.5]βŒͺ . For the line πœ„πœ… 〈[0.4,βˆ’0.2],[0.2,βˆ’0.6]βŒͺ , πΈπœ„πœ… = 〈[0.8, 0.67], [0.67, 0.86]βŒͺ. For the line πœ„πœ… 〈[0.3,βˆ’0.1],[0.1,βˆ’0.5]βŒͺ , πΈπœ„πœ… = 〈[0.6, 0.33], [0.33, 0.71]βŒͺ and for the line 𝜐𝜏 〈[0.7,βˆ’0.3],[0.3,βˆ’0.5]βŒͺ , E𝜐𝜏 = 〈[0.88, 0.75], [0.6, 0.83]βŒͺ . For the point of junction π‘Œ1 , intersecting value Ο‡π‘Œ1 = 〈[0.84, 0.71], [0.63, 0.85]βŒͺ and that for the next point of junction π‘Œ2, Ο‡π‘Œ2 = 〈[0.74, βˆ’0.54], [0.47, 0.77]βŒͺ. Therefore, the m-BPF planarity value for the m-BPFMG show in Fig. 3. is 〈[0.39, βˆ’0.44], [0.48, βˆ’0.38]βŒͺ. From the next theorem, we can compute an m-BPF planarity value for an m-BPFCMG. Theorem 3.1: Let 𝐺 be an m-BPFCMG. An m-BPF planarity value, Ξ“ = 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺ of 𝐺 is given by π‘ƒβ„ŽoΞ“+ = 1 1+ds and π‘ƒβ„ŽoΞ“βˆ’ = βˆ’1 1+ds , β„Ž = 1, 2, β‹― , π‘š , where ds is the number of point of junctions between the lines in 𝐺. Proof: As 𝐺 is complete, we getπ‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 = min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)}, π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒 = max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} for all πœ„, πœ… ∈ 𝑉, β„Ž = 1, 2, β‹― , π‘š and 𝑒 = 1,2, β‹― , 𝑑. Let π‘Œ1, π‘Œ2, β‹― , π‘Œπ‘Ÿ be the points of junction connecting the lines in 𝐺. For a line πœ„πœ… in 𝐺, 〈[π‘ƒβ„ŽoπΈπœ„πœ… + , π‘ƒβ„ŽoπΈπœ„πœ… βˆ’]β„Ž=1 π‘š βŒͺ = 〈[ π‘ƒβ„ŽoΨ𝑅 +(πœ„πœ…)𝑒 min{π‘ƒβ„ŽoΞ¨Q +(πœ„),π‘ƒβ„ŽoΞ¨Q +(πœ…)} , π‘ƒβ„ŽoΨ𝑅 βˆ’(πœ„πœ…)𝑒 max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„),π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} ] β„Ž=1 π‘š βŒͺ = 〈[1 , 1]β„Ž=1 π‘š βŒͺ. Consequently, for the point π‘Œ1 which is the point of junctions connecting the lines πœ„πœ… and 𝜐𝜏, the junction value is Ο‡π‘Œ1 = 〈[1 , 1]β„Ž=1 π‘š βŒͺ. Hence, Ο‡π‘Œβ„Ž = 〈[1 , 1]β„Ž=1 π‘š βŒͺ for β„Ž = 1, 2, β‹― , π‘š. Now for β„Ž = 1, 2, β‹― , π‘š, 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]βŒͺ = 〈[ 1 1 + {π‘ƒβ„ŽoΟ‡π‘Œ1 + + π‘ƒβ„ŽoΟ‡π‘Œ2 + + β‹― + π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ + } , βˆ’1 1 + {π‘ƒβ„ŽoΟ‡π‘Œ1 βˆ’ + π‘ƒβ„ŽoΟ‡π‘Œ2 βˆ’ + β‹― + π‘ƒβ„ŽoΟ‡π‘Œπ‘Ÿ βˆ’ } ]βŒͺ = 〈[ 1 1 + {1 + 1 + β‹― + 1} , βˆ’1 1 + {1 + 1 + β‹― + 1} ]βŒͺ = 〈[ 1 1 + ds , βˆ’1 1 + ds ]βŒͺ. Therefore, m-BPF planarity Ξ“ is given by〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺwhere π‘ƒβ„ŽoΞ“+ = 1 1+ds and π‘ƒβ„ŽoΞ“βˆ’ = βˆ’1 1+ds , β„Ž = 1, 2, β‹― , π‘š. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 268 https://internationalpubls.com Theorem 3.2: Let 𝐺 be an m-BPFPG with m-BPF planarity value 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺ is such that π‘ƒβ„ŽoΞ“+ β‰₯ 0.5, or π‘ƒβ„ŽoΞ“βˆ’ ≀ 0.5 for β„Ž = 1, 2, β‹― , π‘š. Then the number of points of junction between m- BPF strong lines in 𝐺 is at most one. Proof: Let 𝐺 be a strong m-BPFPG. Assume that 𝐺 has at least two point of junctions π‘Œ1 and π‘Œ2 between the two strong lines in 𝐺. For any strong line ( πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ ) π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 β‰₯ 1 2 min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)} , π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒 ≀ 1 2 max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)}. This proves that π‘ƒβ„ŽoΟ‡π‘Œ + β‰₯ 0.5 or π‘ƒβ„ŽoΟ‡π‘Œ βˆ’ ≀ 0.5 for β„Ž = 1, 2, β‹― , π‘š. Thus for 2 intersecting strong lines ( πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ )and ( 𝜐𝜏 〈[π‘ƒβ„ŽoΞ¨R +(𝜐𝜏)𝑠 ,π‘ƒβ„ŽoΞ¨R βˆ’(𝜐𝜏)𝑠] h=1 m βŒͺ ), π‘ƒβ„ŽoπΈπœ„πœ… + +π‘ƒβ„Žo𝐸𝜐𝜏 + 2 β‰₯ 0.5, π‘ƒβ„ŽoπΈπœ„πœ… βˆ’ +π‘ƒβ„Žo𝐸𝜐𝜏 βˆ’ 2 ≀ 0.5, for β„Ž = 1, 2, β‹― , π‘š. That is, π‘ƒβ„ŽoΟ‡π‘Œ1 + β‰₯ 0.5, π‘ƒβ„ŽoΟ‡π‘Œ1 βˆ’ ≀ 0.5. Similarlyπ‘ƒβ„ŽoΟ‡π‘Œ2 + β‰₯ 0.5, π‘ƒβ„ŽoΟ‡π‘Œ2 βˆ’ ≀ 0.5. This implies that1 + π‘ƒβ„ŽoΟ‡π‘Œ1 + + π‘ƒβ„ŽoΟ‡π‘Œ2 + β‰₯ 2, 1 + π‘ƒβ„ŽoΟ‡π‘Œ1 βˆ’ + π‘ƒβ„ŽoΟ‡π‘Œ2 βˆ’ ≀ 2. Therefore, π‘ƒβ„ŽoΞ“+ = 1 1+π‘ƒβ„ŽoΟ‡π‘Œ1 + +π‘ƒβ„ŽoΟ‡π‘Œ2 + ≀ 0.5, π‘ƒβ„ŽoΞ“βˆ’ = βˆ’1 1+π‘ƒβ„ŽoΟ‡π‘Œ1 βˆ’ +π‘ƒβ„ŽoΟ‡π‘Œ2 βˆ’ β‰₯ βˆ’0.5 for β„Ž = 1, 2, β‹― , π‘š. It contradicts the assumption that the m-BPFG is strong m-BPFPG. Therefore, there cannot be two points where strong the lines are intersected. Naturally, the m-BPF planarity value drops as the number of strong m-BPF line junction points rises. Similar to this, if there is just one point where 2 strong lines cross, an m-BPF planarity values are π‘ƒβ„ŽoΞ“+ < 0.5, and π‘ƒβ„ŽoΞ“βˆ’ > βˆ’0.5 for β„Ž = 1, 2, β‹― , π‘š. A strong m-BPFPG is any m-BPFPG that has no crossing between the lines. Thus, we get the conclusion that there is only one position where the strong lines of 𝐺 can connect. Theorem 3.3: Let 𝐺 be an m-BPFPG with an m-BPF planarity value Ξ“ = 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺ. If π‘ƒβ„ŽoΞ“+ β‰₯ 0.67, π‘ƒβ„ŽoΞ“βˆ’ ≀ βˆ’0.67, then 𝐺 do not have any point of junction between at two strong lines. Proof: If possible, let π‘Œ be a point of junction between two m-BPF strong lines ( πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ ) and ( 𝜐𝜏 〈[π‘ƒβ„ŽoΞ¨R +(𝜐𝜏)𝑠 ,π‘ƒβ„ŽoΞ¨R βˆ’(𝜐𝜏)𝑠] h=1 m βŒͺ ). For any m-BPF strong line ( πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 ,π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ ), we have π‘ƒβ„ŽoπΈπœ„πœ… + β‰₯ 0.5, π‘ƒβ„ŽoπΈπœ„πœ… βˆ’ ≀ 0.5 for β„Ž = 1, 2, β‹― , π‘š.For the minimum value of π‘ƒβ„ŽoπΈπœ„πœ… +and π‘ƒβ„Žo𝐸𝜐𝜏 + , π‘ƒβ„ŽoΟ‡π‘Œ + = 0.5 for β„Ž = 1, 2, β‹― , π‘š. Then, π‘ƒβ„ŽoΞ“+ = 1 1+0.5 < 0.67. Similarly, π‘ƒβ„ŽoΞ“βˆ’ > βˆ’0.67 forβ„Ž = 1, 2, β‹― , π‘š, a contradiction. As a result, 𝐺 lacks a spot where two m-BPF strong lines connect. Next, strong m-BPFPG is defined as follows. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 269 https://internationalpubls.com Definition 3.8: An m-BPFPG 𝐺 is called strong m-BPFPG if the m-BPF planarity value 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺ of the graph is π‘ƒβ„ŽoΞ“+ β‰₯ 0.67, π‘ƒβ„ŽoΞ“βˆ’ ≀ βˆ’0.67. Theorem 3.4: Any complete m-BPFG of 5 nodes or complete bipartite m-BPFG of 6 nodes are not strong m-BPFPG. Proof: Let 𝐺 = (𝑉, 𝑄, 𝑅 ) be a complete m-BPFG of 5 nodes, here 𝑉 = {πœ„, πœ…, 𝛼, 𝜐, 𝜏 } and 𝑅 = {( πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…) , π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)]h=1 m βŒͺ ) , |πœ„πœ… ∈ 𝑉 Γ— 𝑉}. βˆ€, πœ„, πœ… ∈ 𝑉, we get , π‘ƒβ„ŽoΞ¨R +(πœ„πœ…) = min{π‘ƒβ„ŽoΞ¨Q +(πœ„), π‘ƒβ„ŽoΞ¨Q +(πœ…)}, π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…) = max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„), π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} for β„Ž = 1, 2, β‹― , π‘š. A complete m-BPFG's m-BPF planarity value, according to Theorem 1, is Ξ“ = 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺwhere π‘ƒβ„ŽoΞ“+ = 1 1+ds and π‘ƒβ„ŽoΞ“βˆ’ = βˆ’1 1+ds , β„Ž = 1, 2, β‹― , π‘š , ds is the number of point of junctions connecting the lines in 𝐺. We know that the geometric representation of the underlying crisp graph of an m-BPF complete graph with 5 nodes is non-planar, and that no representation can avoid one point of junction. So, 〈[π‘ƒβ„ŽoΞ“+, π‘ƒβ„ŽoΞ“βˆ’]β„Ž=1 π‘š βŒͺ = 〈[05, βˆ’0.5]β„Ž=1 π‘š βŒͺ. Hence, 𝐺 is not astrong m-BPFPG. The complete bipartite m-BPFG of 6 nodes cannot be shown to constitute a strong m-BPFPG, however. 4. Faces of an m-Bipolar fuzzy planar graph The parameter of importance is the face of the m-BPFPG. A region enclosed by m-BPF lines is known as the face of an m-BPFPG. Each m-BPF face has m-BPF lines that define its boundaries. Crisp face is created when all of the lines in the boundary of an m-BPF face have relationship values of [1, βˆ’1]. The existence of the m-BPF face is negated if one of these lines is eliminated. Therefore, the minimum energy of m-BPF lines in a border determines whether an m-BPF face exists. Below is a definition of an m-BPF face and its relationship values in an m-BPFG. Definition 4.1: Let 𝐺 be an m-BPFPG and𝑅 = {( πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 , π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ ) , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝑉 Γ— 𝑉}. An m-BPF face of 𝐺 is a region, bordered by the set of m-BPF lines 𝐸′ βŠ‚ 𝑉 Γ— 𝑉, of a geometric sign of 𝐺. The positive and negative values of the m-BPF face is 〈[π‘ƒβ„Žo F+, π‘ƒβ„Žo Fβˆ’]β„Ž=1 π‘š βŒͺ, where π‘ƒβ„Žo F+ = min {( π‘ƒβ„ŽoΨ𝑅 +(πœ„πœ…)𝑒 min{π‘ƒβ„ŽoΞ¨Q +(πœ„),π‘ƒβ„ŽoΞ¨Q +(πœ…)} ) , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝐸′} , π‘ƒβ„Žo Fβˆ’ = max {(βˆ’ π‘ƒβ„ŽoΨ𝑅 βˆ’(πœ„πœ…)𝑒 max{π‘ƒβ„ŽoΞ¨Q βˆ’(πœ„),π‘ƒβ„ŽoΞ¨Q βˆ’(πœ…)} ) , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝐸′} , β„Ž = 1, 2, β‹― , π‘š. Definition 4.2: If an m-BPF face has a positive relationship value π‘ƒβ„Žo F+ > 0.5 or a negative relationship value π‘ƒβ„Žo F+ > -0.5, for β„Ž = 1, 2, β‹― , π‘š, it is said to be a strong m-BPF face; otherwise, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 270 https://internationalpubls.com it is called to be a weak face. The outer m-BPF face is an infinite region found in every m-BPFPG. Remaining faces are call inner m-BPF faces. Example 4: Take an m-BPFPG 𝐺 = (𝑉, 𝑄, 𝑅)as shown in Fig. 4.where 𝑄 = { π‘ž1 〈[0.8,βˆ’0.7],[0.7,βˆ’0.65]βŒͺ , π‘ž2 〈[0.9,βˆ’0.52],[0.8,βˆ’0.7]βŒͺ , π‘ž3 〈[0.7,βˆ’0.9],[0.5,βˆ’0.7]βŒͺ , π‘ž4 〈[0.63,βˆ’0.51],[0.79,βˆ’0.54]βŒͺ }, 𝑅 = { π‘ž1π‘ž2 〈[0.75,βˆ’0.51],[0.65,βˆ’0.54]βŒͺ , π‘ž1π‘ž3 〈[0.62,βˆ’0.63],[0.49,βˆ’0.53]βŒͺ , π‘ž2π‘ž3 〈[0.69,βˆ’0.5],[0.49,βˆ’0.65]βŒͺ , π‘ž1π‘ž4 〈[0.23,βˆ’0.14],[0.6,βˆ’0.2]βŒͺ , π‘ž3π‘ž4 〈[0.59,βˆ’0.49],[0.47,βˆ’0.52]βŒͺ }. Then m-BPFPG has the following faces: β€’ m-BPF face 𝛽1 is bounded by the lines π‘ž1π‘ž4 〈[0.23,βˆ’0.14],[0.6,βˆ’0.2]βŒͺ , π‘ž3π‘ž4 〈[0.59,βˆ’0.49],[0.47,βˆ’0.52]βŒͺ , π‘ž1π‘ž3 〈[0.62,βˆ’0.63],[0.49,βˆ’0.53]βŒͺ , β€’ m-BPF face 𝛽2 is surrounded by the lines π‘ž1π‘ž2 〈[0.75,βˆ’0.51],[0.65,βˆ’0.54]βŒͺ , π‘ž1π‘ž3 〈[0.62,βˆ’0.63],[0.49,βˆ’0.53]βŒͺ , π‘ž2π‘ž3 〈[0.69,βˆ’0.5],[0.49,βˆ’0.65]βŒͺ , β€’ outer m-BPF face 𝛽3 surrounded by the lines π‘ž1π‘ž2 〈[0.75,βˆ’0.51],[0.65,βˆ’0.54]βŒͺ , π‘ž2π‘ž3 〈[0.69,βˆ’0.5],[0.49,βˆ’0.65]βŒͺ , π‘ž1π‘ž4 〈[0.23,βˆ’0.14],[0.6,βˆ’0.2]βŒͺ , π‘ž3π‘ž4 〈[0.59,βˆ’0.49],[0.47,βˆ’0.52]βŒͺ . Fig. 4 Faces in m-Bipolar Fuzzy Planar Graph Clearly the relationship value of m-BPF face 𝛽1 is〈[0.37, βˆ’0.27], [0.86, βˆ’0.37]βŒͺ. The relationship value of m-BPF face 𝛽3is〈[0.37, βˆ’0.27], [0.86, βˆ’0.37]βŒͺ. Thus 𝛽1and 𝛽3 are weak m-BPF faces and 𝛽2 is a strong m-BPF face with the relationship value〈[0.89, βˆ’0.90], [0.93, βˆ’0.82]βŒͺ. We are now introducing dual of m-BPFPG. Each m-BPF line between two nodes corresponds to each line in the border between two faces of the m-BPFPG, and in the m-BPFDG, nodes correspond to the strong m-BPF faces of the m-BPFPG. Below is the formal definition. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 271 https://internationalpubls.com Definition 4.3: Let 𝐺 be an m-BPFPG and let 𝑅 = {( πœ„πœ… 〈[π‘ƒβ„ŽoΞ¨R +(πœ„πœ…)𝑒 , π‘ƒβ„ŽoΞ¨R βˆ’(πœ„πœ…)𝑒] h=1 m βŒͺ ) , 𝑒 = 1,2, β‹― , 𝑑|πœ„πœ… ∈ 𝑉 Γ— 𝑉}. Let 𝛽1, 𝛽2, β‹― , π›½π‘Ÿbe the strong m-BPF faces of 𝐺. An m-BPFDG of 𝐺 is an m-BPFPG 𝐺𝑑 = (𝑉𝑑, 𝑄𝑑 , 𝑅𝑑), where 𝑉𝑑 = {π‘žπ‘–, 𝑖 = 1, 2, β‹― , π‘Ÿ}, and the node π‘žπ‘–of 𝐺𝑑 is measured for the face 𝛽𝑖of 𝐺. The positive and negative relationship values of nodes are taken by the mapping 𝑄𝑑 = 〈[π‘ƒβ„ŽoΞ¨ 𝑄𝑑 + , π‘ƒβ„ŽoΞ¨ 𝑄𝑑 βˆ’ ] h=1 m βŒͺ , π‘ƒβ„ŽoΞ¨ 𝑄𝑑 + : 𝑉𝑑 β†’ [0, 1] and π‘ƒβ„ŽoΞ¨ 𝑄𝑑 βˆ’ : 𝑉𝑑 β†’ [βˆ’1, 0] such that π‘ƒβ„ŽoΞ¨ 𝑄𝑑 + (π‘žπ‘–) = max{π‘ƒβ„ŽoΞ¨ 𝑅𝑑 + (πœ„πœ…)𝑒, 𝑒 = 1,2, β‹― , 𝑑 | πœ„πœ… 𝑖𝑠 π‘Ž 𝑙𝑖𝑛𝑒 π‘œπ‘“ π‘‘β„Žπ‘’ π‘π‘œπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘‘π‘Ÿπ‘œπ‘›π‘” π‘š βˆ’ 𝐡𝑃𝐹 π‘“π‘Žπ‘π‘’ 𝛽𝑖} , π‘ƒβ„ŽoΞ¨ 𝑄𝑑 βˆ’ (π‘žπ‘–) = min{π‘ƒβ„ŽoΞ¨ 𝑅𝑑 βˆ’ (πœ„πœ…)𝑒, 𝑒 = 1,2, β‹― , 𝑑| πœ„πœ… 𝑖𝑠 π‘Ž 𝑙𝑖𝑛𝑒 π‘œπ‘“ π‘‘β„Žπ‘’ π‘π‘œπ‘’π‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘œπ‘“ π‘‘β„Žπ‘’ π‘ π‘‘π‘Ÿπ‘œπ‘›π‘” π‘š βˆ’ 𝐡𝑃𝐹 π‘“π‘Žπ‘π‘’ 𝛽𝑖}. There may exist excess of one common line corresponding to two faces 𝛽𝑖 and 𝛽𝑗 of 𝐺. Therefore, there may be in excess of one line corresponding to two nodesπ‘žπ‘– and π‘žπ‘— in m-BPFDG 𝐺𝑑 .Let 〈[π‘ƒβ„ŽoΞ¨R +r (π‘žπ‘–π‘žπ‘—), π‘ƒβ„ŽoΞ¨R βˆ’r (π‘žπ‘–π‘žπ‘—)] β„Ž=1 π‘š βŒͺ denotes the positive and negative relationship values of the rthline between π‘žπ‘–and π‘žπ‘— , the positive and negative relationship values of the m-BPF lines of the m- BPFDG are given byπ‘ƒβ„ŽoΞ¨ 𝑅𝑑 +r (π‘žπ‘–π‘žπ‘—)π‘Ÿ = π‘ƒβ„ŽoΨ𝑅 +r (πœ„πœ…)𝑒,π‘ƒβ„ŽoΞ¨ 𝑅𝑑 βˆ’r (π‘žπ‘–π‘žπ‘—)π‘Ÿ = π‘ƒβ„ŽoΨ𝑅 βˆ’r (πœ„πœ…)𝑒, where(πœ„πœ…)𝑒is an line in the boundary corresponding 2 strong m-BPF faces 𝛽𝑖 and 𝛽𝑗and π‘Ÿ = 1,2, β‹― , 𝑠,where 𝑠is the number of common lines in the boundary between 𝛽𝑖 and 𝛽𝑗 or the number of lines between π‘žπ‘– and π‘žπ‘— . If there be any strong pendant line in the m-BPFPG, then there will be a self loop in 𝐺𝑑 corresponding to this pendant line. The line positive and negative relationship value of the self loop is equal to the positive and negative relationship values of the pendant line. m-BPFDG of m-BPFPG does not contain point of junction of the lines for a certain representation, so it is m-BPFPG with planarity Value 〈[1 , βˆ’1]β„Ž=1 π‘š βŒͺ. Hence the m-BPF face of m-BPFDG can be similarly described as in m-BPFPGs. Example 5: Consider an m-BPFPG 𝐺 = (𝑉, 𝑄, 𝑅)of πΊβˆ— = (𝑉, 𝐸)as given away in Fig. 5 such that 𝑉 = {𝑝, π‘ž, π‘Ÿ, 𝑠}, 𝐸 = {π‘π‘ž, π‘žπ‘Ÿ, π‘Ÿπ‘ , 𝑠𝑝, π‘π‘Ÿ} Fig. 5.2-BPFG and it’s 2-BPFDG Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 272 https://internationalpubls.com 𝑄 = { 𝑝 〈[0.9,βˆ’0.7],[0.8,βˆ’0.5]βŒͺ , π‘ž 〈[0.7,βˆ’0.8],[0.8,βˆ’0.9]βŒͺ , π‘Ÿ 〈[0.5,βˆ’0.23],[0.7,βˆ’0.9]βŒͺ , 𝑠 〈[0.6,βˆ’0.8],[0.7,βˆ’0.35]βŒͺ }, 𝑅 = { π‘π‘ž 〈[0.65,βˆ’0.69],[0.75,βˆ’0.45]βŒͺ , π‘žπ‘Ÿ 〈[0.49,βˆ’0.21],[0.65,βˆ’0.75]βŒͺ , π‘Ÿπ‘  〈[0.49,βˆ’0.2],[0.61,βˆ’0.32]βŒͺ , 𝑠𝑝 〈[0.55,βˆ’0.45],[0.69,βˆ’0.23]βŒͺ , π‘π‘Ÿ 〈[0.41,βˆ’0.2],[0.66,βˆ’0.49]βŒͺ , π‘žπ‘Ÿ 〈[0.46,βˆ’0.2],[0.65,βˆ’0.81]βŒͺ }. The m-BPFPG has the following faces: β€’ m-BPF face 𝛽1 is bounded by π‘π‘ž 〈[0.65,βˆ’0.69],[0.75,βˆ’0.45]βŒͺ , π‘π‘Ÿ 〈[0.41,βˆ’0.2],[0.66,βˆ’0.49]βŒͺ , π‘žπ‘Ÿ 〈[0.49,βˆ’0.21],[0.65,βˆ’0.75]βŒͺ β€’ m-BPF face 𝛽2 bounded by the lines π‘π‘Ÿ 〈[0.41,βˆ’0.2],[0.66,βˆ’0.49]βŒͺ , 𝑠𝑝 〈[0.55,βˆ’0.45],[0.69,βˆ’0.23]βŒͺ , π‘Ÿπ‘  〈[0.49,βˆ’0.2],[0.61,βˆ’0.32]βŒͺ β€’ m-BPF face 𝛽3 is bounded by the lines π‘žπ‘Ÿ 〈[0.49,βˆ’0.21],[0.65,βˆ’0.75]βŒͺ , π‘žπ‘Ÿ 〈[0.46,βˆ’0.2],[0.65,βˆ’0.81]βŒͺ and β€’ outer m-BPF 𝛽4 is surrounded by π‘π‘ž 〈[0.65,βˆ’0.69],[0.75,βˆ’0.45]βŒͺ , π‘žπ‘Ÿ 〈[0.46,βˆ’0.2],[0.65,βˆ’0.81]βŒͺ , π‘Ÿπ‘  〈[0.49,βˆ’0.2],[0.61,βˆ’0.32]βŒͺ , 𝑠𝑝 〈[0.55,βˆ’0.45],[0.69,βˆ’0.23]βŒͺ . According to routine calculations, all faces are strong m-BPF faces. We consider into an account a node for the m-BPFDG for each strong m-BPF face.So the node set𝑉𝑑 = {π‘ž1, π‘ž2, π‘ž3, π‘ž4}, where the node π‘žπ‘— is given consequent to the strong m-BPF face 𝛽𝑗 , 𝑗 = 1, 2,3,4. Thus, 𝑄𝑑 = { π‘ž1 〈[0.65,βˆ’0.69],[0.75,βˆ’0.75]βŒͺ , π‘ž2 〈[0.55,βˆ’0.45],[0.69,βˆ’0.49]βŒͺ , π‘ž3 〈[0.49,βˆ’0.21],[0.65,βˆ’0.81]βŒͺ , π‘ž4 〈[0.65,βˆ’0.69],[0.75,βˆ’0.81]βŒͺ }, The faces 𝛽2and 𝛽4 in 𝐺share two common lines, 𝑝𝑠 and π‘Ÿπ‘ . Consequently, there are two lines in the m-BPFDG of G between the nodes π‘ž2 and π‘ž4. That line’s positive and negative relationship values are given by { π‘ž2π‘ž4 〈[0.49,βˆ’0.2],[0.61,βˆ’0.32]βŒͺ , π‘ž2π‘ž4 〈[0.55,βˆ’0.45],[0.69,βˆ’0.23]βŒͺ }. The other lines of the m- BPFDG's of positive and negative relationship values are computed as follows. { π‘ž1π‘ž3 〈[0.49,βˆ’0.21],[0.65,βˆ’0.75]βŒͺ , π‘ž1π‘ž2 〈[0.41,βˆ’0.2],[0.66,βˆ’0.49]βŒͺ , π‘ž1π‘ž4 〈[0.65,βˆ’0.69],[0.75,βˆ’0.45]βŒͺ , π‘ž3π‘ž4 〈[0.46,βˆ’0.2],[0.65,βˆ’0.81]βŒͺ }. Thus, the line set of m-BPFDG is 𝑅𝑑 = { π‘ž2π‘ž4 〈[0.49,βˆ’0.2],[0.61,βˆ’0.32]βŒͺ , π‘ž2π‘ž4 〈[0.55,βˆ’0.45],[0.69,βˆ’0.23]βŒͺ , π‘ž1π‘ž3 〈[0.49,βˆ’0.21],[0.65,βˆ’0.75]βŒͺ , π‘ž1π‘ž2 〈[0.41,βˆ’0.2],[0.66,βˆ’0.49]βŒͺ , π‘ž1π‘ž4 〈[0.65,βˆ’0.69],[0.75,βˆ’0.45]βŒͺ , π‘ž3π‘ž4 〈[0.46,βˆ’0.2],[0.65,βˆ’0.81]βŒͺ }. In Fig. 5, the m-BPFDG 𝐺𝑑of 𝐺 is drawn by dotted line. In m-BPFDGs, weak lines in planar graphs are not taken into an account for any computations. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 273 https://internationalpubls.com Theorem 4.1: Let 𝐺 be an m-BPFPG whose number of nodes, the number of m-BPF lines and number of strong faces are denoted by π‘₯, 𝑦, 𝑧 respectively. Let 𝐺𝑑be the m-BPFDG of 𝐺. Then: i. no. of nodes of 𝐺𝑑= π‘₯, ii.no. of lines of 𝐺𝑑= 𝑦, iii.no. of m-BPF faces of 𝐺𝑑= 𝑧. Proof: The definition of m-BPFDG provides evidence for (i) (ii), and (iii). Theorem 4.2: Let 𝐺 be a strong m-BPFPG without weak lines and the m-BPFDG of 𝐺 be 𝐺𝑑. Then the positive relationship and negative relationship values of m-BPF lines of 𝐺𝑑are equal to positive relationship and negative relationship values of m-BPF lines of 𝐺. Proof: The dual graph 𝐺𝑑 of 𝐺 is a strong m-BPFPG as there is no point of junction between any lines. Let {𝛽1, 𝛽2, β‹― , π›½π‘Ÿ}be the set of strong faces of 𝐺. By the definition of m-BPFDG we know that π‘ƒβ„ŽoΞ¨ 𝑅𝑑 +r (π‘žπ‘–π‘žπ‘—) = π‘ƒβ„ŽoΨ𝑅 +r (πœ„πœ…) , π‘ƒβ„ŽoΞ¨ 𝑅𝑑 βˆ’r (π‘žπ‘–π‘žπ‘—) = π‘ƒβ„ŽoΨ𝑅 βˆ’r (πœ„πœ…)where πœ„πœ…is a common line between 2 strong m-BPF faces 𝛽𝑖and 𝛽𝑗 and π‘Ÿ = 1,2, β‹― , 𝑠, where 𝑠 being the no. of common lines in the boundary between 𝛽𝑖and 𝛽𝑗 . The no. of m-BPF lines of two graphs 𝐺 and 𝐺𝑑are same as 𝐺 has no weak lines. So, for each m-BPF line of 𝐺 there is an m-BPF line in 𝐺𝑑 with the same relationship value. We're now studying at isomorphism between m-BPFPGs. Definition 4.4: Let πΊπ‘Ž = (π‘‰π‘Ž , π‘„π‘Ž, π‘…π‘Ž)and 𝐺𝑏 = (𝑉𝑏 , 𝑄𝑏 , 𝑅𝑏)be two m-BPFPGs of the graphs πΊπ‘Ž βˆ— = (π‘‰π‘Ž , πΈπ‘Ž )and 𝐺𝑏 βˆ— = (𝑉𝑏 , 𝐸𝑏 ) respectively. (i) An isomorphism between πΊπ‘Ž and 𝐺𝑏 is a bijective transformation ΞΆ: π‘‰π‘Ž β†’ 𝑉𝑏such that for each β„Ž = 1, 2, β‹― , π‘š (a) π‘ƒβ„ŽoΞ¨Qa + (πœ„π‘Ž) = π‘ƒβ„ŽoΞ¨Qb + (ΞΆ(πœ„π‘Ž)), π‘ƒβ„ŽoΞ¨Qa βˆ’ (πœ„π‘Ž) = π‘ƒβ„ŽoΞ¨Qb βˆ’ (ΞΆ(πœ„π‘Ž))for all πœ„π‘Ž ∈ π‘‰π‘Ž (b) π‘ƒβ„ŽoΞ¨Ra + (πœ„π‘Žπœ…π‘Ž) = π‘ƒβ„ŽoΞ¨Rb + (ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž)), π‘ƒβ„ŽoΞ¨Ra βˆ’ (πœ„π‘Žπœ…π‘Ž) = π‘ƒβ„ŽoΞ¨Rb βˆ’ (ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž)) for all πœ„π‘Žπœ…π‘Ž ∈ π‘‰π‘Ž 2 ⃑ . (ii) A weak isomorphism between πΊπ‘Ž and 𝐺𝑏is a bijective transformation ΞΆ: π‘‰π‘Ž β†’ 𝑉𝑏 such that for each β„Ž = 1, 2, β‹― , π‘š (a) ΞΆ is a homomorphism (b) π‘ƒβ„ŽoΞ¨Qa + (πœ„π‘Ž) = π‘ƒβ„ŽoΞ¨Qb + (ΞΆ(πœ„π‘Ž)), π‘ƒβ„ŽoΞ¨Qa βˆ’ (πœ„π‘Ž) = π‘ƒβ„ŽoΞ¨Qb βˆ’ (ΞΆ(πœ„π‘Ž))for all πœ„π‘Ž ∈ π‘‰π‘Ž. (iii) A co-weak isomorphism betweenπΊπ‘Ž and 𝐺𝑏 is a bijective transformation ΞΆ: π‘‰π‘Ž β†’ 𝑉𝑏such that for each β„Ž = 1, 2, β‹― , π‘š (a) ΞΆ is a homomorphism (b) π‘ƒβ„ŽoΞ¨Ra + (πœ„π‘Žπœ…π‘Ž) = π‘ƒβ„ŽoΞ¨Rb + (ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž)), π‘ƒβ„ŽoΞ¨Ra βˆ’ (πœ„π‘Žπœ…π‘Ž) = π‘ƒβ„ŽoΞ¨Rb βˆ’ (ΞΆ(πœ„π‘Ž)ΞΆ(πœ…π‘Ž)) for all πœ„π‘Žπœ…π‘Ž ∈ π‘‰π‘Ž 2 ⃑ . That isomorphism between two m-BPFPGs is an equivalence relation is simple to confirm. However, if two m-BPFGs are isomorphic so that one is m-BPFPG, the other will be m-BPFG. As evidence, consider the following. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 274 https://internationalpubls.com Theorem 4.3: Let πΊπ‘Ž be an m-BPFPG and let 𝐺𝑏 be an m-BPFG. If there exists an isomorphismΞΆ: πΊπ‘Ž β†’ 𝐺𝑏 , 𝐺𝑏 can be represented as m-BPFPG with the same planarity value of πΊπ‘Ž . Proof: Line and node weights are preserved through isomorphism. Additionally, isomorphic m- BPFGs maintain their order and size. 𝐺𝑏 will therefore have the same size and order as πΊπ‘Ž. After then, 𝐺𝑏 can be represented similarly to πΊπ‘Ž. As a result, 𝐺𝑏 will have the same number of junctions where lines cross and the same planarity value as πΊπ‘Ž. Accordingly, 𝐺𝑏may be represented as m-BPFPG with the same planarity value as πΊπ‘Ž. Theorem 4.4: Let 𝐺𝑏 be the m-BPFDG of m-BPFDG of a strong m-BPFDG 𝐺 without weak lines. Then there exists a co-weak isomorphism connecting 𝐺 and 𝐺𝑏. Proof: Assume that 𝐺𝑏is an m-BPFDG of πΊπ‘Žand that πΊπ‘Žis an m-BPFDG of 𝐺. The no. of nodes of 𝐺𝑏 are equal to the strong m-BPF faces of πΊπ‘Žand the no. of strong m-BPF faces in πΊπ‘Žis equal to the no. of nodes in 𝐺. The no. of nodes in 𝐺𝑏 and 𝐺 are therefore equal. Additionally, an m-BPFPG and its dual have the same number of lines. The line relationship value of a line in a dual graph is identical to the line relationship value of a line in m-BPFG, according to the definition of m-BPFDG. The co-weak isomorphism between 𝐺 and 𝐺𝑏can be created. Hence the outcome. With the same number of nodes, two m-BPFPGs may be isomorphic. However, the following relationships may exist between the m-BPF planarity values of two m-BPFPGs. Theorem 4.5: Let πΊπ‘Žand 𝐺𝑏be two isomorphic m-BPFGs with the corresponding m-BPF planarity values Ξ“πΊπ‘Ž and Γ𝐺𝑏 . ThenΞ“πΊπ‘Ž = Γ𝐺𝑏 . Proof: Obvious. We then make the following claims without providing any proof. Theorem 4.6: Let πΊπ‘Ž and 𝐺𝑏 be two weak isomorphic m-BPFGs with the corresponding m-BPF planarity values of Ξ“πΊπ‘Ž and Γ𝐺𝑏 . If the line positive relationship and a line negative relationship values of the respective intersecting lines are the same, then Ξ“πΊπ‘Ž = Γ𝐺𝑏 . Theorem 4.7: Let πΊπ‘Žand 𝐺𝑏 be two co-weak isomorphic m-BPFGs with the corresponding m-BPF planarity values of Ξ“πΊπ‘Ž and Γ𝐺𝑏 . If the terminal nodes of the respective intersecting lines have the same minimum, positive relationship and maximum negative relationship values, then Ξ“πΊπ‘Ž =Γ𝐺𝑏 . Conclusions Unpredictable behavior and opinions of voters leads to vagueness and uncertainty when traditional graph theory is used for analyzing the voter sentiment and shifting loyalties. We already know that bipolar fuzzy sets are capable of handling fuzzy sets with vague and uncertainty data to some extent. By modifying them to include multiple parameters on bipolar. In this research, we apply the idea of m-bipolar fuzzy set to multigraph and planar graph because m-bipolar fuzzy sets have shown advantages over bipolar fuzzy sets in addressing ambiguity and uncertainty to some extent. We can generate m-BPFGs to provide better analysis. 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