EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6216 ISSN 1307-5543 – ejpam.com Published by New York Business Global Domination in Bipolar Fuzzy Rough Digraphs with Applications to Decision-Making Aneesa Arif1, Aliya Fahmi1, Aziz Khan2, Aiman Mukheimer2, Thabet Abdeljawad4,2,5,∗, Rajermani Thinakaran3 1 Department of Mathematics, Faculty of Science, University of Faisalabad, Faisalabad, Pakistan 2 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia 3 Faculty of Data Science and Information Technology, INTI International University, Negeri Sembilan, Malaysia 4 Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, Avcılar-Istanbul, 34310, Turkiye 5 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa Abstract. Fuzzy Rough Digraphs are insufficient for scenarios involving both positive and neg- ative influences. To address the limitations of Fuzzy Rough Digraphs in modeling conflicting information, this paper introduces the Bipolar Fuzzy Rough Digraph (BFRD) as a new frame- work for decision-making under uncertainty. We define its fundamental properties, including the strength of paths, connectedness, vertex degree, the Regular BFRD. From these, we establish the concepts of the minimum dominating set and domination number. An algorithm is then developed to apply this framework to practical problems. The model’s efficacy is demonstrated through a real-world application: identifying an optimal set of rural areas for establishing medicine supply markets by finding the minimum dominating set. This work provides a robust mathematical tool for solving complex problems involving bipolarity and as a process innovation. 2020 Mathematics Subject Classifications: 03E72, 03B52, 68T37, 05C69, 05C72, 05C85 Key Words and Phrases: Theory of fuzzy sets, fuzzy sets, fuzzy logic, applications of fuzzy set theory, dominating sets, fuzzy graph theory, graph algorithms ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6216 Email addresses: aliyafahmi@gmail.com, akhan@psu.edu.sa (A. Fahmi, A. Khan), mukheimer@psu.edu.sa (A. Mukheimer), tabdeljawad@psu.edu.sa (T. Abdeljawad), rajermani.thina@newinti.edu.my (R. Thinakaran) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 2 of 36 1. Introduction The introduction of Fuzzy Set theory by Zadeh [1, 2] provided a revolutionary mathe- matical framework to handle the inherent vagueness of real-world data, moving beyond the binary limitations of classical set theory. In classical set theory, an element either belongs to a set or does not, represented by binary membership (0 or 1). However, in the real world, many concepts and phenomena are not strictly black or white; they often exhibit shades of gray or degrees of membership. The fuzzy set theory addresses this limitation by allowing elements to belong to a set with a degree of membership that ranges between 0 and 1. The objective of fuzzy set theory is to provide a formal framework that accom- modates the inherent fuzziness of real-world data and phenomena. In 1983, Chakraborty and Das [3–5] extended studies of fuzzy relations and fuzzy equivalence relations and other [83–86]. This framework found a natural and powerful application in graph theory, leading to the development of Fuzzy Graphs, a concept pioneered by Kauffmann [6], Yeh and Bang [7], and Rosenfeld [8]. Kauffmann studied the extension of classical graph theory to fuzzy graph theory in 1973. In classical graph theory, edges and vertices are either present or absent, leading to a binary representation. In contrast, fuzzy graph theory allows for edges and vertices to have degrees of membership, indicating the strength or degree of connection between nodes. This innovation spurred extensive research into the structural properties of these graphs, with key contributions on fuzzy groups by Bhattacharya [9], connectedness by Banerjee [10], fuzzy vertex graphs by Koczy [11]. Udupa and Sama- rasekera [12] used the concept of fuzzy connectedness in image segmentation. Bhutani [13, 14] presented the ideas of the cut node and fuzzy edge node, and node connectivity is presented by Mathew and Sunitha [15, 16]. Binu et al. [17] studied the applications of the connectivity status of fuzzy graphs in network science. A particularly vital concept that emerged is domination in fuzzy graphs, introduced by Somasundaram and Somasundaram [18], a topic that has been further developed to in- clude domination numbers and independent sets by researchers like Gani and Vadivel [19], Manjusha and Sunitha [20], and Talebi et al. [21]. The idea of domination in fuzzy graphs is used to analyze the control or influence that certain vertices have over others in a fuzzy network, taking into account the uncertainty or ambiguity in the relationships between vertices. However, standard fuzzy models are limited in scenarios involving conflicting information. To address this, Zhang [22] introduced Bipolar Fuzzy Sets (BFSs), which assign each element both positive and negative membership degrees, indicating the degree to which it belongs and does not belong to the set, respectively. An element’s negative degree of membership falls between [−1, 0], where −1 denotes complete non-membership. This concept was quickly extended to graph theory by Akram [23], leading to the creation of Bipolar Fuzzy Graphs. Akram et al. [24–26] examined the uses of BFSs in the theory of graphs, the operations on bipolar fuzzy graphs, and their applications in decision-making problems. Paulik and Ghorai [27] studied the applications of the connectivity index of bipolar fuzzy graphs. Akram et al. [28, 29] examined the various forms of bipolar fuzzy A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 3 of 36 graphs for handling uncertainty. Gong and Hua [30] studied the practical application of fuzzy edge connectivity. Karunambigai et al. [31] and Akram et al. [33] introduced the idea of domination in bipolar fuzzy graphs. Muneera et al. [32] studied the domination in various forms of bipolar fuzzy graphs. A parallel line of inquiry for managing uncertainty stems from Pawlak’s Rough Set the- ory [34–36], which provides a formal method for approximating concepts from incomplete information. In rough set theory, information is organized into granules or sets of objects. These sets can represent concepts, categories, or classes within a dataset. RSs define lower and upper approximations of sets based on the available information. The lower approx- imation represents the set of elements that are certainly within the set, while the upper approximation includes elements that may or may not be part of the set. The boundary region of a set in rough set theory consists of elements that are uncertain or indeterminate about their membership status. These elements lie on the boundary between the lower and upper approximations of the set. This approach was shown to be effective for knowl- edge discovery and data analysis by Kryszkiewicz [37]. To harness the strengths of both paradigms, Dubois and Prade [38, 39] introduced hybrid models like Rough Fuzzy Sets, which were extensively studied and generalized by Yao [40], Radzikowska and Kerre [41], and Yeung et al. [42]. This hybridization was later applied to bipolar fuzzy environments by Yang et al. [43, 44]. After that, researchers [54–57] extended the research of rough graphs and studied different types, including directed rough graphs, and vertex rough graphs, and discussed their properties. These concepts were then extended to network structures. Rough graph theory extends rough set theory and graph theory to analyze and characterize imprecise or uncertain data in graph-based systems. He et al. [49–52] introduced the notion of rough graphs and their types, including weighted rough graphs and S-rough graphs. Liang et al. [53] studied the type of rough graph, known as an edge rough graph. Akram and Arshad [58] presented fuzzy rough graph theory. While fuzzy graphs focus on representing uncertainty and vagueness in graph-based data through fuzzy member- ship values, fuzzy rough graphs extend this concept by incorporating RS approximations to handle uncertainty and approximation simultaneously. Researchers [59, 60] studied properties and extensions of rough fuzzy graphs in the Neutrosophic Sets. The study of directed rough fuzzy graphs was expanded upon by Ahmad and Nawaz [62, 63], who also examined its practical applications in networks related to human trafficking and trade. In their study of vertex degree concepts, Nawaz and Ahmad [64] defined several opera- tions, such as union, cartesian product, and composition of directed rough fuzzy graphs. Akram and Zafar [65] initialized the concept of connectivity between vertices and edges of rough fuzzy directed graphs and discovered its applications in decision-making problems. Ahmad et al. [66, 67] introduced the concepts of strength of connectivity, neighborhood connectivity index, and domination in rough fuzzy directed graphs. Khan et al. [68, 69] established a non-linear system of variable order of fractional differen- tial equations and a fractal-fractional hybrid model to calculate the impact of fast-moving greenhouse gas emissions on climate change and coastal ecosystems. Kundu et al. [70] studied the complexity of habitat in a discrete predator-prey model. Alzabut et al. [71] A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 4 of 36 studied a discrete fractional equation model with an initial condition built to investigate tumor-immune interactions. This brings us to a critical and unaddressed research gap. The literature shows two pow- erful, but separate, streams of development: Bipolar Fuzzy Graphs and Fuzzy Rough Digraphs. To date, no framework unifies these capabilities. The motivation for this re- search is to fill this void. The urgency of this integration is underscored by the widespread success of bipolar fuzzy logic in various decision-making domains, as demonstrated in re- cent studies. The literature provides extensive evidence for the efficacy of Bipolar Fuzzy Sets (BFSs) in advancing Multi-Criteria Decision Making methodologies, as summarized in the work of Akram et al. [72]. This framework’s practical utility is demonstrated through its successful integration with established MCDM techniques to solve complex, real-world problems. For example, Akram and Shumaiza [75] developed a bipolar fuzzy PROMETHEE process for green supplier selection, while Akram et al. [76] extended the TOPSIS and ELECTRE-I methods to handle nuanced diagnostic problems. Beyond direct application, a significant research thrust has focused on optimizing these models for efficiency. Work by Ali et al. has been pivotal in this area, introducing meth- ods for both attribute reduction in bipolar fuzzy relational systems [73] and parameter reduction in bipolar fuzzy soft sets [77], thereby making the decision-making algorithms more computationally manageable. The adaptability of the BFS framework is further highlighted by its application to specialized data structures, such as the analysis of bipo- lar fuzzy N-soft information by Akram et al. [74]. While the concept of domination in Fuzzy Rough Digraphs (FRDs) has useful applications, it suffers from a critical problem: it cannot effectively model situations involving conflict- ing positive and negative information. At the same time, bipolar fuzzy sets have been extensively applied to algebraic decision-making to handle precisely this kind of duality. This reveals a significant research gap, as there is currently no mathematical framework that integrates the power of bipolar fuzzy sets with the structural uncertainty of rough digraphs to model networks under conditions of both bipolarity and roughness simultane- ously. The motivation for this research stems directly from this gap. Real-world decision sce- narios are frequently characterized by this kind of bipolar uncertainty, and the lack of a suitable model hinders our ability to analyze these problems comprehensively. Therefore, the primary objective of this paper is to introduce and formalize the concept of the Bipolar Fuzzy Rough Digraph (BFRD) to address this need. The novelty of our research is: • To develop the notions of the strength of a path, strength of connectedness, and domination in BFRDs. • To develop an effective domination model based on BRFDs. • To study a type of BFRDs known as Regular Bipolar Fuzzy Rough Digraphs. • To develop a proposed algorithm and apply it to practical decision-making problems. A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 5 of 36 The flowchart of domination in digraphs and its extension to the domination in the Bipolar Fuzzy Rough Digraph is given in Figure A. Domination in Digraphs Domination in Fuzzy Digraphs Domination in Bipolar Fuzzy Digraphs Domination in Fuzzy Rough Digraphs Domination in Bipolar Fuzzy Rough Digraphs Figure A: Flowchart of domination in digraphs and its types The list of abbreviations used in this paper is listed in Table 1: Abbreviations Name RS Rough Set BFRD Bipolar Fuzzy Rough Digraph FRD Fuzzy Rough Digraph FS Fuzzy Set BFS Bipolar Fuzzy Set FRS Fuzzy Rough Set BFRS Bipolar Fuzzy Rough Set BFTR Bipolar Fuzzy Tolerance Relation DS Dominating Set MDS Minimum Dominating Set Table 1: List of abbreviations The paper is organized as follows. Section 2 includes the crucial BFRD preliminary in- formation. Section 3 presents findings about domination in BFRDs along with examples, A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 6 of 36 theorems, and some types of BFRDs, including Regular Bipolar Fuzzy Rough Digraphs and Totally Regular Bipolar Fuzzy Rough Digraphs. Section 4 presents an algorithm and practical uses of domination in BFRDs for decision-making. Section 5 presents a succinct summary of the paper. 2. Preliminaries In this section, we will cover the foundations of domination in Fuzzy Graphs to estab- lish the concept of domination in Bipolar Fuzzy Rough Digraphs. Definition 1.[39] Consider a fuzzy equivalence relation J on the universal set X and let O be a FS on X, then the lower approximation JO and the upper approximation JO are given by: { JO = ∧[(1− J(α, β)) ∨O(β)] JO = ∨[J(α, β) ∧O(β)], ∀α ∈ X Such that JO − JO ̸= ∅ then the ordered pair (JO, JO) is called the FRS. Definition 2.[58] Consider a fuzzy tolerance relation J on the universal set X, B be a fuzzy tolerance relation on A∗ ⊆ X ×X, O be a FS on X, JO = (JO, JO) be a FRS on X, and V be a FS on A∗ such that:{ B(α1α2, β1β2) ≤ J(α1β1) ∧ J(α2β2), ∀ α1α2, β1β2 ∈ A∗ V (αβ) ≤ (JO)(α) ∧ (JO)(β), ∀αβ ∈ A∗ Then the lower approximation BV and the upper approximation BV are given by:{ (BV )(αβ) = ∧[(1−B(α1β1, α2β2)) ∨ V (α2β2)] (BV )(αβ) = ∨[B(α1β1, α2β2) ∧ V (α2β2)], ∀α1β1 ∈ A∗ The pair BV = (BV,BV ) is a fuzzy rough relation on X. Definition 3. [58] Consider a fuzzy tolerance relation J on the universal set X, B be a fuzzy tolerance relation on A∗ ⊆ X × X, O be a FS on X, JO = (JO, JO) be a FRS on X, and BV = (BV,BV ) be a fuzzy rough relation on X, then the FRD is given by: G = (O, JO, V,BV ) Where G = (JO,BV ) is the lower approximation of graph G and G = (JO,BV ) is the upper approximation of G such that:{ BV (αβ) ≤ ∧ [JO(α), JO(β)] BV (αβ) ≤ ∧ [JO(α), JO(β)], ∀ αβ ∈ A∗. Definition 4.[61] Let O = (O,O) be a FRD on a universal set X then P : τ0 → A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 7 of 36 τ1 → · · · → τn is a directed path of length n from a = τ0 to b = τn in O and O where µO(τi−1, τi) > 0 and µO(τi−1, τi) > 0 for i = 1, 2, .., n then strength of the path is defined as: [ µO(a, b) ]n = ∧ [µO(a, τ1), µO(τ1, τ2), . . . , µO(τn−1, b)][ µO(a, b) ]n = ∧ [µO(a, τ1), µO(τ1, τ2), . . . , µO(τn−1, b)] Such that the strength of a path in O = (O,O) is defined by: [µ(a, b)]n = [ µO(a, b) ]n + [ µO(a, b) ]n Definition 5.[61] Let O = (O,O) be a FRD on a universal set X . The strength of connectedness from point τ0 to τ1 in O and O is defined by: CONNO(τ0, τ1) = supn∈N[µO(τ0, τ1)] n CONNO(τ0, τ1) = supn∈N[µO(τ0, τ1)] n Definition 6.[61] An edge τ0τ1 in FRD O = (O,O) is considered as a strong edge in O and O if: CONNO−τ0τ1(τ0, τ1) ≤ µO(τ0, τ1) CONNO−τ0τ1 (τ0, τ1) ≤ µO(τ0, τ1) If an edge τ0τ1 is strong in both O and O then it is strong in O = (O,O). Definition 7.[67] Let O = (O,O) be a FRD on a universal set X . Let τ0, τ1 ∈ X then vertex τ0 dominates the vertex τ1 in O if directed edge from τ0 to τ1 is a strong edge in O such that: CONNO−τ0τ1(τ0, τ1) ≤ µO(τ0, τ1) Similarly, τ0 dominates the vertex τ1 in O if directed edge from τ0 to τ1 is a strong edge in O such that: CONNO−τ0τ1 (τ0, τ1) ≤ µO(τ0, τ1) If directed edge from τ0 to τ1 is a strong edge in both O and O then τ0 dominates the vertex τ1 in O = (O,O). A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 8 of 36 Definition 8. [46] Let X be universal set and Υ = (Υ+,Υ−) be a bipolar fuzzy equiv- alence relation on X. Let W = (W+,W−) be a BFS on X. The lower approximation of W under Υ is denoted by ΥW and the upper approximation of W under Υ is denoted by ΥW such that: ΥW = (ΥW+,ΥW−) ΥW = (ΥW+,ΥW−) ΥW and ΥW are defined by: ΥW+(ϵ0) = ∧ [1−Υ+(ϵ0, ϵ1) ∨W+(β)] ΥW−(ϵ0) = ∨ [−1−Υ−(ϵ0, ϵ1) ∧W−(ϵ1)] ΥW+(ϵ0) = ∨ [Υ+(ϵ0, ϵ1) ∧W+(ϵ1)] ΥW−(ϵ0) = ∧ [Υ−(ϵ0, ϵ1) ∨W−(ϵ1)] , ∀ϵ0, ϵ1 ∈ X The pair (ΥW,ΥW ) is called BFRS if ΥW −ΥW ̸= ∅ . Definition 9. [47] Let X be universal set and Υ = (Υ+,Υ−) be a bipolar fuzzy tolerance equivalence relation on X. Let W = (W+,W−) be a BFS on X and (ΥW,ΥW ) is a BFRS on X. Let A ⊆ X ×X and L = (L−, L+) be a BFTR on A such that: L+(αβ, γθ) ≤ Υ+(α, γ) ∧Υ+(β, θ) L−(αβ, γθ) ≥ Υ−(α, γ) ∨Υ−(β, θ) Let a BFS Λ = (Λ+,Λ−) on A such that:{ Λ+(γθ) ≤ (ΥW+)(γ) ∧ΥW+(θ) Λ−(γθ) ≥ (ΥW−)(γ) ∨ΥW−(θ) The lower approximation is denoted by LΛ and upper approximation is denoted by LΛ such that: LΛ = (LΛ+, LΛ−), LΛ = (LΛ+, LΛ−) And are defined by: LΛ+(αβ) = ∧ [(1− L+(αβ, γθ)) ∨ Λ+(γθ)] LΛ−(αβ) = ∨ [(−1− L−(αβ, γθ)) ∧ Λ−(γθ)] LΛ+(αβ) = ∨ [L+(αβ, γθ) ∧ Λ+(γθ)] LΛ−(αβ) = ∧ [(L−(αβ, γθ) ∨ Λ−(γθ)], ∀γθ ∈ A Then the pair (LΛ, L,Λ) is called a Bipolar fuzzy rough relation. A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 9 of 36 3. Domination in Bipolar Fuzzy Rough Digraphs In this section, we present the ideas of Bipolar Fuzzy Rough Digraphs, the union of Bipolar Fuzzy Rough Digraph, the strength of a path, the strength of the connectedness, domination, minimum dominating set, domination number, degree of the vertices, and the notions of Regular, Totally Regular Bipolar Fuzzy Rough Digraphs. Definition 10. Let X be a universal set. If Υ is a BFTR on X, L is a BFTR on A ⊆ X ×X, W be a BFS on X, ΥW = (ΥW,ΥW ) is a BFRS on X, and LΛ = (LΛ, LΛ) is a bipolar fuzzy rough relation on X, then the BFRD is defined by: Γ = (W,ΥW,Λ, LΛ) Where Γ = (ΥW,LΛ) is the lower approximation of Γ and Γ = (ΥW,LΛ) is the upper approximation of Γ such that: LΛ+(αβ) ≤ ∧ [ΥW+(α),ΥW+(β)] LΛ−(αβ) ≥ ∨ [ΥW−(α),ΥW−(β)] LΛ+(αβ) ≤ ∧ [ΥW+(α),ΥW+(β)] LΛ−(αβ) ≥ ∨ [ΥW−(α),ΥW−(β)], ∀ αβ ∈ A. Example 1. Let X = {ϵ, η, ζ} be a universal set and Υ = (Υ+,Υ−) be a BFTR on X defined in Tables 2 and 3. Υ+ ϵ η ζ ϵ 1 0.1 0.3 η 0.1 1 0.5 ζ 0.3 0.5 1 Table 2: Υ+ of relation Υ Υ− ϵ η ζ ϵ −1 −0.1 −0.4 η −0.1 −1 −0.2 ζ −0.4 −0.2 −1 Table 3: Υ− of relation Υ Let W = {(ϵ, 0.2,−0.1), (η, 0.3,−0.1), (ζ, 0.1,−0.3)} be a BFS on X. Then the lower ap- proximation of W with respect to Υ is given by: ΥW = {(ϵ, 0.2,−0.1), (η, 0.3,−0.1), (ζ, 0.1,−0.3)} The upper approximation Υ is given by: ΥW = {(ϵ, 0.2,−0.3), (η, 0.3,−0.2), (ζ, 0.3,−0.3)} Since ΥW −ΥW ̸= ∅, therefore, (ΥW,ΥW ) is a BFRS. Let L = (L+, L−) be a BFTR on A ⊆ X ×X and defined in the Tables 4 and 5: A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 10 of 36 L+ ϵϵ ϵη ηζ ϵζ ϵϵ 1 0.1 0.1 0.2 ϵη 0.1 1 0.05 0.02 ηζ 0.1 0.05 1 0.03 ϵζ 0.2 0.02 0.03 1 Table 4: L− of relation L L− ϵϵ ϵη ηζ ϵζ ϵϵ −1 −0.1 −0.05 −0.3 ϵη −0.1 −1 −0.1 −0.1 ηζ −0.05 −0.1 −1 −0.05 ϵζ −0.3 −0.1 −0.05 −1 Table 5: L− of relation L Let Λ = (Λ+,Λ−) be a BFS on A defined by: Λ = {(ϵϵ, 0.2.− 0.2), (ϵη, 0.1,−0.1), (ηζ, 0.2,−0.1), (ϵζ, 0.1,−0.2)} The set of lower approximation of Λ is: LΛ = {(ϵϵ, 0.2,−0.2), (ϵη, 0.1,−0.1), (ηζ, 0.2,−0.1), (ϵζ, 0.1,−0.2)} The set of upper approximation of Λ is: LΛ = {(ϵϵ, 0.2,−0.2), (ϵη, 0.1,−0.1), (ηζ, 0.2,−0.1), (ϵζ, 0.2,−0.2)} The graphs of Γ and Γ of Γ are given in Figure 1: ϵ (0.2,−0.1) η (0.3,−0.1) ζ (0.1,−0.3) (0.1, -0.1) (0.1, -0.2) (0.2, -0.2) (0.2, -0.1) Γ = (ΥW,LΛ) ϵ (0.2,−0.3) η (0.3,−0.2) ζ (0.3,−0.3) (0.1, -0.1) (0.2, -0.2) (0.2, -0.2) (0.2, -0.1) Γ = (ΥW,LΛ) Figure 1: Γ = (Γ,Γ) Definition 11. The union of two BFRDs Γ1 = (Γ1,Γ1) and Γ2 = (Γ2,Γ2) on universal set X is defined as Γ1 ∪Γ2 = (Γ1 ∪Γ2,Γ1 ∪Γ2) where Γ1 ∪Γ2 = (ΥW1 ∪ΥW2, LΛ1 ∪LΛ2) and Γ1 ∪ Γ2 = (ΥW1 ∪ΥW2, LΛ1 ∪ LΛ2) such that: (ΥW1 ∪ΥW2)(x) = { ΥW+ 1 (x) ∨ΥW+ 2 (x) ΥW− 1 (x) ∧ΥW− 2 (x), ∀x ∈ supp(W1 ∪W2) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 11 of 36 (ΥW1 ∪ΥW2)(x) = { ΥW+ 1 (x) ∨ΥW+ 2 (x) ΥW− 1 (x) ∧ΥW− 2 (x), ∀x ∈ supp(W1 ∪W2) (LΛ1 ∪ LΛ2)(xy) = { LΛ+ 1 (xy) ∨ LΛ+ 2 (xy) LΛ− 1 (xy) ∧ LΛ− 2 (xy), ∀xy ∈ supp(Λ1 ∪ Λ2) (LΛ1 ∪ LΛ2)(xy) = { LΛ+ 1 (xy) ∨ LΛ+ 2 (xy) LΛ− 1 (xy) ∧ LΛ− 2 (xy), ∀xy ∈ supp(Λ1 ∪ Λ2) Definition 12. Let Γ = (Γ,Γ) be a BFRD on a universal set X then P : ϵ0 → ϵ1 → · · · → ϵn is a directed path of length n from a = ϵ0 to b = ϵn in Γ and Γ where (LΛ+(ϵi−1, ϵi) > 0, LΛ−(ϵi−1, ϵi) < 0) and (LΛ+(ϵi−1, ϵi) > 0, LΛ−(ϵi−1, ϵi) < 0) for i = 1, 2, .., n then: [ LΛ+(a, b) ]n = ∧ [LΛ+(a, ϵ1), LΛ +(ϵ1, ϵ2), . . . , LΛ +(ϵn−1, b)][ LΛ−(a, b) ]n = ∨ [LΛ−(a, ϵ1), LΛ −(ϵ1, ϵ2), . . . , LΛ −(ϵn−1, b)][ LΛ+(a, b) ]n = ∧ [LΛ+(a, ϵ1), LΛ +(ϵ1, ϵ2), . . . , LΛ +(ϵn−1, b)][ LΛ−(a, b) ]n = ∨ [LΛ−(a, ϵ1), LΛ −(ϵ1, ϵ2), . . . , LΛ −(ϵn−1, b)] Such that the strength of a path in Γ = (Γ,Γ) is defined by:[ LΛ+(a, b) ]n = [ LΛ+(a, b) ]n + [ LΛ+(a, b) ]n[ LΛ−(a, b) ]n = [ LΛ−(a, b) ]n + [ LΛ−(a, b) ]n Definition 13. Let Γ = (Γ,Γ) be a BFRD on a universal set X. The strength of connectedness from point ϵ0 to ϵ1 in Γ and Γ is defined by: CONN+ Γ (ϵ0, ϵ1) = supn∈N[LΛ +(ϵ0, ϵ1)] n CONN− Γ (ϵ0, ϵ1) = infn∈N[LΛ −(ϵ0, ϵ1)] n CONN+ Γ (ϵ0, ϵ1) = supn∈N[LΛ +(ϵ0, ϵ1)] n CONN− Γ (ϵ0, ϵ1) = infn∈N[LΛ −(ϵ0, ϵ1)] n and it is denoted by: CONNΓ(ϵ0, ϵ1) = (CONN+ Γ (ϵ0, ϵ1),CONN− Γ (ϵ0, ϵ1)) CONNΓ(ϵ0, ϵ1) = (CONN+ Γ (ϵ0, ϵ1),CONN− Γ (ϵ0, ϵ1)) Definition 14. An edge ϵ0ϵ1 in BFRD Γ = (Γ,Γ) is considered as a strong edge in Γ and Γ if: CONN+ Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≤ LΛ+(ϵ0, ϵ1) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 12 of 36 CONN− Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≥ LΛ−(ϵ0, ϵ1) CONN+ Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≤ LΛ+(ϵ0, ϵ1) CONN− Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≥ LΛ−(ϵ0, ϵ1) If an edge ϵ0ϵ1 is strong in both Γ and Γ then it is strong in Γ = (Γ,Γ). Example 2. Let Γ = (Γ,Γ) be a BFRD defined on the universal set X = {ξ1, ξ2, ξ3, ξ4} given in Figure 2: ξ1 (0.1,−0.2) ξ2 (0.2,−0.1) ξ3 (0.3,−0.2) ξ4 (0.3,−0.3) (0.1, -0.1) (0.2, -0.1) (0.2, -0.1) (0.1, -0.05) Γ = (ΥW,LΛ) ξ1 (0.4,−0.6) ξ2 (0.3,−0.4) ξ3 (0.5,−0.4) ξ4 (0.6,−0.2) (0.3, -0.3) (0.3, -0.3) (0.2, -0.2) (0.3, -0.1) Γ = (ΥW,LΛ) Figure 2: Γ = (Γ,Γ) CONN+ Γ−ξ1ξ2 (ξ1, ξ2) =0 ≤ 0.1 = LΛ+(ξ1, ξ2) CONN− Γ−ξ1ξ2 (ξ1, ξ2) =0 ≥ −0.1 = LΛ−(ξ1, ξ2) CONN+ Γ−ξ2ξ3 (ξ2, ξ3) =0 ≤ 0.2 = LΛ+(ξ2, ξ3) CONN− Γ−ξ2ξ3 (ξ2, ξ3) =0 ≥ −0.1 = LΛ−(ξ2, ξ3) CONN+ Γ−ξ3ξ4 (ξ3, ξ4) =0 ≤ 0.2 = LΛ+(ξ3, ξ4) CONN− Γ−ξ3ξ4 (ξ3, ξ4) =0 ≥ −0.1 = LΛ−(ξ3, ξ4) CONN+ Γ−ξ1ξ4 (ξ1, ξ4) =0.1 ≤ 0.1 = LΛ+(ξ1, ξ4) CONN− Γ−ξ1ξ4 (ξ1, ξ4) =− 0.1 ≤ −0.05 = LΛ−(ξ1, ξ4) CONN+ Γ−ξ1ξ2 (ξ1, ξ2) =0 ≤ 0.3 = LΛ+(ξ1, ξ2) CONN− Γ−ξ1ξ2 (ξ1, ξ2) =0 ≥ −0.3 = LΛ−(ξ1, ξ2) CONN+ Γ−ξ1ξ2 (ξ2, ξ3) =0 ≤ 0.3 = LΛ+(ξ2, ξ3) CONN− Γ−ξ2ξ3 (ξ2, ξ3) =0 ≥ −0.3 = LΛ−(ξ2, ξ3) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 13 of 36 CONN+ Γ−ξ3ξ4 (ξ3, ξ4) =0 ≤ 0.2 = LΛ+(ξ3, ξ4) CONN− Γ−ξ3ξ4 (ξ3, ξ4) =0 ≥ −0.2 = LΛ−(ξ3, ξ4) CONN+ Γ−ξ1ξ4 (ξ1, ξ4) =0.2 ≤ 0.3 = LΛ+(ξ1, ξ4) CONN− Γ−ξ1ξ4 (ξ1, ξ4) =− 0.2 ≤ −0.1 = LΛ−(ξ1, ξ4) The strong edges in Γ are (ξ1, ξ2), (ξ2, ξ3), and (ξ3, ξ4). The strong edges in Γ are (ξ1, ξ2), (ξ2, ξ3), and (ξ3, ξ4). Therefore (ξ1, ξ2), (ξ2, ξ3), and (ξ3, ξ4) are strong edges in Γ = (Γ,Γ). Definition 15. Let Γ = (Γ,Γ) be a BFRD on a universal set X. Let ϵ0, ϵ1 ∈ X then vertex ϵ0 dominates the vertex ϵ1 in Γ if directed edge from ϵ0 to ϵ1 is a strong edge in Γ such that: CONN+ Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≤ LΛ+(ϵ0, ϵ1) CONN− Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≥ LΛ−(ϵ0, ϵ1) Similarly, ϵ0 dominates the vertex ϵ1 in Γ if directed edge from ϵ0 to ϵ1 is a strong edge in Γ such that: CONN+ Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≤ LΛ+(ϵ0, ϵ1) CONN− Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≥ LΛ−(ϵ0, ϵ1) ϵ0 dominates the vertex ϵ1 in Γ = (Γ,Γ) if directed edge from ϵ0 to ϵ1 is a strong edge both in Γ and Γ. Definition 16. The fuzzy cardinality of set of points V in Γ or order of Γ is defined as: |V(Γ)| = ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 The fuzzy cardinality of set of points V in Γ or order of Γ is defined as: |V(Γ)| = ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 Example 3. Let Γ = (Γ,Γ) be a BFRD defined on the universal set X = {η, ξ, ϵ} given in Figure 1: The fuzzy cardinality of vertices in Γ are: |V(Γ)| = ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 |V(Γ)| =1 +ΥW+(η) + ΥW−(η) 2 + 1 + ΥW+(ξ) + ΥW−(ξ) 2 + 1 + ΥW+(ϵ) + ΥW−(ϵ) 2 A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 14 of 36 |V(Γ)| =1 + 0.3 + (−0.1) 2 + 1 + (0.1) + (−0.3) 2 + 1 + (0.2) + (−0.1) 2 |V(Γ)| =1.55 |V(Γ)| = ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 |V(Γ)| =1 +ΥW+(η) + ΥW−(η) 2 + 1 + ΥW+(ξ) + ΥW−(ξ) 2 + 1 + ΥW+(ϵ) + ΥW−(ϵ) 2 |V(Γ)| =1 + 0.3 + (−0.2) 2 + 1 + (0.3) + (−0.3) 2 + 1 + (0.2) + (−0.2) 2 |V(Γ)| =1.55 Definition 17. Let Γ = (Γ,Γ) be a BFRD. A subset γF of ΥW is said to be the lower dominating set of Γ if for every ϵ0 ∈ ΥW −γF there exists ϵ1 ∈ γF such that ϵ1 dominates ϵ0. A subset γF of ΥW is said to be the upper dominating set of Γ if for every ϵ0 ∈ ΥW −γF there exists ϵ1 ∈ γF such that ϵ1 dominates ϵ0. A set of vertices is called a DS in Γ = (Γ,Γ) if it is both upper and lower dominating set. Definition 18. Let Γ = (Γ,Γ) be a BFRD. A DS in Γ is called a lower minimal dominating set if there are no proper subsets of it in Γ. A DS in Γ is called an upper minimal dominating set if there are no proper subsets of it in Γ. A DS that is both lower and upper minimal dominating set is called a minimal dominating set in Γ = (Γ,Γ). A minimal dominating set for which the sum of fuzzy cardinalities in Γ and Γ is least among all minimal dominating sets of Γ = (Γ,Γ) is called a MDS D(Γ). |D(Γ)|+ |D(Γ)| = ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 + ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 Definition 19. Let Γ = (Γ,Γ) be a BFRD. The fuzzy cardinality of a minimal dom- inating set D(Γ) in Γ is called lower domination number ΩD(Γ). The fuzzy cardinality of a minimal dominating set D(Γ) in Γ is called upper domination number ΩD(Γ). The sum of the lower and upper domination number of a MDS D(Γ) is called the domination number of BFRD Γ = (Γ,Γ): ΩD(Γ) = ΩD(Γ) + ΩD(Γ) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 15 of 36 Example 4. Let Γ = (Γ,Γ) be a BFRD defined on the universal set X = {ξ, κ, ε, δ} given in Figure 3: ξ(0.3,−0.2) κ (0.2,−0.3) ε (0.1,−0.2) δ (0.2,−0.2) (0.2, -0.2) (0.1, -0.2) (0.1, -0.05) (0.2, -0.1) Γ = (ΥW,LΛ) ξ(0.4,−0.5) κ (0.5,−0.3) ε (0.6,−0.2) δ (0.3,−0.5) (0.4, -0.3) (0.5, -0.2) (0.2, -0.4) (0.3, -0.3) Γ = (ΥW,LΛ) Figure 3: Γ = (Γ,Γ) CONN+ Γ−ξκ(ξ, κ) =0 ≤ 0.2 = LΛ+(ξ, κ) CONN− Γ−ξκ(ξ, κ) =0 ≥ −0.2 = LΛ−(ξ, κ) CONN+ Γ−κϵ(κ, ϵ) =0 ≤ 0.1 = LΛ+(κ, ϵ) CONN− Γ−κϵ(κ, ϵ) =0 ≥ −0.2 = LΛ−(κ, ϵ) CONN+ Γ−κδ(κ, δ) =0 ≤ 0.2 = LΛ+(κ, δ) CONN− Γ−κδ(κ, δ) =0 ≥ −0.1 = LΛ−(κ, δ) CONN+ Γ−ξδ(ξ, δ) =0.2 ≥ 0.1 = LΛ+(ξ, δ) CONN− Γ−ξδ(ξ, δ) =− 0.1 ≤ −0.05 = LΛ−(ξ, δ) CONN+ Γ−ξκ (ξ, κ) =0 ≤ 0.4 = LΛ+(ξ, κ) CONN− Γ−ξ,κ (ξ, κ) =0 ≥ −0.3 = LΛ−(ξ, κ) CONN+ Γ−κϵ (κ, ϵ) =0 ≤ 0.5 = LΛ+(κ, ϵ) CONN− Γ−κϵ (κ, ϵ) =0 ≥ −0.2 = LΛ−(κ, ϵ) CONN+ Γ−κδ (κ, δ) =0 ≤ 0.3 = LΛ+(κ, δ) CONN− Γ−κδ (κ, δ) =0 ≥ −0.3 = LΛ−(κ, δ) CONN+ Γ−ξδ (ξ, δ) =0.3 ≥ 0.2 = LΛ+(ξ, δ) CONN− Γ−ξδ (ξ, δ) =− 0.3 ≤ −0.4 = LΛ−(ξ, δ) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 16 of 36 Therefore ξκ, κϵ, and κδ are strong edges but ξδ is not a strong edge. The lower mini- mum dominating set in Γ is D = {κ, ξ} and the upper minimum dominating set in Γ is D = {κ, ξ}. The fuzzy cardinality of MDS in Γ is: ΩD(Γ) = 1 + ΥW+(κ) + ΥW−(κ) 2 + 1 + ΥW+(ξ) + ΥW−(ξ) 2 = 1 + 0.2 + (−0.3) 2 + 1 + 0.3 + (−0.2) 2 = 1 The fuzzy cardinality of MDS in Γ is: ΩD(Γ) = 1 + ΥW+(κ) + ΥW−(κ) 2 + 1 + ΥW+(ξ) + ΥW−(ξ) 2 = 1 + 0.5 + (−0.3) 2 + 1 + 0.4 + (−0.5) 2 = 1.05 The domination number of BFRD is: ΩD(Γ) = ΩD(Γ) + ΩD(Γ) = 1 + 1.05 = 2.05 Minimum dominating set Bounds. Let Γ = (Γ,Γ) be a BFRD. Let ϵ0ϵ1 be a strong edge in Γ, ∀ϵ1 ∈ Γ then the MDS is: D = {ϵ0} Therefore, a MDS has a minimum of 1 vertex. A MDS is always a subset of a vertex set with n vertices X such that: D ⊆ X Therefore, a MDS has a maximum of n vertices. Remark. Every MDS is a minimal dominating set in a BFRD. Theorem 1. Ω(Γ = (Γ,Γ)) ≤ |V (Γ)| where |V (Γ)| = |V (Γ)| + |V (Γ)| is the order of Bipolar Fuzzy Rough Digraphs. Proof: By definition of order of BFRDs: |V (Γ)| = |V (Γ)|+ |V (Γ)| Where { |V(Γ)| = ∑ ϵi∈V 1+ΥW+(ϵi)+ΥW−(ϵi) 2 |V(Γ)| = ∑ ϵi∈V 1+ΥW+(ϵi)+ΥW−(ϵi) 2 A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 17 of 36 such that |V(Γ)|, |V(Γ)| > 0. In BFRD the MDS D(Γ) of Γ = (Γ,Γ) is a subset V of Γ = (Γ,Γ). Case 1: If Γ = (Γ,Γ) has no strong edge then D(Γ) = V: ΩD(Γ) =|V(Γ)| = ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 ΩD(Γ) =|V(Γ)| = ∑ ϵi∈V 1 + ΥW+(ϵi) + ΥW−(ϵi) 2 ΩD(Γ) =ΩD(Γ) + ΩD(Γ) ΩD(Γ) =|V(Γ)|+ |V(Γ)| ΩD(Γ) =|V(Γ)| Case 2: If there is at least one strong edge in Γ = (Γ,Γ) which is strong edge in Γ and Γ then D(Γ) ⊂ V: ΩD(Γ) <|V (Γ)| ΩD(Γ) <|V (Γ)| ΩD(Γ) =ΩD(Γ) + ΩD(Γ) ΩD(Γ) <|V (Γ)|+ |V (Γ)| ΩD(Γ) <|V (Γ)| Hence Ω(Γ = (Γ,Γ)) ≤ |V (Γ)| is proved. Definition 20. The fuzzy edge cardinality of set of edges E in Γ is defined as: |E(Γ)| = ∑ ϵiϵj∈E 1 + LΛ+(ϵiϵj) + LΛ−(ϵiϵj) 2 The fuzzy cardinality of set of edges E in Γ is defined as: |E(Γ)| = ∑ ϵiϵj∈E 1 + LΛ+(ϵiϵj) + LΛ−(ϵiϵj) 2 Theorem 2. Let Γ1 = (Γ1,Γ1) and Γ2 = (Γ2,Γ2) be two Bipolar Fuzzy Rough Digraphs such that V1 ∩ V2 = ∅ where V1 and V2 are set of vertices of Γ1 and Γ2 respectively then Ω(Γ1 ∪ Γ2) = Ω(Γ1) + Ω(Γ2). Proof : Let D1 be the MDS of Γ1 and D2 be the MDS of Γ2 such that ΩD1(Γ1) is domination number of Γ1 and ΩD2(Γ2) is domination number of Γ2. Since V1 ∩ V2 = ∅, therefore D1+D2 is the MDS of Γ1∪Γ2 such that ΩD1+D2(Γ1∪Γ2) is domination number of Γ1 ∪ Γ2, then: ΩD1+D2(Γ1 ∪ Γ2) = ΩD1(Γ1) + ΩD2(Γ2) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 18 of 36 It follows that: Ω(Γ1 ∪ Γ2) = Ω(Γ1) + Ω(Γ2) Example 5. Let Γ1 = (Γ1,Γ1) and Γ2 = (Γ2,Γ2) be two BFRDs given in Figures 4 and 5: ξ1 (0.1,−0.2) ξ2 (0.2,−0.1) ξ3 (0.3,−0.2) ξ4 (0.3,−0.3) (0.1, -0.1) (0.2, -0.1) (0.2, -0.1) (0.1, -0.2) Γ1 = (ΥW1, LΛ1) ξ1 (0.4,−0.6) ξ2 (0.3,−0.4) ξ3 (0.5,−0.4) ξ4 (0.6,−0.2) (0.3, -0.3) (0.3, -0.3) (0.2, -0.2) (0.3, -0.1) Γ1 = (ΥW1, LΛ1) Figure 4: Γ = (Γ,Γ) κ1 (0.3,−0.1) κ2 (0.1,−0.3) κ3 (0.2,−0.1) (0.1, -0.3) (0.05, -0.1)(0.1, -0.1) Γ2 = (ΥW2, LΛ2) κ1 (0.3,−0.2) κ2 (0.4,−0.5) κ3 (0.5,−0.4) (0.2, -0.1) (0.3, -0.2)(0.2, -0.2) Γ2 = (ΥW2, LΛ2) Figure 5: Γ = (Γ,Γ) Then the lower and upper approximations of union of graphs Γ1 ∪ Γ2 are given in Figure 6 and 7: A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 19 of 36 ξ1 (0.1,−0.2) ξ2 (0.2,−0.1) ξ3 (0.3,−0.2) ξ4 (0.3,−0.3) (0.1, -0.1) (0.2, -0.1) (0.2, -0.1) (0.1, -0.2) κ1 (0.3,−0.1) κ2 (0.1,−0.3) κ3 (0.2,−0.1) (0.1, -0.3) (0.05, -0.1)(0.1, -0.1) Figure 6: Γ1 ∪ Γ2 ξ1 (0.4,−0.6) ξ2 (0.3,−0.4) ξ3 (0.5,−0.4) ξ4 (0.6,−0.2) (0.3, -0.3) (0.3, -0.3) (0.2, -0.2) (0.3, -0.1) κ1 (0.3,−0.2) κ2 (0.4,−0.5) κ3 (0.5,−0.4) (0.2, -0.1) (0.3, -0.2)(0.2, -0.2) Figure 7: Γ1 ∪ Γ2 The minimal dominating sets of Γ1 are D1 = {ξ2, ξ4} and D2 = {ξ1, ξ3}. The sum of fuzzy cardinalities of D1 = {ξ2, ξ4} in Γ1 and Γ1 is: ΩD1(Γ1) + ΩD1(Γ1) = 1 + 0.2 + (−0.1) 2 + 1 + 0.3 + (−0.3) 2 + 1 + 0.3 + (−0.4) 2 + 1 + 0.6 + (−0.2) 2 ΩD1(Γ1) + ΩD1(Γ1) =2.2 The sum of fuzzy cardinalities of D2 = {ξ1, ξ3} in Γ1 and Γ1 is: ΩD2(Γ1) + ΩD1(Γ1) = 1 + 0.1 + (−0.2) 2 + 1 + 0.3 + (−0.2) 2 + 1 + 0.4 + (−0.6) 2 + 1 + 0.5 + (−0.4) 2 ΩD2(Γ1) + ΩD1(Γ1) =1.95 D2 = {ξ1, ξ3} has minimum sum of fuzzy cardinalities, therefore {ξ1, ξ3} is a MDS and the domination number is Ω(Γ1) = 1.95. A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 20 of 36 The minimal dominating sets are D1 = {κ1, κ2}, D2 = {κ1, κ3}, and D3 = {κ2, κ3}. The sum of fuzzy cardinalities of D1 = {κ1, κ2} in Γ2 and Γ2 is: ΩD1(Γ2) + ΩD2(Γ2) = 1 + 0.3 + (−0.1) 2 + 1 + 0.1 + (−0.3) 2 + 1 + 0.3 + (−0.2) 2 + 1 + 0.4 + (−0.5) 2 ΩD1(Γ2) + ΩD1(Γ2) =2 The sum of fuzzy cardinalities of D2 = {κ1, κ3} in Γ2 and Γ2 is: ΩD2(Γ2) + ΩD2(Γ2) = 1 + 0.3 + (−0.1) 2 + 1 + 0.2 + (−0.1) 2 + 1 + 0.3 + (−0.2) 2 + 1 + 0.5 + (−0.4) 2 ΩD2(Γ2) + ΩD2(Γ2) =2.25 The sum of fuzzy cardinalities of D3 = {κ2, κ3} in Γ2 and Γ2 is: ΩD3(Γ2) + ΩD3(Γ2) = 1 + 0.2 + (−0.1) 2 + 1 + 0.1 + (−0.3) 2 + 1 + 0.5 + (−0.4) 2 + 1 + 0.4 + (−0.5) 2 ΩD3(Γ2) + ΩD3(Γ2) =1.95 D3 = {κ2, κ3} has minimum sum of fuzzy cardinalities, therefore {κ2, κ3} is a MDS and the domination number is Ω(Γ2) = 1.95. By using the same procedure, the minimal dom- inating set of Γ1 ∪ Γ2 is {ξ1, ξ3, κ2, κ3} and the domination number is Ω(Γ1 ∪ Γ2) = 3.9. Notice that Ω(Γ1 ∪ Γ2) = Ω(Γ1) + Ω(Γ2). Definition 21. Let Γ = (Γ,Γ) be a BFRD, the degree of a vertex ϵ0 ∈ V in Γ is defined as: dΓ(ϵ0) =(d+Γ (ϵ0), d − Γ (ϵ0)) =((( ∑ ϵ0 ̸=ϵ1 LΛ+(ϵ0, ϵ1) + ∑ ϵ0 ̸=ϵ1 LΛ+(ϵ1, ϵ0) + ∑ ϵ0=ϵ1 LΛ+(ϵ0, ϵ1)), ( ∑ ϵ0 ̸=ϵ1 LΛ−(ϵ0, ϵ1) + ∑ ϵ0 ̸=ϵ1 LΛ−(ϵ1, ϵ0)) + ∑ ϵ0=ϵ1 LΛ−(ϵ1, ϵ0))) dΓ(ϵ0) =(d+ Γ (ϵ0), d − Γ (ϵ0)) =((( ∑ ϵ0 ̸=ϵ1 LΛ+(ϵ0, ϵ1) + ∑ ϵ0 ̸=ϵ1 LΛ+(ϵ1, ϵ0) + ∑ ϵ0=ϵ1 LΛ+(ϵ0, ϵ1)), ( ∑ ϵ0 ̸=ϵ1 LΛ−(ϵ0, ϵ1) + ∑ ϵ0 ̸=ϵ1 LΛ−(ϵ1, ϵ0)) + ∑ ϵ0=ϵ1 LΛ−(ϵ1, ϵ0))) such that: dΓ(ϵ0) = dΓ(ϵ0) + dΓ(ϵ0) = (d+Γ (ϵ0) + d+ Γ (ϵ0), d − Γ (ϵ0) + d− Γ (ϵ0)) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 21 of 36 Example 6. Let Γ = (Γ,Γ) be a BFRD defined in Figure 1 then the degree of ϵ is given by: dΓ(ϵ) =(d+Γ (ϵ), d − Γ (ϵ)) = ( (LΛ+(ϵ, ϵ) + LΛ+(ϵ, η) + LΛ+(ϵ, ξ)), (LΛ−(ϵ, ϵ) + LΛ−(ϵ, η) + LΛ−(ϵ, ξ)) ) =(0.4,−0.5) dΓ(ϵ) =(d+ Γ (ϵ), d− Γ (ϵ)) = ( (LΛ+(ϵ, ϵ) + LΛ+(ϵ, η) + LΛ+(ϵ, ξ)), (LΛ−(ϵ, ϵ) + LΛ−(ϵ, η) + LΛ−(ϵ, ξ)) ) =(0.5,−0.5) dΓ(ϵ) =(d+Γ (ϵ) + d+ Γ (ϵ), d−Γ (ϵ) + d− Γ (ϵ)) =(0.9,−1.0) Definition 22. Let Γ = (Γ,Γ) be a BFRD, the total degree of a vertex ϵ0 ∈ V in Γ is defined as: tdΓ(ϵ0) =(td+Γ (ϵ0), td − Γ (ϵ0)) td+Γ (ϵ0) =d+Γ (ϵ0) + ΥW+(ϵ0) td−Γ (ϵ0) =d−Γ (ϵ0) + ΥW−(ϵ0) tdΓ(ϵ0) =(td+ Γ (ϵ0), td − Γ (ϵ0)) td+ Γ (ϵ0) =d+ Γ (ϵ0) + ΥW+(ϵ0) td− Γ (ϵ0) =d− Γ (ϵ0) + ΥW−(ϵ0) such that: tdΓ(ϵ0) = tdΓ(ϵ0) + tdΓ(ϵ0) = (td+Γ (ϵ0) + td+ Γ (ϵ0), td − Γ (ϵ0) + td− Γ (ϵ0)) Example 7. Let Γ = (Γ,Γ) be a BFRD on universal set X = {ξ1, ξ2, ξ3, ξ4} defined below in Figure 8: The total degree of vertex ξ3 is given by: td+Γ (ξ3) =0.1 + 0.1 + 0.1 + 0.2 = 0.5 td−Γ (ξ3) =− 0.1 + (−0.1) + (−0.1) + (−0.2) = −0.5 td+ Γ (ξ3) =0.2 + 0.2 + 0.1 + 0.3 = 0.8 td− Γ (ξ3) =− 0.2 + (−0.1) + (−0.2) + (−0.4) = −0.9 A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 22 of 36 ξ1 (0.1,−0.2) ξ2 (0.2,−0.1) ξ3 (0.2,−0.2) ξ4 (0.1,−0.1) (0.1, -0.1) (0.1, -0.1) (0.1, -0.1) (0.1, -0.1) Γ = (ΥW,LΛ) ξ1 (0.3,−0.1) ξ2 (0.2,−0.3) ξ3 (0.3,−0.4) ξ4 (0.4,−0.3) (0.1, -0.1) (0.1, -0.2) (0.2, -0.2) (0.2, -0.1) Γ = (ΥW,LΛ) Figure 8: Γ = (Γ,Γ) tdΓ(ξ3) =(0.5 + 0.8,−0.5 + (−0.9)) = (1.3,−1.4) Definition 23. A BFRD Γ = (Γ,Γ) is said to be Regular if dΓ(ϵ0) = (α, β) ∀ϵ0 ∈ V, where α, β are real numbers. Definition 24. A BFRD Γ = (Γ,Γ) is said to be Totally Regular if tdΓ(ϵ0) = (α, β) ∀ϵ0 ∈ V, where α, β are real numbers. Example 8. Let Γ = (Γ,Γ) be a BFRD on universal set X = {ξ1, ξ2, ξ3, ξ4} de- fined in Figure 9: Notice that the graph Γ is both regular and totally Regular Bipolar Fuzzy Rough Digraph. ξ1 (0.1,−0.2) ξ2 (0.1,−0.2) ξ3 (0.1,−0.2) ξ4(0.1,−0.2) (0.1, -0.1) (0.2, -0.1) (0.1, -0.1) (0.2, -0.1) Γ1 = (ΥW1, LΛ1) ξ1 (0.4,−0.6) ξ2 (0.4,−0.6) ξ3 (0.4,−0.6)ξ4 (0.4,−0.6) (0.3, -0.3) (0.3, -0.1) (0.3, -0.3) (0.3, -0.1) Γ1 = (ΥW1, LΛ1) Figure 9: Γ = (Γ,Γ) A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 23 of 36 Theorem 3. Let Γ = (Γ,Γ) be a Bipolar Fuzzy Rough Digraphs and ΥW = (ΥW+,ΥW−) be a constant function in Γ and Γ then Γ is Regular iff it is Totally Regular bipolar fuzzy rough digraph. Proof. Let ΥW = (ΥW+,ΥW−) be a constant function such that ΥW+(ϵ0) = ρ1 and ΥW−(ϵ0) = ρ2, ∀ϵ0 ∈ V. Consider Γ be a Regular Bipolar Fuzzy Rough Digraph then: d+Γ (ϵ0) = α, d−Γ (ϵ0) = β ∀ϵ0 ∈ V td+Γ (ϵ0) =d+Γ (ϵ0) + ΥW+(ϵ0), td−Γ (ϵ0) = d−Γ (ϵ0) + ΥW−(ϵ0) ∀ϵ0 ∈ V =⇒ td+Γ (ϵ0) =α+ ρ1, td−Γ (ϵ0) = β + ρ2 ∀ϵ0 ∈ V Hence Γ is a Totally Regular Bipolar Fuzzy Rough Digraph. Conversely, suppose that Γ is a Totally Regular Bipolar Fuzzy Rough Digraph such that: td+Γ (ϵ0) =µ1, td−Γ (ϵ0) = µ2 ∀ϵ0 ∈ V =⇒ d+Γ (ϵ0) + ΥW+(ϵ0) = µ1, d−Γ (ϵ0) + ΥW−(ϵ0) = µ2 ∀ϵ0 ∈ V =⇒ d+Γ (ϵ0) + ρ1 = µ1, d−Γ (ϵ0) + ρ2 = µ2 ∀ϵ0 ∈ V =⇒ d+Γ (ϵ0) = µ1 − ρ1, d−Γ (ϵ0) = µ2 − ρ2 ∀ϵ0 ∈ V Hence Γ is a Regular Bipolar Fuzzy Rough Digraph. Theorem 4. Let Γ = (Γ,Γ) be a Regular as well as Totally Regular Bipolar Fuzzy Rough Digraph then ΥW = (ΥW+,ΥW−) is a constant function in Γ and Γ. Proof. Let Γ = (Γ,Γ) be a Regular and Totally Regular Bipolar Fuzzy Rough Digraph such that: d+Γ (ϵ0) =α, d−Γ (ϵ0) = β ∀ϵ0 ∈ V td+Γ (ϵ0) =µ1, td−Γ (ϵ0) = µ2 ∀ϵ0 ∈ V =⇒ d+Γ (ϵ0) + ΥW+(ϵ0) = µ1, d−Γ (ϵ0) + ΥW−(ϵ0) = µ2 ∀ϵ0 ∈ V =⇒ α+ΥW+(ϵ0) = µ1, β +ΥW−(ϵ0) = µ2 ∀ϵ0 ∈ V =⇒ ΥW+(ϵ0) = µ1 − α, ΥW−(ϵ0) = µ2 − β ∀ϵ0 ∈ V Hence ΥW = (ΥW+,ΥW−) is a constant function. 4. Applications 4.1. Selecting a minimum set of rural areas to set up medicine market Rural areas often face greater challenges in accessing medical facilities due to a com- bination of geographical, infrastructural, economic, and demographic factors. As a re- sult, residents in rural areas may experience difficulties in obtaining essential medications A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 24 of 36 needed to manage acute and chronic health conditions. Lack of access to medicines can lead to delayed treatment, exacerbation of illnesses, and preventable health complications. Moreover, rural populations are often disproportionately affected by certain health is- sues, such as chronic diseases, infectious diseases, and maternal and child health concerns. Providing access to medicines in rural areas helps to improve health outcomes, reduce morbidity and mortality rates, and enhance the overall quality of life for rural residents. Additionally, ensuring medication supply in rural communities contributes to economic development, as healthy individuals are better able to participate in the workforce and contribute to local economies. Consider a set of rural areas X = {R1, R2, R3, R4, R5, R6}. A medicine company wants to set up a supply market in a minimum number of rural areas such that medicines can be supplied to each rural area of the set to meet the needs of each rural area. A vertex’s positive membership value in the graph indicates how much the ru- ral area has the supply of medicines in the set of rural areas and the negative membership value represents how much the rural area needs the supply of medicines in the set of rural areas. Every directed edge depicts the transport of medications between rural areas. The directed edge’s positive membership value from one rural area to another represents the positive supply relationship from one rural area to the other. The directed edge’s negative membership value from one rural area to another represents a certain level of uncertainty or ambiguity in the supply relationship due to factors like transportation challenges, stock availability, and demand fluctuations. Let Υ = (Υ+,Υ−) be a BFTR on X defined in the Tables 6 and 7. Υ+ R1 R2 R3 R4 R5 R6 R1 1 0.2 0.3 0.5 0.1 0.4 R2 0.2 1 0.1 0.2 0.3 0.5 R3 0.3 0.1 1 0.3 0.4 0.2 R4 0.5 0.2 0.3 1 0.1 0.2 R5 0.1 0.3 0.4 0.1 1 0.5 R6 0.4 0.5 0.2 0.2 0.5 1 Table 6: Υ+ of relation Υ Υ− R1 R2 R3 R4 R5 R6 R1 −1 −0.1 −0.2 −0.4 −0.3 −0.25 R2 −0.1 −1 −0.3 −0.2 −0.5 −0.4 R3 −0.2 −0.3 −1 −0.3 −0.1 −0.6 R4 −0.4 −0.2 −0.3 −1 −0.35 −0.45 R5 −0.3 −0.5 −0.1 −0.35 −1 −0.2 R6 −0.25 −0.4 −0.6 −0.45 −0.2 −1 Table 7: Υ− of relation Υ LetW = {(R1, 0.2,−0.1), (R2, 0.3,−0.3), (R3, 0.1,−0.3), (R4, 0.4,−0.2), (R5, 0.3,−0.5), (R6, 0.4,−0.4)} be a BFS on X. The lower approximation of W with respect to Υ is: ΥW = {(R1, 0.2,−0.1), (R2, 0.3,−0.3), (R3, 0.1,−0.3), (R4, 0.4,−0.2), (R5, 0.3,−0.5), (R6, 0.4,−0.4)} The upper approximation of W with respect to Υ is: ΥW = {(R1, 0.4,−0.3), (R2, 0.4,−0.5), (R3, 0.3,−0.4), (R4, 0.4,−0.4), (R5, 0.4,−0.5), (R6, 0.4,−0.4)} Let L = (L+, L−) be a BFTR on A ⊆ X ×X and defined by the Tables 8 and 9: A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 25 of 36 L+ R1R6 R2R1 R2R5 R3R4 R5R6 R2R6 R5R1 R1R6 1 0.2 0.2 0.1 0.1 0.2 0.05 R2R1 0.2 1 0.1 0.075 0.3 0.4 0.3 R2R5 0.2 0.1 1 0.1 0.3 0.5 0.1 R3R4 0.1 0.075 0.1 1 0.2 0.1 0.05 R5R6 0.1 0.3 0.3 0.2 1 0.1 0.4 R2R6 0.2 0.4 0.5 0.1 0.1 1 0.3 R5R1 0.05 0.3 0.1 0.05 0.4 0.3 1 Table 8: L+ of relation L L− R1R6 R2R1 R2R5 R3R4 R5R6 R2R6 R5R1 R1R6 −1 −0.1 −0.05 −0.2 −0.3 −0.1 −0.25 R2R1 −0.1 −1 −0.3 −0.2 −0.25 −0.1 −0.5 R2R5 −0.05 −0.3 −1 −0.3 −0.2 −0.1 −0.3 R3R4 −0.2 −0.2 −0.3 −1 −0.1 −0.3 −0.05 R5R6 −0.3 −0.25 −0.2 −0.1 −1 −0.5 −0.25 R2R6 −0.1 −0.1 −0.1 −0.3 −0.5 −1 −0.1 R5R1 −0.25 −0.5 −0.3 −0.05 −0.25 −0.1 −1 Table 9: L+ of relation L Let Λ = (Λ+,Λ−) be a BFS on A defined by: Λ = {(R1R6, 0.2.− 0.1), (R2R1, 0.1,−0.1), (R2R5, 0.3,−0.3), (R3R4, 0.1,−0.2), (R5R6, 0.1,−0.2), (R2R6, 0.3,−0.2), (R5R1, 0.2,−0.1)} The set of lower approximations of Λ is: LΛ = {(R1R6, 0.2,−0.1), (R2R1, 0.1,−0.1), (R2R5, 0.3,−0.3), (R3R4, 0.1,−0.2), (R5R6, 0.1,−0.2), (R2R6, 0.3,−0.2), (R5R1, 0.2,−0.1)} The set of upper approximation of Λ is: LΛ = {(R1R6, 0.2,−0.2), (R2R1, 0.3,−0.3), (R2R5, 0.3,−0.3), (R3R4, 0.1,−0.3), (R5R6, 0.3,−0.2), (R2R6, 0.3,−0.2), (R5R1, 0.3,−0.3)} Using this information, the lower and upper approximations of the BFRDs are shown in Figures 10 and 11 respectively. Notice that R5R6 is not a strong arc in Γ but it is a strong arc in Γ. Therefore, all the arcs except R5R6 are strong in Γ = (Γ,Γ). The strong arcs show that a maximum amount of medicines can be supplied between two rural areas most cost-effectively. The upper and lower minimum dominating set is D = {R2, R3}, therefore the MDS of A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 26 of 36 R1 (0.2,−0.1) R2 (0.3,−0.3) R3 (0.1,−0.3) R4 (0.4,−0.2) R5 (0.3,−0.5) R6 (0.4,−0.4) (0.2, -0.1) (0.1, -0.1) (0.3, -0.3) (0.1, -0.2) (0.1, -0.2) (0.3, -0.2) (0.2, -0.1) Figure 10: Γ = (ΥW,LΛ) Γ = (Γ,Γ) is D = {R2, R3}. The lower domination number is: ΩD(Γ) = 1 + (0.3) + (−0.3) 2 + 1 + (0.1) + (−0.3) 2 =0.9 The upper domination number is: ΩD(Γ) = 1 + (0.4) + (−0.5) 2 + 1 + (0.1) + (−0.3) 2 =0.85 The domination number is: ΩD(Γ) = ΩD(Γ) + ΩD(Γ) = 0.9 + 0.85 = 1.75 The MDS is the optimum set of the minimum number of rural areas that can supply the medicines to the other rural areas in the most cost-effective manner. The domination number of BFRD shows that the net cost to supply medicines to the rural areas of the A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 27 of 36 R1 (0.4,−0.3) R2 (0.4,−0.5) R3 (0.3,−0.4) R4 (0.4,−0.4) R5 (0.4,−0.5) R6 (0.4,−0.4) (0.2, -0.2) (0.3, -0.3) (0.3, -0.3) (0.1, -0.3) (0.3, -0.2) (0.3, -0.2) (0.3, -0.3) Figure 11: Γ = (ΥW,LΛ) MDS. Our analysis of the rural medicine supply network yielded a clear and actionable strategic plan. The algorithm identified the MDS as D = {R2, R3} with an overall domination number of 1.75. It means that he selection of {R2, R3} as the optimal set of supply hubs is not arbitrary; it is a direct result of their strategic position within the Bipolar Fuzzy Rough Digraph. These two locations were identified as the strongest positive dominators, meaning they possess the most effective combination of high supply capacity which means they have a strong positive capacity to serve other areas and their connections to other areas are characterized by high positive weights (representing efficiency) and low negative weights (representing minimal uncertainty or risk). In practical terms, R2 might be located at a major highway intersection, ensuring reliable transport, while R3 might have superior storage infrastructure. Our model prioritizes these locations over others that might be more geographically central but have less reliable supply lines. By identifying a minimum dominating set of just two locations, the company can avoid the significant expense of establishing depots in three, four, or more areas. This centralization drastically reduces costs related to infrastructure, staffing, inventory management, and bulk transportation, thereby maximizing the return on investment. In summary, the results of our BFRD model go beyond simply identifying important A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 28 of 36 nodes; they provide a multi-faceted, evidence-based strategy for optimizing a supply chain, managing risk, and minimizing operational costs. The algorithm of the proposed method is given in Table 10. 4.2. A comparison of Bipolar Fuzzy Rough Digraph and Fuzzy Rough Digraph To clarify the incremental contribution of our work, this section provides a direct comparison between the analytical process of existing Fuzzy Rough Domination (FRD) models and our proposed Bipolar Fuzzy Rough Domination (BFRD) methodology. 4.2.1. The Process and Limitations of Existing Fuzzy Rough Digraph Domi- nation Existing models for domination in Fuzzy Rough Digraphs follow a single-channel process. They model a network using one fuzzy value per relationship to represent its strength. The analysis then computes a single lower and upper approximation of the graph to identify a minimum dominating set. The critical limitation of this process is the ambiguity of its output. While it can identify a set of influential nodes, it provides no information about the nature of that influence. A node identified as ”dominant” could be a positive leader or a negative bottleneck, but the model cannot distinguish between the two, leaving decision-makers with an incomplete picture. 4.2.2. The Incremental Contribution of the Proposed BFRD Algorithm Our proposed methodology, detailed in the algorithm Table 10, introduces a dual-channel analytical process that provides a far richer and more actionable output. The incremental contribution is embedded in the specific steps of our algorithm: (i) Unlike FRD models, our algorithm begins by capturing both positive and negative influences explicitly, using Bipolar Fuzzy sets for vertices W and edges Λ (Steps 2-6). This establishes a two-dimensional foundation for the entire analysis from the outset. (ii) The core novelty lies in the computation of separate lower (ΥW,LΛ) and upper (ΥW,LΛ) approximations for both the positive and negative relationships (Steps 7-8). This constructs a multi-layered Bipolar Fuzzy Rough Digraph Γ = (Γ,Γ) that preserves the distinction between positive and negative ties, rather than collapsing them into a single measure of strength. (iii) Crucially, the domination analysis is performed on these distinct layers. Our algo- rithm finds minimal dominating sets independently in the lower approximation Γ (representing certain influence) and the upper approximation Γ (representing po- tential influence) (Steps 11-13). This multi-step process allows us to identify nodes A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 29 of 36 based on the specific nature and certainty of their influence—a level of granularity impossible with existing methods. (iv) The final minimum dominating set (Dm) and domination number (ΩD) (Steps 14- 16) are derived from this richer, multi-layered analysis. The final interpretation (Step 17) is therefore not just an identification of ”who is influential,” but a precise, evidence-based recommendation on which nodes form the optimal positive-influence core of the network. To understand the practical difference, consider a scenario of evaluating a project team’s performance. Team dynamics are complex, influenced by positive factors like collaboration and expertise, as well as negative factors like conflict and poor communication. A standard Fuzzy Rough Digraph (FRD) can capture the strength of the relationships. For example, it might assign the connection between Team Member A and Team Member B a high value of 0.8, indicating a strong bond. However, the FRD cannot distinguish the nature of this bond. Is it a strong, positive collaboration, or a tense, negative rivalry that forces them to interact? The model’s inability to handle this bipolarity leaves the analysis incomplete. This is where the Bipolar Fuzzy Rough Digraph (BFRD) provides a far more insightful analysis. The BFRD can represent the same relationship using two distinct values: • A positive membership of 0.8 to reflect their strong collaboration. • A negative membership of − 0.3 to account for some minor communication issues. This dual representation offers a much clearer and more realistic understanding of the team’s dynamics, capturing both the supportive and conflicting aspects of their relation- ship. A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 30 of 36 1. Consider the set X of vertices ϵ1, ϵ2, . . . , ϵn 2. Consider the Bipolar Fuzzy Vertex set W on X 3. Consider the set A of edges q1, q2, . . . , qu where qr = ϵsϵt for some 1 ≤ s, t ≤ n 4. Consider the Bipolar Fuzzy Edge set Λ on A 5. Insert the Bipolar Fuzzy tolerance relation L = (L+, L−) on A ⊆ X ×X 6. Insert the Bipolar fuzzy tolerance relation Υ on X 7. Compute lower approximation ΥW and upper approximation ΥW by using definition:{ ΥW+(α) = ∧ [1−Υ+(α, β) ∨W+(β)] , ΥW−(α) = ∨ [−1−Υ−(α, β) ∧W−(β)] ΥW+(α) = ∨ [Υ+(α, β) ∧W+(β)] , ΥW−(α) = ∧ [Υ−(α, β) ∨W−(β)] , ∀α, β ∈ X 8. Compute lower approximation LΛ and upper approximation LΛ by using definitions:{ LΛ+(αβ) = ∧ [(1− L+(αβ, γθ)) ∨ Λ+(γθ)], LΛ−(αβ) = ∨ [(−1− L−(αβ, γθ)) ∧ Λ−(γθ)] LΛ+(αβ) = ∨ [L+(αβ, γθ) ∧ Λ+(γθ)], LΛ−(αβ) = ∧ [(L−(αβ, γθ) ∨ Λ−(γθ)], ∀γθ ∈ A 9. Draw Bipolar Fuzzy Rough Digraph Γ = (Γ,Γ) 10. Find out strong arcs in lower approximation Γ and in upper approximation Γ by using the definition:{ CONN+ Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≤ LΛ+(ϵ0, ϵ1),CONN− Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≥ LΛ−(ϵ0, ϵ1) CONN+ Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≤ LΛ+(ϵ0, ϵ1),CONN− Γ−ϵ0ϵ1 (ϵ0, ϵ1) ≥ LΛ−(ϵ0, ϵ1) 11. Find the minimal dominating sets Di, i = 1, 2, . . . , k in Γ 12. Find the minimal dominating sets Di, i = 1, 2, . . . , k in Γ 13. Find the minimal dominating sets Di, i = 1, 2, . . . , k in Γ = (Γ,Γ) 14. Find the minimum dominating sets Dm in Γ = (Γ,Γ) 14. Find the lower domination number ΩD(Γ) of a minimum dominating set 15. Find the upper domination number ΩD(Γ) of a minimum dominating set 16. Find the domination number ΩD(Γ) = ΩD(Γ) + ΩD(Γ) of BFRD 17. The minimum dominating set is the optimum set of the minimum number of rural areas that can supply the medicines to the other rural areas in the most cost-effective manner Table 10: Algorithm for finding the optimum set of vertices 5. Conclusion This research addressed a critical limitation in existing Fuzzy Rough Digraph (FRD) models: their inability to handle the conflicting positive and negative preferences inherent in many real-world decision problems. To overcome this, we introduced the Bipolar Fuzzy Rough Digraph (BFRD), a novel framework designed to model and analyze complex net- works under conditions of bipolar uncertainty. The primary motivation for this research was the critical inability of existing Fuzzy Rough Digraphs (FRDs) to model conflicting information in decision-making scenarios. Our work addresses this gap through the principal novelty of introducing the Bipolar Fuzzy Rough Digraph (BFRD), a new mathematical structure designed to handle uncertainty with both positive and negative preferences. Our proposed method centers on a new domination theory developed specifically for BFRDs. We established the foundational concepts of the domination number, vertex degree, and Regular BFRDs, which together provide the mathematical tools to identify key influential nodes within networks characterized by dualistic properties. The primary significance of this work lies in its ability to provide a more robust and realistic model for decision analysis. By formally incorporating both supporting and opposing factors, the BFRD framework enables a more nuanced and accurate assessment of complex systems, which is a crucial advancement for decision-support systems dealing with real-world am- biguity. The practical utility of our theoretical framework was validated through a real-world appli- cation. We developed a novel algorithm based on our domination model to solve a critical logistics problem: identifying an optimal set of rural areas for medicine distribution. The A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 31 of 36 results were clear and impactful: our algorithm successfully identified the minimum dom- inating set of the BFRD, which directly corresponds to the smallest number of supply hubs needed to efficiently serve the entire network. This tangible outcome confirms that our BFRD-based method is not just a theoretical construct but an effective, practical tool that can be used to generate cost-effective solutions for complex optimization challenges. Looking ahead, this work opens several avenues for future research. The immediate next steps will involve extending this concept to more complex, multi-layered structures, such as Bipolar Fuzzy Soft Graphs and Intuitionistic Bipolar Fuzzy Rough Soft Digraphs. Fur- thermore, there is significant potential in developing dynamic BFRD models to analyze evolving networks and in exploring the integration of our domination-based algorithms with machine learning techniques for predictive analysis. Acknowledgements A.K, A.M and T.A would like to thank Prince Sultan University for the support through APC TAS research lab. References [1] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. [2] L. A. Zadeh. Similarity relations and fuzzy orderings. Information Sciences, 3(2):177– 200, 1971. [3] M. K. Chakraborty and M. Das. Studies in fuzzy relations over fuzzy subsets. Fuzzy Sets and Systems, 9(1–3):79–89, 1983. [4] M. K. Chakraborty and M. Das. On fuzzy equivalence i. Fuzzy Sets and Systems, 11(1):185–193, 1983. [5] M. K. Chakraborty and M. Das. On fuzzy equivalence ii. Fuzzy Sets and Systems, 11(1–3):299–307, 1983. [6] A. Kauffmann. Introduction à la théorie des sous-ensembles flous, 1, volume 1. Mas- son, 1973. [7] R. T. Yeh and S. Y. Bang. Fuzzy relations, fuzzy graphs, and their applications to clustering analysis. In Fuzzy Sets and Their Applications to Cognitive and Decision Processes, pages 125–149. Academic Press, 1975. [8] A. Rosenfeld. Fuzzy graphs. In Fuzzy Sets and Their Applications to Cognitive and Decision Processes, pages 77–95. Academic Press, 1975. [9] P. Bhattacharya. Some remarks on fuzzy graphs. Pattern Recognition Letters, 6(5):297–302, 1987. [10] S. Banerjee. An optimal algorithm to find the degrees of connectedness in an undi- rected edge-weighted graph. Pattern Recognition Letters, 12(7):421–424, 1991. [11] L. Kóczy. Fuzzy graphs in the evaluation and optimization of networks. Fuzzy Sets and Systems, 46(3):307–319, 1992. [12] J. K. Udupa and S. Samarasekera. Fuzzy connectedness and object definition: Theory, A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 32 of 36 algorithms, and applications in image segmentation. Graphical Models and Image Processing, 58(3):246–261, 1996. [13] K. R. Bhutani and A. Rosenfeld. Fuzzy end nodes in fuzzy graphs. Information Sciences, 152:323–326, 2003. [14] K. R. Bhutani and A. Rosenfeld. Geodesics in fuzzy graphs. Electronic Notes in Discrete Mathematics, 15:49–52, 2003. [15] S. Mathew and M. S. Sunitha. Node connectivity and arc connectivity of a fuzzy graph. Information Sciences, 180(4):519–531, 2010. [16] M. S. Sunitha and S. Mathew. Fuzzy graph theory: a survey. Annals of Pure and Applied Mathematics, 4(1):92–110, 2013. [17] M. Binu, S. Mathew, and J. N. Mordeson. Connectivity status of fuzzy graphs. Information Sciences, 573:382–395, 2021. [18] A. Somasundaram and S. Somasundaram. Domination in fuzzy graphs–i. Pattern Recognition Letters, 19(9):787–791, 1998. [19] A. N. Gani and P. Vadivel. On domination, independence and irredunance in fuzzy graph. International Review of Fuzzy Mathematics, 3(2):191–198, 2008. [20] O. T. Manjusha and M. S. Sunitha. Notes on domination in fuzzy graphs. Journal of Intelligent & Fuzzy Systems, 27(6):3205–3212, 2014. [21] A. A. Talebi, G. Muhiuddin, S. H. Sadati, and H. Rashmanlou. New concepts of domination in fuzzy graph structures with application. Journal of Intelligent & Fuzzy Systems, 42(4):3705–3718, 2022. [22] W. R. Zhang. Bipolar fuzzy sets and relations: a computational framework for cog- nitive modeling and multiagent decision analysis. In NAFIPS/IFIS/NASA’94. Pro- ceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intelligent Systems Conference, pages 305–309. IEEE, December 1994. [23] M. Akram. Bipolar fuzzy graphs. Information Sciences, 181(24):5548–5564, 2011. [24] M. Akram. Bipolar fuzzy graphs with applications. Knowledge-Based Systems, 39:1– 8, 2013. [25] M. Akram and R. Akmal. Certain operations on bipolar fuzzy graph structures. Applications and Applied Mathematics: An International Journal (AAM), 11(1):30, 2016. [26] M. Akram and N. Waseem. Novel applications of bipolar fuzzy graphs to decision making problems. Journal of Applied Mathematics and Computing, 56:73–91, 2018. [27] S. Poulik and G. Ghorai. Certain indices of graphs under bipolar fuzzy environment with applications. Soft Computing, 24:5119–5131, 2020. [28] M. Akram, M. Sarwar, and W. A. Dudek. Special types of bipolar fuzzy graphs. In Graphs for the Analysis of Bipolar Fuzzy Information, pages 127–159. 2021. [29] M. Akram, M. Sarwar, and W. A. Dudek. Bipolar fuzzy sets and bipolar fuzzy graphs. In Graphs for the Analysis of Bipolar Fuzzy Information, pages 1–80. 2021. [30] S. Gong and G. Hua. Fuzzy edge connectivity in bipolar fuzzy networks and the appli- cations in topology design. International Journal of Intelligent Systems, 37(8):5425– 5442, 2022. A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 33 of 36 [31] M. G. Karunambigai, M. Akram, K. Palanivel, and S. Sivasankar. Domination in bipolar fuzzy graphs. In 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pages 1–6. IEEE, 2013. [32] A. Muneera, T. N. Rao, J. V. Rao, and R. S. Rao. Domination in regular and irregular bipolar fuzzy graphs. Journal of Critical Reviews, 7(11):793–796, 2020. [33] M. Akram, M. Sarwar, and W. A. Dudek. Domination in bipolar fuzzy graphs. In Graphs for the Analysis of Bipolar Fuzzy Information, pages 253–280. 2021. [34] Z. Pawlak. Rough sets. International journal of computer & information sciences, 11:341–356, 1982. [35] Z. Pawlak. Rough set approach to knowledge-based decision support. European journal of operational research, 99(1):48–57, 1997. [36] Z. Pawlak. Rough set theory and its applications to data analysis. Cybernetics & Systems, 29(7):661–688, 1998. [37] M. Kryszkiewicz. Rough set approach to incomplete information systems. Information sciences, 112(1-4):39–49, 1998. [38] D. Dubois and H. Prade. Rough fuzzy sets and fuzzy rough sets. International Journal of General System, 17(2-3):191–209, 1990. [39] D. Dubois and H. Prade. Putting rough sets and fuzzy sets together. In Intelligent decision support: Handbook of applications and advances of the rough sets theory, pages 203–232. Springer Netherlands, Dordrecht, 1992. [40] Y. Y. Yao. A comparative study of fuzzy sets and rough sets. Information sciences, 109(1-4):227–242, 1998. [41] A. M. Radzikowska and E. E. Kerre. A comparative study of fuzzy rough sets. Fuzzy sets and systems, 126(2):137–155, 2002. [42] D. S. Yeung, D. Chen, E. C. Tsang, J. W. Lee, and X. Wang. On the generalization of fuzzy rough sets. IEEE Transactions on Fuzzy Systems, 13(3):343–361, 2005. [43] H. L. Yang, S. G. Li, S. Wang, and J. Wang. Bipolar fuzzy rough set model on two different universes and its application. Knowledge-Based Systems, 35:94–101, 2012. [44] H. L. Yang, S. G. Li, Z. L. Guo, and C. H. Ma. Transformation of bipolar fuzzy rough set models. Knowledge-Based Systems, 27:60–68, 2012. [45] N. Malik and M. Shabir. Rough fuzzy bipolar soft sets and application in decision- making problems. Soft Computing, 23:1603–1614, 2019. [46] Y. Han, P. Shi, and S. Chen. Bipolar-valued rough fuzzy set and its applications to the decision information system. IEEE Transactions on Fuzzy Systems, 23(6):2358–2370, 2015. [47] S. A. Shanthi and M. Saranya. On bipolar fuzzy rough connected spaces. In AIP Conference Proceedings, volume 2177, page 020002. AIP Publishing LLC, 2019. [48] A. Mubarak, M. Shabir, and W. Mahmood. Pessimistic multigranulation rough bipo- lar fuzzy set and their application in medical diagnosis. Computational and Applied Mathematics, 42(6):249, 2023. [49] T. He and K. Shi. Rough graph and its structure. Journal of Shandong University, 41(6):46–50, 2006. [50] T. He, Y. Chen, and K. Shi. Weighted rough graph and its application. In Sixth A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 34 of 36 International Conference on Intelligent Systems Design and Applications, volume 1, pages 486–491, 2006. [51] T. He. S-rough graph and its properties. In Proceedings of 2011 International Confer- ence on Computer Science and Network Technology, volume 1, pages 344–347, 2011. [52] T. He. Representation form of rough graph. Applied Mechanics and Materials, 157:874–877, 2012. [53] M. Liang, B. Liang, L. Wei, and X. Xu. Edge rough graph and its application. In 2011 Eighth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD), volume 1, pages 335–338, 2011. [54] T. He. Rough properties of rough graph. Applied Mechanics and Materials, 157- 158:517–520, 2012. [55] H. Tong and R. Yang. Different kinds of rough graph and their representation forms. DEStech Transactions on Engineering and Technology Research, 2017. no. mdm, DEStech Publications. [56] T. He and S. Jia. Properties of directed rough graph. DEStech Transactions on Computer Science and Engineering, 2017. [57] B. Mathew, S. J. John, and H. Garg. Vertex rough graphs. Complex & Intelligent Systems, 6:347–353, 2020. [58] M. Akram and M. Arshad. Fuzzy rough graph theory with applications. International Journal of Computational Intelligence Systems, 12(1):90–107, 2018. [59] J. Chen, J. Mi, and Y. Lin. A graph approach for fuzzy-rough feature selection. Fuzzy Sets and Systems, 391:96–116, 2020. [60] B. Said, M. Lathamaheswari, P. K. Singh, A. A. Ouallane, A. Bakhouyi, A. Bakali, and N. Deivanayagampillai. An intelligent traffic control system using neutrosophic sets, rough sets, graph theory, fuzzy sets and its extended approach: A literature review. Neutrosophic Sets and Systems, 50:10–26, 2022. [61] M. Akram and F. Zafar. Hybrid soft computing models applied to graph theory, volume 380. Springer International Publishing, Cham, Switzerland, 2020. [62] U. Ahmad and I. Nawaz. Directed rough fuzzy graph with application to trade networking. Computational and Applied Mathematics, 41(8):366, 2022. [63] U. Ahmad and I. Nawaz. Wiener index of a directed rough fuzzy graph and application to human trafficking. Journal of Intelligent & Fuzzy Systems, 2023. Preprint, 1–17. [64] I. Nawaz and U. Ahmad. Certain concepts in directed rough fuzzy graphs and appli- cation to mergers of companies. Fuzzy Information and Engineering, 15(3), 2023. [65] M. Akram and F. Zafar. A new approach to compute measures of connectivity in rough fuzzy network models. Journal of Intelligent & Fuzzy Systems, 36(1):449–465, 2019. [66] U. Ahmad, I. Nawaz, and S. Broumi. Connectivity index of directed rough fuzzy graphs and its application in traffic flow network. Granular Computing, pages 1–22, 2023. [67] U. Ahmad and T. Batool. Domination in rough fuzzy digraphs with application. Soft Computing, 27(5):2425–2442, 2023. [68] H. Khan, S. Ahmed, J. Alzabut, A. T. Azar, and J. F. Gómez-Aguilar. Nonlinear A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 35 of 36 variable order system of multi-point boundary conditions with adaptive finite-time fractional-order sliding mode control. International Journal of Dynamics and Control, pages 1–17, 2024. [69] H. Khan, M. Aslam, A. H. Rajpar, Y. M. Chu, S. Etemad, S. Rezapour, and H. Ah- mad. A new fractal-fractional hybrid model for studying climate change on coastal ecosystems from the mathematical point of view. Fractals, page 2440015, 2024. [70] S. Kundu, J. Alzabut, M. E. Samei, and H. Khan. Habitat complexity of a discrete predator-prey model with hassell-varley type functional response. Community and Ecology, 1(1), 2023. [71] J. Alzabut, R. Dhineshbabu, A. G. M. Selvam, J. F. Gómez-Aguilar, and H. Khan. Existence, uniqueness and synchronization of a fractional tumor growth model in discrete time with numerical results. Results in Physics, 54:107030, 2023. [72] M. Akram, A. J. C. R. Shumaiza, and J. C. R. Alcantud. Multi-criteria decision making methods with bipolar fuzzy sets. In Mathematics, pages 214–226. Springer, Singapore, 2023. [73] G. Ali, M. Akram, and J. C. R. Alcantud. Attributes reductions of bipolar fuzzy relation decision systems. Neural Computing and Applications, 32(14):10051–10071, 2020. [74] M. Akram, U. Amjad, and B. Davvaz. Decision-making analysis based on bipolar fuzzy n-soft information. Computational and Applied Mathematics, 40(6):182, 2021. [75] M. Akram and A. K. Shumaiza. Multi-criteria group decision-making for selection of green suppliers under bipolar fuzzy promethee process. Symmetry, 12(1):77, 2020. [76] M. Akram, Shumaiza, and M. Arshad. Bipolar fuzzy topsis and bipolar fuzzy electre-i methods to diagnosis. Computational and Applied Mathematics, 39(1):7, 2020. [77] G. Ali, M. Akram, A. N. Koam, and J. C. R. Alcantud. Parameter reductions of bipolar fuzzy soft sets with their decision-making algorithms. Symmetry, 11(8):949, 2019. [78] Aliya Fahmi, A. Hashmi, A. Khan, A. Mukheimer, Thabet Abdeljawad, and R. Thi- nakaran. Triangular intuitionistic fuzzy frank aggregation for efficient renewable energy project selection. European Journal of Pure and Applied Mathematics, 18(3):6227–6227, 2025. [79] Aliya Fahmi, A. Khan, A. Hashmi, A. Mukheimer, Thabet Abdeljawad, and R. Thi- nakaran. Domination in rough m-polar fuzzy digraphs based on trade networking. European Journal of Pure and Applied Mathematics, 18(3):6301–6301, 2025. [80] Aliya Fahmi, A. Khan, Thabet Abdeljawad, M. A. Hassan, A. Mukheimer, and R. Thinakaran. Analyzing global economic shifts due to the afghan-america war using complex cubic fuzzy todim method. European Journal of Pure and Applied Mathematics, 18(3):5866–5866, 2025. [81] H. Khan, W. F. Alfwzan, R. Latif, J. Alzabut, and R. Thinakaran. Ai-based deep learning of the water cycle system and its effects on climate change. Fractal and Fractional, 9(6):361, 2025. [82] H. Khan, J. Alzabut, D. K. Almutairi, H. Gulzar, and W. K. Alqurashi. Data analysis of fractal-fractional co-infection covid-tb model with the use of artificial intelligence. A. Arif et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6216 36 of 36 Fractals, 33(4):1–21, 20255. [83] H. Khan, M. Abdel-Aty, D. K. Almutairi, J. F. Gómez-Aguilar, and J. Alzabut. Artificial intelligence neural networking for data clustering of carbon dioxide model. Ain Shams Engineering Journal, 16(8):103460, 2025. [84] H. Khan, J. Alzabut, D. K. Almutairi, W. K. Alqurashi, S. Pinelas, O. Tuna, and M. A. Azim. A coupled nonlinear system of integro-differential equations using mod- ified abc operator. Fractals, 33(6), 2025. [85] M. Ullah, M. Sarwar, H. Khan, T. Abdeljawad, and A. Khan. Near-coincidence point results in metric interval space and hyperspace via simulation functions. Advances in Difference Equations, 2020(1):291, 2020. [86] Rajermani Thinakaran, Somasekar Jalari, Vikram Neerugatti, Ravindra Raman Cholla, and K. Nagendra Rao. Smart energy management system using iot for real- time monitoring and theft detection. In Proceedings of the 2024 9th International Conference on Information Technology and Digital Applications (ICITDA), pages 1– 5. IEEE, 2024.