1 Introduction

1.1 Research background and related works

In various real-world scenarios, we often describe complex systems using models composed of nodes and connecting lines that represent specific relationships between pairs of nodes. Graph theory enables us to illustrate the existing connections within networks or systems. The application of graph theory has become standard practice across different contexts, including computer networks, electric grids, and more.

Some investigates the modeling and analysis of transportation and telecommunications networks using fuzzy graph theory. Due to various factors like traffic congestion, varied communication quality, or insufficient data, transportation and communication networks frequently deal with uncertainties and imprecise information. By introducing fuzzy sets into graph theory, where edges and vertices are given membership values to indicate varying degrees of connection or dependability, fuzzy graphs offer a useful framework for addressing these issues. The goal of this research is to employ fuzzy graphs to create models for the best possible path selection, network dependability, and efficiency in transportation and telecommunication systems. The outcomes show how fuzzy graph approaches may manage the complexity of the real world and provide better answers for network analysis and uncertain decision-making.

In modern settings, analysts recognize that the data that supports our perception of the world is rife with uncertainty. This uncertainty has a big impact on a lot of different areas of science and technology. Although the fuzzy set structure provides a way to express this uncertainty using membership values that range from 0 to 1, there are some situations in which conventional numerical representations are insufficient. Consider the challenge of determining the minimal number of traffic inspectors needed to effectively monitor an urban road network in the presence of multipolar fuzzy information. These situations are outside the purview of traditional fuzzy set theory and are closely related to the idea of a strong edge geodesic foundation. Therefore, it becomes necessary to present new approaches that can handle these complex, unpredictable datasets. It is this necessity that motivates the notion of a strong edge geodesic number in the context of m-polar fuzzy graphs. This conceptual breakthrough offers a sophisticated toolkit designed to handle the intricacies present in unpredictable multipolar information settings.

Zadeh [38] identified doubtfulness as a phenomenon and the ambiguity of real-life events in 1965. He presented a fuzzy set that altered people’s perceptions of science and technology. Fuzzy graphs were first proposed by Kafmann [18] using Zadeh’s fuzzy relation. Later, Rosenfeld [33] introduced the possibility of adding nodes and edges, in addition to other theoretical notions related to fuzziness, such as routes, connectedness, cycles, etc. Fuzzy graphs are then the topic of a lengthy discussion [26]. Numerous definitions and practical applications have been examined in [36]. Mordeson and Nair [27] also discuss some new concepts related to fuzzy graphs. The concept of a geodesic number of crisp graphs for vertices was first proposed by Frank Harary et al. [12,13,16]. Suvarna and Sunitha [35] expanded on the concept of geodesic distance in the context of fuzziness, while Linda and Sunitha [19] used μ-distance. Geodetic numbers of graphs and digraphs were introduced by Chang [17]. On perfect geodesic fuzzy graphs, Rehmani and Sunitha conducted research [34]. Atici introduced the graph’s edge geodetic number [9]. Originally proposed by Chen et al. [14], the m-polar fuzzy graph (mPFG). Density on mPFG was originally introduced later by Ghorai and Pal [15]. Moreover, Adeel and Akram have deeded on mPFGs and line graphs [1]. Akram et al. [2] concentrated on a few edge properties specific to mPFG. They also carried out their work on more different types of fuzzy graphs [4–8,37]. Mahapatra and Pal [20] later introduced fuzzy coloring of mPFG. Mahapatra and associates looked at fuzzy fractional coloring on fuzzy graphs more recently [21].Later on, they carried out their work on some different types of its generalization [22,23,29,30]. Mandal et al. [25] researched mPFG’s strong arcs.

Authors contributation towards geodesic as well as their generalization given in Table 1.

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Table 1. Authors contributation towards geodesic as well as their generalization.

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1.4 Notations and symbols

In this section, we introduce and define key notations essential for the development of the theories presented throughout the paper. The table below, Table 2, outlines these notations along with their respective meanings for clarity and reference.

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Table 2. Aberration of some useful terms.

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2 Preliminaries

Here, introduction of some basic definitions is provided.

Let be a graph, where V (non-empty set) is called vertex set and E (empty or non-empty set) is called edge set. If no edge incident with a vertex, then the vertex is said to be isolated vertex, otherwise, it is said to be non-isolated vertex.

In this article indicates the i^th, s^th content of the projection mapping, with standing in for .

Definition 1. [38] A fuzzy set A on the universal set X is characterized by a mapping , which is called the membership function. A fuzzy set is denoted by .

Definition 2. [18] A fuzzy graph is a triplet with underlying crisp graph where is a fuzzy set in V and is a fuzzy set in such that , for all and , for all . Here σ and μ are the fuzzy vertex and fuzzy edge of G respectively.

Definition 3. [36] The fuzzy graph is said to be a fuzzy subgraph of if , for all and for all .

Definition 4. [14] An mPFS A (or a [0,1]) on X is a mapping . m(X) indicates the set of all m-polar fuzzy sets on X.

Definition 5. [14] Let A be an mPFS. Then height of A is indicated by h(A) and is defined by

Definition 6. [15] An mPFG with UCG , as well as represents an mPFS of V and respectively and which obey the rule such that , for all as well as for all . and indicates i^th component of MV of node a and edge (a,c).

Definition 7. [15] is considered a complete mPFG if holds for every and .

Definition 8. [15] The mPF strong graph is identified when

For each and , holds.

Definition 9. [15] Let and be two mPFGs of UCGs and respectively. A homomorphism from H₁ to H₂ is denoted by a mapping such that

and

for all , and .

Definition 10. [15] A bijective homomorphism between H₁ and H₂ is represented as a mapping with the property

is called a weak isomorphism or (weak) node -isomorphism.

Definition 11. [15] A bijective homomorphism between H₁ and H₂ is a mapping with the property , for all and is called a co-weak isomorphism or (weak) line-isomorphism.

If f from H₁ to H₂ is a (weak) line-isomorphism as well as a (weak) node-isomorphism, it is referred to as an isomorphism between H₁ and H₂.

Definition 12. [20] When an edge of the mPFG is is . It is regarded as independently strong. It’s known as an independently weak edge if it isn’t.

Definition 13. [20] The definition of the edge strength (t,w) is

for .

Definition 14. [3] Given a mPFG with UCG , let . Consider a subgraph with UCG belonging to H. If is a cycle and there isn’t a unique (the edge set of P) such that , for , then P is referred to as a mPF cycle.

Definition 15. [24] Let be a path in H, and let be a mPFG. Next, we use S(P) to indicate the path P’s strength, which may be found as follows: .

The path between d₁ and d_k has the following SC: , where .

Definition 16. [25] For every s = 1 to m, a graph is referred to as a mPF tree if there is a spanning mPF subgraph that is also a mPF tree. is implied in .

Definition 17. [3] Suppose is an mPFG. Then mPF open-nbd of a node x is indicated as N(x) and defined as . Then the closed-nbd of a node x is indicated as N[x] and given by .

3 Strong geodesic number of mPFG

The concept of edge geodesic numbers is a critical parameter in graph theory, used to determine the minimum number of edges that intersect all shortest paths (geodesics) between pairs of vertices in a graph. This metric has valuable real-life applications, particularly in optimizing network designs, improving transportation systems, and enhancing communication networks. In logistics and urban planning, edge geodesic numbers help identify critical routes or connections that must be maintained to ensure efficient movement across a network. In computer networks, they assist in determining key edges for data flow optimization and fault tolerance. Similarly, in social networks and biological systems, understanding edge geodesics aids in analyzing influential connections that influence system behavior. By applying edge geodesic numbers, real-world networks can be made more resilient, cost-effective, and efficient, highlighting the importance of this graph-theoretic parameter in practical applications.

In this section, a strong geodesic number of mPFG by using strong geodesic distance has been established. This is different from geodesic distance as here we want to calculate the geodesic distance only on a strong path that is only on α-strong as well as β-strong edges. Some useful properties for strong geodesic numbers have been discussed here.

Definition 18. Let be an mPFG and a,d be two nodes of G. Let P be a strong path from a to d. If there does not exist any shorter strong path other than P in between a to d, then the path P is called geodesic and the length of P is said to be a strong geodesic distance of a and d and which is indicated by d_g(a,d).

Theorem 1. In a mPFG, strong geodesic distance On the set of vertices, is a metric space.

Proof. Let e, t be two nodes in V. Then the strong geodesic distance between two nodes e,t is indicated as d_g(e,t). To prove a strong geodesic distance is a metric, we have to show that it satisfies four properties of metric space.

1. Let be the shortest strong path in between e and t. Then , for and . This shows that the length of P is non-null. Hence d_g(e,t) > 0. Therefore, d_g(e,t) > 0, for all .
2. d_g(e,t) = 0 holds iff e = t. Hence, d_g(e,t) = 0 holds iff e, t.
3. satisfied as the shortest strong path from e to t is the same as the shortest strong path from t to e. Therefore, .
4. Let e, t, c be three nodes in V. We are to show that . If possible, let . Let Q be a strong mPF path from e to c. Let Q₁ be a strong mPF path from e to t and Q₂ be a strong mPF path from t to c. Then be a strong mPF path from e to c which has the least length from the path Q, a contradiction. Therefore, .

Therefore, a strong geodesic distance in an mPFG is a metric space.

Definition 19. Let’s consider a connected mPFG denoted by . Suppose we have a subset of nodes . We define the strong geodesic closure of U, denoted as (U), as the set comprising all nodes in U along with those nodes lying on strong geodesics connecting nodes within U. If , we refer to U as a strong geodesic set or a strong geodesic cover. A strong geodesic basis is any minimal subset of nodes within a strong geodesic cover, and the cardinality of this basis is termed the strong geodesic number of G, indicated by gn(G).

An example that demonstrates every definition above is given below.

Example 1. Here, we examine a 3PFG illustrated in Fig 1 to illustrate the aforementioned definitions.

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Fig 1. 3PFG G.

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We give the MV of nodes as well as edges of G in tabular form in Table 3 and 4.

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Table 3. Vertex membership value.

https://doi.org/10.1371/journal.pone.0327882.t003

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Table 4. Edge membership value.

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After some routine calculation, we can say that all the edges in G are either α-strong or β- strong. If we take , then we can easily see that U includes all the nodes in U and any other nodes that not include in U must lie on the strong geodesics connecting the nodes within U. Therefore, its fulfill the condition of definition 19. Hence, . Therefore, U is a strong geodesic cover of G. It is evident that U is a minimally node-counting strong geodesic cover. Hence, U is a strong geodesic basis of G and a strong geodesic number is 4 that is, .

Remark 1. If all the edges of an mPFG whose UCG is a Peterson graph are strong, then .

Theorem 2. The strong geodesic number of a complete mPFG having a n number of nodes is n.

Proof. Suppose G as a complete mPFG. Consequently, it is devoid of δ-strong arcs, implying that all edges within G are strong edges. Take two nodes, x and y, within G. As G^* is complete, there exists an edge connecting x and y, forming a strong geodesic (x,y). This strong geodesic doesn’t contain any arcs between other pairs of nodes within G, indicating that serves as a strong geodesic cover for G. To establish it as a strong geodesic basis, we need to demonstrate that U has minimal cardinality. Suppose there exists another strong geodesic cover T of G with fewer nodes compared to U. In this case, at least one node in U is absent in T. Let’s denote this node as a. Consequently, the edges incident to a do not lie on any strong geodesic connecting pairs of nodes within W, hence violating the condition of T being a geodesic cover of G. Therefore, U stands as a geodesic basis of G. Hence, .

Theorem 3. The strong geodesic number is the same for two mPFG , where .

Proof. Suppose as well as be two mPFGs. Since , then there exists a mapping such that for each

1. , .
2. , .

Suppose H have a strong geodesic number n. Then there is a node-set such that S contains n number of nodes and . Suppose . As , then the isomorphic image of the vertices a_i, is also cover the node set . Let . Then we can claim that is a strong geodesic basis of . If possible, let is not a strong geodesic basis of . Since ϕ is a bijective mapping between , therefore there exists which is also a bijective mapping. Then we must have , which shows that S is not a strong geodesic basis of H, a contradiction. Hence, is a strong geodesic basis of . Thus .

4 Strong edge geodesic number of mPFG

In this section, strong edge geodesic numbers by using strong geodesic distance for mPFG have been established. In previous discussions, we have seen that strong geodesic distance is different from ordinary geodesic distance. Therefore, it is totally a new concept which will be discussed thoroughly in this section. Some necessary and sufficient conditions for strong edge geodesic numbers have also been studied here.

Definition 20. Let us consider a mPFG, represented by , where the collection of nodes is indicated by V. Let represent the underlying crisp graph (UCG) of H. Let S represent a group of nodes in the mPFG. The strong geodesic closure of S, shown by (S)_e, is the set of edges that lie on the strong geodesics linking the nodes in S. A set (S)_e = E denotes the strong edge geodesic cover of H if all of the edges in the mPFG are included.

Definition 21. The number of elements in a strong edge geodesic basis is known as the strong edge geodesic number, and it is denoted by gn_e(H). A strong edge geodesic cover with the lowest cardinality is referred to as a strong edge geodesic basis.

The aforementioned definitions are exemplified through the following example provided below.

Example 2. Here, we consider an 3PFG having UCG to depict the above definitions.

If we consider , then we see that (S)_e = {{a,b},{b,c},{c,d},{a,d}} = E and , Therefore gn_e(H) = 2 and . Hence, we may say that gn_e(H) = gn(H), for some certain mPFG H. One simple observation is that if we consider S₁ = {b,d}, then also and . In this case, we also have gn_e(H) = 2 and . Here, we can also see that a strong edge geodesic basis is not unique.

Next, we consider another 3PFG having UCG whose vertex set and having 5 edges shown in Fig 3 where we can not find any strong edge geodesic basis for H₁.

Here, we see that no such set exists for which holds.

Note 1. One simple observation from the Fig 2 is that it has no δ-strong arcs and from the Fig 3 we see that it has a δ-strong arcs, namely (a,c).

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Fig 2. 3PFG H having four nodes and five edges to illustrate the definition 19.

https://doi.org/10.1371/journal.pone.0327882.g002

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Fig 3. 3PFG H₁ having four node and five edges.

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Thus, the following theorem is established.

Theorem 4. A connected mPFG must have a strong edge geodesic cover iff it has no δ-strong arcs.

Proof. Let us consider have a strong edge geodesic cover. Suppose, U is a strong edge geodesic cover of H. We are to show that it has no δ-strong arcs. If possible, let us assume that it has a δ-strong arcs, namely (x,y). In light of the definition of a strong edge geodesic cover, we may therefore conclude that, in contrast to the assertion that U is a strong edge geodesic cover of H, the arc (x,y) does not contain in any strong edge geodesic path connecting the nodes of U. Therefore, there are no δ-strong arcs in H.

On the other hand, consider that H lacks any δ-strong arcs. As such, it only includes arcs that are β or α-strong. As a result, every arc in H is strong. As a result, a strong edge geodesic cover of H is formed by V.

Theorem 5. The strong edge geodesic number is the same for two mPFG , where .

Proof. Suppose as well as be two mPFGs. Since , then there exists a mapping such that for each .

1. , .
2. , .

Suppose H has a strong edge geodesic number n. Then there exists a node set such that U contains n number of nodes and . Suppose . As , then the isomorphic image of the vertices a_i, is also cover the node set . Let . Then we can claim that is a strong edge geodesic basis of . If possible, let is not a strong edge geodesic basis of . Since ϕ is a bijective mapping between , therefore there exists which is also a bijective mapping. Then we must have , which shows that U is not a strong edge geodesic basis of H, a contradiction. Hence, is a strong edge geodesic basis of . Thus .

Theorem 6. Given a connected mPFG with no δ-strong arcs and a full mPFG K_n with n nodes, the strong edge geodesic number of H is n according to UCG .

Proof. Since there are no δ-strong arcs in H, all of its edges are strong edges. Assume that H has two nodes, x,y. Given the completion of H^*, an edge connects x,y. The geodesic (x,y) is strong for the nodes x,y. In H, there isn’t an arc that is on the strong geodesic connecting any two nodes. Therefore, a strong edge geodesic cover of H is . We must demonstrate that U has the least cardinality to establish that it is a strong edge geodesic basis. Assume that W is an additional strong edge geodesic cover of H that has fewer nodes than U. Afterwards, at least one node in U does not exist inside W. Let such a node be a. Hence, W is not an edge geodesic cover of H since the edges incident with a do not lie on any strong geodesic linking pairs of nodes in W. As a result, an edge geodesic basis of H is U. Hence, strong edge geodesic number which is denoted by gn_e(H) as stated in definition 21 must be n.

Proposition 1. For any complete mPFG , the strong edge geodesic number is n, where .

Proof. This proposition clearly follows from the above theorem.

The converse part of the Theorem 6 as well as Proposition 1 may not always be true. This is depicted through an example.

Example 3. Here, we consider a 3PFG H having four nodes whose UCG is K₄.

In Fig 4, we see that all the edges of H are strong. If we take , then and S is a strong edge geodesic basis of H. Hence, gn_e(H) = 4 but the given graph is not a complete 3PFG as for the edges (a,d) and (a,b), not holds.

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Fig 4. 3PFG H.

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Remark 2. From the Theorem 2 and 6, we can conclude that for any complete mPFG H, we have gn(H) = gn_e(H).

Next, we attempt to determine the strong edge geodesic number of a linked mPFG, both its largest lower bound and its least upper bound.

Theorem 7. For any connected mPFG having no δ-arcs, we have , where n is the number of nodes of H.

Proof. Clearly, any strong edge geodesic cover must have at least two nodes. So .

As H is free from any δ-arcs, therefore the strong edge geodesic cover set can have at most n nodes, where n is the number of nodes of H. As, if we consider a complete mPFG then the strong edge geodesic cover set has at n nodes. In all other cases, the strong edge geodesic cover set can have at most n nodes. Hence, .

Combining all the above cases, we have .

Theorem 8. For any complete bipartite mPFG , where and , , gn_e(H) = min{p,q}.

Proof. Three scenarios will be examined in order to demonstrate this theory.

Case 1. Let . Then by using the Proposition 1, we can prove the theorem.

Case 2. Let and . This part can be proved by using the Theorem 6.

Case 3. Let and . Without sacrificing generality, we can presume that p<q. Let and . We have to prove that is an edge geodesic cover of H. Any edge lies on the geodesic joining the nodes of U. Thus U is a strong edge geodesic cover of H.

To prove the theorem it only remains to show that U has the minimum cardinality. If possible, let S be another collection of nodes having less cardinality compared to U. Consider the following cases:

Case a: Let . Then there exists at least one node say a_t in V₁ such that . For any edge , the only strong geodesic containing and . Consequently, no strong geodesic connecting two nodes of S may include . As a result, S is not a geodesic cover of H with a strong edge.

Case b: Let . Then in a similar way as stated above, It is demonstrable that S is not a geodesic cover with a strong edge on H.

Case c: Let . This case only arises when S contains a node from V₁ and a node from V₂. As , their exists a node and a node such that . At this point, the edge does not connect any two nodes of S with a strong geodesic. Hence, S is not a strong edge geodesic cover of H.

Thus, U is a strong edge geodesic basis of H. Hence gn_e(H) = p. Thus, gn_e(H) = min{p,q}.

Next, we find some relation between strong geodesic basis as well as strong edge geodesic basis of a connected mPFG. After finding their relation, we can easily find the least upper bound of the strong edge geodesic basis of any connected mPFG.

Theorem 9. Suppose H is an mPFG. Each strong edge geodesic cover of H is also a strong geodesic cover of H.

Proof. Let U be a geodesic cover of H with a strong edge. Assume that x is an incident node of H with respect to y. The edge (x,y) then rests on a strong geodesic linking to one or more nodes in U. Subsequently, it becomes apparent that x represents a node situated on a strong geodesic connecting a pair of nodes within U. U is a strong geodesic cover of H since x is arbitrary. Thus, the theorem.

Theorem 10. Suppose H is an mPFG. Then .

Proof. Let U be a strong edge geodesic basis of H. Then, gn_e(H) = |U|. Again, from the Theorem 9, we can say that U is a strong geodesic cover of H. Therefore, . Hence, , for any mPFG H.

Theorem 11. Suppose H is an mPF tree having no δ-strong arcs. Then H has a unique strong edge geodesic basis containing mPF end nodes.

Proof. Assume that U is the set of all mPF end nodes in H. According to [24]’s Theorem 3.19, H lacks β-strong arcs because it is a mPF tree. Once again, H has no δ-strong arcs according to the criteria specified. As a result, H has α-strong arcs in every direction. Consequently, H is a mPF tree, and H* is a tree as well. H* and H have identical end nodes. Once more, we are aware that there is only one path connecting any two nodes in H*. Then each arc in H has to be a part of a strong geodesic connecting two nodes in U. Therefore, U is a geodesic cover of H with a strong edge.

It just has to be demonstrated that U has the lowest cardinality of all the strong edge geodesic covers of H to establish that U is a strong edge geodesic basis of H. Let T be an additional strong edge geodesic cover of H, if at all possible, with less cardinality than U. Then, one node in U, let’s call a, exists and is not a part of T. Given that H* is a tree, the edges that incident with a do not form a strong geodesic connecting any two nodes in U. This leads to a contradiction: U is not a strong edge geodesic cover of H. Thus, a strong edge geodesic basis of H is U.

Corollary 1. Let H be an mPFG where H* (the underlying crisp graph of H) is a tree. If H* is a star graph, then gn_e(H) = n − 1, where .

Proof. As H* (the underlying crisp graph of H) is a star graph, therefore we can think of it as K_1,n−1 as it’s exactly one node connected with remaining with n–1 nodes. Since H is an mPF tree, therefore it is free from δ-strong arcs. Then by Theorem 11, we know that all the end nodes of H together form a unique strong edge geodesic basis of H. As H contains n–1 end nodes therefore gn_e(H) = n − 1.

This corollary can be generalized for a special type of mPFG. The generalized version is given below as follows:

Theorem 12. Suppose H be an mPFG which is free from δ-strong arcs and every vertex of H has MV

. If there exists a vertex whose degree for each component is n–1, then .

Proof. Considering that H is a mPFG devoid of δ-strong arcs. We therefore have a unique strong edge geodesic cover of H according to the Theorem of 11. Let a represent the node whose degree is n–1 for every component. Assume . We must first demonstrate that U is a geodesic cover with a strong edge of H. We have two cases for this.

Case 1: a edges the incidence.

Assume that the edge incident with a is (a,c). Given that a is the only degree node whose MV’s individual components are n–1, therefore a is incident with every vertex in H and there exists at least one node, say d such that d is a neighbour of a and c,d are not adjacent as a is the only node of degree whose each component of MV is n–1. Thus the edge (a,c) lies on the strong geodesic d–a–c joining the nodes of U. In similar manner we can show that U cover all the edges of G.

Case 2: Edges not incident with a.

Consider the edge (e,f) which is not incident with a. As , therefore the edge (e,f) lies on the strong geodesic connecting the nodes e,f. Thus U covers the edge (e,f). In the same way, we can prove that U covers all the edges of H which are not incident with a.

Combining both cases, we can conclude that U is a strong edge geodesic cover of H.

To show gn_e(H) = n − 1, we have to show that U is a strong edge geodesic basis of U that is, we have to show that there does not exist another node set having cardinality n–2. If possible, let S be a strong edge geodesic cover of G having cardinality n–2. Let .

Case a: Let .

(a,x) or (a,z) does not lie on any strong geodesic connecting nodes of S since . Hence, S is not a geodesic cover of H with a strong edge.

Case b: and if . Then, S is not a strong edge geodesic cover of H since there is no geodesic joining the nodes of S via (a,x) or (a,z). From all of the cases, we can say that U is a strong edge geodesic basis of H. Thus gn_e(H) = n − 1.

Next, we will focus on strong edge geodesic cover in light of the neighbourhood concept of an mPFG.

Theorem 13. Consider a mPFG with no δ-strong arcs, denoted by . For some , let be such that . Then, on each strong edge geodesic cover of H, v lies.

Proof. Assume that S is the geodesic strong edge cover of H and . We must demonstrate that . Let if at all possible. Given that , u,v are adjacent. The edge (u,v) must lie on a strong geodesic P uniting some pair of nodes, say a,b of S, as S is a strong edge geodesic cover of H. Undoubtedly, . Let t be an additional node that is next to v and distinct from u. We observe that u,t are also adjacent as , which runs counter to the idea that P is a strong geodesic of H. As so, v lies on each geodesic cover of H with a strong edge.

Theorem 14. If be an mPFG having no δ-strong arcs where . Then gn_e(H) = n if and only if for each node a of H, for some .

Proof. Assume that gn_e(H) = n. We’ll use a paradoxical approach to demonstrate this point. Let any node , if at all feasible, have a node c of H such that . Assume . We attempt to establish that S is a geodesic cover of H with a strong edge.

Let an edge of H be (p,q). if . Consequently, S encompasses the edge (p,q).

Case 1: Let c = q. Then . By the given hypothesis we have . Therefore, there exists a node t in N(c) such that t is not in N[p]. Clearly, . Therefore the edge (p,q) lies on p–q–t. Thus S is a strong edge geodesic cover of H. Therefore, , a contradiction.

Case 2: Let p = c. By continuing as above, we can demonstrate that S is a geodesic cover with a strong edge for H. Consequently, there is a contradiction .

By combining everything said above, we obtain for each node a of H for some .

On the other hand, consider that for every node a of H, given that . Next, we have c sits on each strong edge geodesic cover of H, derived from Theorem 13. Consequently, gn_e(H) = n.

5 Application

A key mathematical tool for representing real-world occurrences that are connected by graphical networks, in which nodes and edges represent m-polar fuzzy information, is the mPFG. Here, we apply the notion of a strong edge geodesic number inside a mPFG to a particular kind of covering problem.

5.1 Model construction

These days, a nation’s economic progress is greatly influenced by its transportation system. Well-organized planning will help to prevent loss and ensure that traffic moves smoothly. As a result, the covering problem is the core issue with the current research topic. Path covering is one of the more significant issues among them. Here, we talk about a path covering issue that has strong edge geodesic properties. Here, we study a road network represented as a mPFG, in which every node is a bus depot and an edge connects two nodes if bus routes exist between these depots. Our goal is to ascertain the bare minimum of traffic inspectors required to supervise and examine the bus routes on the network of metropolitan roads. At the same time, we want to find routes that passengers find less important so that they can be eliminated and reduce the losses that transportation companies suffer from inadequate collection. One may wonder if we can assign the MV of nodes and edges when we use 3PFG to represent this road network. Given that it has nine nodes and is a 3PFG. As a result, three factors will be needed to determine the maximum value of nodes, and three unique variables all of which must be uncertain?will be needed to display the maximum value of edges. Three factors will be taken into account while calculating the nodes MV: {the depot’s capacity, location, and services rendered}. The efficiency of bus route transportation is significantly influenced by the depot’s capacity, location, and services rendered. The depot serves as the operational hub where buses are maintained, refueled, and dispatched, making its capacity a crucial determinant of fleet size and service frequency. A strategically located depot minimizes deadhead miles distance traveled without passengers thereby reducing fuel costs, time wastage, and environmental impact. Additionally, the range of services rendered, such as maintenance, cleaning, and driver facilities, ensures operational reliability, vehicle longevity, and improved service quality. Collectively, these factors directly impact route efficiency, resource utilization, and passenger satisfaction. Optimizing depot characteristics is therefore fundamental for sustainable and cost-effective bus transportation systems. Each of the three factors varies depending on the depot and is characterized by uncertainty. Three factors will be taken into account when calculating the edges MV: {the state of the bus route, the signal delay, and the satisfied number of passengers}. The 3PFG model is presented in Fig 5.

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Fig 5. Model 3PFG G for bus route network system of some road network.

https://doi.org/10.1371/journal.pone.0327882.g005

Here, the edges membership value is given in tabular form in Table 5 for better visualization of the model 3PFG G.

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Table 5. Edge membership value of the proposed model.

https://doi.org/10.1371/journal.pone.0327882.t005

5.2 Illustration of membership values

The connectivity between the town nodes in the model network system which consists of nine nodes and twelve edges?is investigated about a specific 3PFG. First, the emphasis is on identifying if the relationships between the towns are α-strong, β-strong, or δ-strong.

We determine which edges are α-strong, β-strong, or δ-strong by calculating CONN_G−(a,b) for all , where E is the collection of edges in the model 3PFG. Then, using conventional calculations, the edges are categorized appropriately. For reference, the final edge categorization is shown in tabular form in Table 6.

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Table 6. Classification of edges given in tabular form.

https://doi.org/10.1371/journal.pone.0327882.t006

Here, we see that some arcs are δ-strong. From Theorem 4, we say that if a 3PFG has a strong edge geodesic number then it must not have a δ-strong arcs. Hence, we delete the δ-strong arcs from Fig 5 and we get the new 3PFG which is shown in Fig 6, where the existing nodes as well as edges have the same MV as of G described in the Fig 5. Since, we have to find out the solution of the problem stated above we must have an mPFG which has no δ-strong arcs and for our model δ-strong arcs are less priority route or the routes where minimum number of passengers will travel through bus. So, we delete all δ-strong arcs in the Fig 5 to convert the Fig 6.

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Fig 6. New 3PFG G₁ corresponding to the Fig 5.

https://doi.org/10.1371/journal.pone.0327882.g006

6 Advantages, disadvantages and limitations of the proposed work

In real-life scenarios, many problems are tackled using data collected from diverse sources, leading to multi-polarity. Such data, encompassing various origins, poses challenges for traditional fuzzy models, intuitionistic fuzzy models, or bipolar fuzzy models. For instance,we study a road network represented as a mPFG, in which every node is a bus depot and an edge connects two nodes if bus routes exist between these depots. Our goal is to ascertain the bare minimum of traffic inspectors required to supervise and examine the bus routes on the network of metropolitan roads. At the same time, we want to find routes that passengers find less important so that they can be eliminated and reduce the losses that transportation companies suffer from inadequate collection. Three factors will be taken into account while calculating the nodes MV: {the depot’s capacity, location, and services rendered}. The 3-polar fuzzy model is employed. Unlike traditional fuzzy models that handle single-component concepts, and bipolar or intuitionistic FG models limited to two components per edge or node, the mPFG model accommodates three components for each node and edge, offering more efficient fuzziness results. The development and analysis of such mPFGs, along with associated examples and theorems, enhance existing concepts and provide reliable solutions for complex real-world problems.

Additional advantages of the proposed model include:

1. (i) The ability for anyone to analyze MVs within a multi-polar fuzzy environment in a specific manner.
2. (ii) Presentation of a real application of ascertain the bare minimum of traffic inspectors required to supervise and examine the bus routes on the network of metropolitan roads. The mPF model utilizing the concept of geodesic.
3. (iii) Exploration of the concepts of multi-polarity in geodesic cover, basis within the mPFG environment.

The proposed model also has some disadvantages:

1. (i) It cannot accommodate negative MVs for the characters within this environment.
2. (ii) Non-heterogeneous types of data cannot be utilized within this framework.

Here are some limitations of this study:

1. (i) The mPFIG cannot be effectively applied if the MVs of the characters are provided in different interval-valued mPF environments.
2. (ii) The suggested methodology’s wider applicability across several disciplines, like the m-polar fuzzy graph notion, is limited by its primary application to networking systems.

7 Comparative study

At first, Bhutani and Rosenfeld [10] introduced a strong arcs for fuzzy graph theory. Next, they studied geodesic distance for crisp graph [11]. All of them studied strong arcs as well as edge geodesic distance for crisp graph and fuzzy graph theory. Later on, Suvarna and Sunitha [35] first introduced the concept of geodesic number for fuzzy graph. This concept is applicable only for fuzzy graphs. Later on, Reshmani and Sunitha [34] proposed a model to design edge geodesic number for fuzzy graph concept. So, all the earlier results are not applicable when the model is considered in another environment like in m-polar fuzzy sets. This is why the proposed model in this paper plays a significant role in such situations to give better results.

8 Conclusions

This paper introduces and defines the concepts of geodesic numbers and strong edge geodesic numbers within the context of mPFGs. It investigates the strong edge geodesic number for several well-known mPFGs and establishes an upper bound for this parameter. Additionally, the paper explores relationships between strong geodesic and strong edge geodesic numbers, while also delving into intriguing properties of the latter. The concept of edge geodesic numbers plays a critical role in graph theory, particularly in determining the minimum number of edges that intersect all shortest paths (geodesics) within a graph. The concept of the edge geodesic number, which quantifies the minimum number of edges required to maintain geodesic connectivity in a graph, has gained increasing interest in classical and fuzzy graph theory. In recent years, the extension of this parameter to fuzzy and m-polar fuzzy graphs has opened new avenues for exploration due to their ability to model uncertainty and multi-polarity in complex systems. This abstract proposes future directions in the study of the edge geodesic number on m-polar fuzzy graphs, emphasizing the development of novel algorithms to compute this invariant under varying degrees of membership, non-membership, and hesitation. Potential applications in network resilience, data mining, and decision-making systems are discussed, where multi-polarity plays a critical role. Further, we outline the theoretical challenges in defining edge geodesic preservation under m-polar constraints, propose generalizations to weighted and directed m-polar fuzzy graphs, m-polar intuitionistic fuzzy graphs, m-polar interval-valued intuitionistic fuzzy graphs and suggest comparative studies with other fuzzy graph parameters. This direction of study promises to deepen our understanding of structural properties in fuzzy environments and provide robust tools for real-world modeling.

Acknowledgments

The authors are highly grateful to the learned reviewers for their constructive comments which led to improve the quality of the paper strongly.