Ratio Mathematica Volume 42, 2022 Some Edge Domination Parameters in Bipolar Hesitancy Fuzzy Graph Jahir Hussain Rasheed* Mujeeburahman Thacharakavil Chemmala † Dhamodharan Durairaj‡ Abstract In this article, we establish edge domination in Bipolar Hesitancy Fuzzy Graph(BHFG). Various domination parameters such as inverse edge domination and total edge domination in BHFG are determined. Some theorems related to edge domination and examples are also dis- cussed. Keywords: Bipolar fuzzy graph; Hesitant fuzzy graph; Edge domi- nation number; Total edge domination; Inverse edge domination; 2020 AMS subject classifications: 05C72, 05C69, 94D05.1 *(PG and Research Department of Mathematics, Jamal Mohamed College(Autonomous), Af- filiated to Bharathidasan University, Tiruchirapplli-620020, India); hssn jhr@yahoo.com. †(PG and Research Department of Mathematics, Jamal Mohamed College(Autonomous), Af- filiated to Bharathidasan University, Tiruchirapplli-620020, India); mohdmujeebtc@gmail.com. ‡(PG and Research Department of Mathematics, Jamal Mohamed College(Autonomous), Af- filiated to Bharathidasan University, Tiruchirapplli-620020, India); dharan raj28@yahoo.co.in. 1Received on March 26th, 2022. Accepted on June 26th, 2022. Published on June 30th, 2022. doi: 10.23755/rm.v41i0.732. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 157 R. Jahir Hussain, T.C. Mujeeburahman, D. Dhamodharan 1 Introduction The concept of fuzzy sets was first originated by L.A. Zadeh [Zadeh [1965]]. In 1973, Kaufmann established fuzzy graph using Zadeh’s fuzzy relation. The domination concept in fuzzy graph was first established by A. Somasundaram and S. Somasundaram [Somasundaram and Somasundaram [1998]]. The edge domi- nation in fuzzy graphs was initiated by S. Velammal and K.Thiagarajan [Vellamal and Thiagarajan [2012]]. The notion of Bipolar Fuzzy Graph(BFG) was estab- lished by M.Akram [Akram [2002]].The approach of domination in bipolar fuzzy graphs was proposed by M.G. Karunambigai, Palanivel and Akram [Karunambi- gai et al. [2013]].The book by Akram, Sarwar and Dudek entitled ”Graphs for the Analysis of Bipolar Fuzzy Information” [Akram et al. [2021]] is a great tool for understanding the concepts of domination in BFGs. S. Ramya and S. La- vanya developed edge domination in bipolar fuzzy graphs [Ramya and Lavanya [2017]].The notion of hesitant fuzzy sets was first introduced by V.Torra [Torra [2010]] in the year 2010. Hesitancy fuzzy graph, a new approach to fuzzy graph theory was first established by T. Pathinathan,et.al [Pathinathan et al. [2015]].The idea of domination in hesitancy fuzzy graph was investigated by R. Sakthivel et.al,[Sakthivel et al. [2019]]. In the year 2021, K. Anantha Kanaga Jothi and K. Balasangu [Anantha Kanaga Jothi and Balasangu [2021]]defined the idea of irregular and totally irregular bipolar hesitancy fuzzy graphs and some of its prop- erties. 2 Preliminaries Definition 2.1 (Akram [2002]). Let X be a non empty set. A bipolar fuzzy set B in X is an object having the form B = {(x, µPB(x), µ N B (x))|x ∈ X} where, µPB : X → [0, 1] and µ N B : X → [−1, 0] are mapppings. Definition 2.2 (Akram [2002]). A Bipolar Fuzzy Graph (BFG) is of the form G = (V, E) where 1. V = {v1, v2, ...vn} such that µP1 : V → [0, 1] and µN1 : V → [−1, 0] 2. E ⊂ V × V where µP2 : V × V → [0, 1] and µN2 : V × V → [−1, 0] such that µP2 (vi, vj) ≤ min(µ P 1 (vi), µ P 1 (vj)) and µN2 (vi, vj) ≥ max(µ N 1 (vi), µ N 1 (vj)) for all (vi, vj) ∈ E. 158 Some Edge Domination Parameters in Bipolar Hesitancy Fuzzy Graph Definition 2.3 (Akram [2002]). Let G = (V, E) be a BFG is said to be strong then µP2 = min(µ P 1 (vi), µ P 1 (vj)) and µ N 2 = max(µ N 1 (vi), µ N 1 (vj)) ∀vi, vj ∈ V. Definition 2.4 (Akram [2002]). Let G = (V, E) be a BFG is said to be complete then, µP2 (vi, vj) = min(µ P 1 (vi), µ P 1 (vj)) µN2 (vi, vj) = max(µ N 1 (vi), µ N 1 (vj)) for all vi, vj ∈ V. Definition 2.5 (Karunambigai et al. [2013]). An arc (a, b) is said to be strong edge in a BFG, if µP2 (a, b) ≥ (µ P 2 ) ∞(a, b) and µN2 (a, b) ≥ (µ N 2 ) ∞(a, b) whereas (µP2 ) ∞(a, b) = max{(µP2 )k(a, b)|k = 1, 2, ..., n} and (µN2 ) ∞(a, b) = min{(µN2 )k(a, b)|k = 1, 2, ..., n}. Definition 2.6 (Karunambigai et al. [2013]). Let G = (V, E) be a BFG, then cardinality of G is defined as |G| = ∑ vi∈V (1 + µP1 (vi) + µ N 1 (vi)) 2 + ∑ (vi,vj)∈E (1 + µP2 (vi, vj) + µ N 2 (vi, vj)) 2 . Definition 2.7 (Karunambigai et al. [2013]). The cardinality of V, i.e., amount of nodes is termed as the order of G = (V, E) and is signified by |V|(or O(G)) and determined by O(G) = |V| = ∑ vi∈V (1 + µP1 (vi) + µ N 1 (vi)) 2 The no. of elements in a set of S, i.e., amount of edges is termed as size of G = (V, E) and signified as |S|(or S(G)) and determined by S(G) = |S| = ∑ (vi,vj)∈E (1 + µP2 (vi, vj) + µ N 2 (vi, vj)) 2 for all (vi, vj) ∈ E. 159 R. Jahir Hussain, T.C. Mujeeburahman, D. Dhamodharan 3 Bipolar hesitancy fuzzy graph Definition 3.1 (Anantha Kanaga Jothi and Balasangu [2021]). Let X be a non- empty set. A Bipolar hesitancy fuzzy set B = {x, µP1 (x), µ N 1 (x), γ P 1 (x), γ N 1 (x), β P 1 (x), β N 1 (x)/x ∈ X} where µP1 , γ P 1 , β P 1 : X → [0, 1] and µN1 , γN1 , βN1 : X → [−1, 0] are mappings such that, 0 ≤ µP1 (x) + γ P 1 (x) + β P 1 (x) ≤ 1 and −1 ≤ µN1 (x) + γ N 1 (x) + β N 1 (x) ≤ 0 . Definition 3.2 (Anantha Kanaga Jothi and Balasangu [2021]). Let X be a non empty set.Then we call mappings µP2 , γ P 2 , β P 2 : X × X → [0, 1], µN2 , γN2 , βN2 : X × X → [−1, 0] are bipolar hesitancy fuzzy relation on X such that, µP2 (x, y) ≤ µP1 (x) ∧ µP1 (y); µN2 (x, y) ≥ µN1 (x) ∨ µN1 (y); γP2 (x, y) ≤ γP1 (x) ∧ γP1 (y); γ N 2 (x, y) ≥ γN1 (x) ∨ γN1 (y); βP2 (x, y) ≤ βP1 (x) ∧ βP1 (y); βN2 (x, y) ≥ βN1 (x) ∨ βN1 (y). Definition 3.3 (Anantha Kanaga Jothi and Balasangu [2021]). A bipolar hesi- tancy fuzzy relation A on X is called symmetric relation if µP2 (x, y) = µP2 (x, y), µN2 (x, y) = µ N 2 (x, y) ,γ P 2 (x, y) = γ P 2 (x, y), γ N 2 (x, y) = γ N 2 (x, y), β P 2 (x, y) = βP2 (x, y), β N 2 (x, y) = β N 2 (x, y) for all (x, y) ∈ X Definition 3.4 (Pathinathan et al. [2015]). A Hesitancy fuzzy graph is of the form G = (V, E) where, V = {v!, v2, . . . vn} such that µ1, γ1, β1 : V → [0, 1] denote the degree of mem- bership, non-membership and hesitancy of the vertex vi ∈ V respectively and µ1(vi)+γ1(vi)+β1(vi) = 1 for every vi ∈ V where β1(vi) = 1−[µ1(vi)+γ1(vi)] and E ⊆ V × V where µ2, γ2, β2 : V × V → [0, 1] denote the degree of membership, non-membership and hesitancy of the edge (vi, vj) ∈ E respectively such that, µ2(vi, vj) ≤ µ1(vi) ∧ µ1(vj); γ2(vi, vj) ≤ γ1(vi) ∨ γ1(vj); β2(vi, vj) ≤ β1(vi) ∧ β1(vj) and 0 ≤ µ2(vi, vj) + γ2(vi, vj) + β2(vi, vj) ≤ 1 for every (vi, vj) ∈ E. Definition 3.5 (Anantha Kanaga Jothi and Balasangu [2021]). A Bipolar Hesi- tancy Fuzzy Graph (BHFG) is of the form G = (V, E) where (i) V = {v1, v2, . . . , vn} such that µP1 , γP1 , βP1 : V → [0, 1] denote the degree of positive membership, positive non-membership and positive hesitancy of 160 Some Edge Domination Parameters in Bipolar Hesitancy Fuzzy Graph the vertex vi ∈ V respectively, µN1 , γN1 , βN1 : V → [−1, 0] denote the degree of negative membership,negative non-membership and negative hesitancy of the vertex vi ∈ V. For every vi ∈ V, µP1 (vi) + γ P 1 (vi) + β P 1 (vi) = 1 and µ N 1 (vi) + γ N 1 (vi) + β N 1 (vi) = −1 βP1 (vi) = 1 − [µP1 (vi) + γP1 (vI)] and βN1 (vi) = −1 − [µN1 (vi) + γN1 (vi)] (ii) E ⊆ V×V where, µP2 , γP2 , βP2 : V×V → [0, 1]; µN2 , γN2 , βN2 : V×V → [−1, 0] are mappings such that µP2 (vi, vj) ≤ µ P 1 (vi) ∧ µ P 1 (vj) µN2 (vi, vj) ≥ µ N 1 (vi) ∨ µ N 1 (vj) γP2 (vi, vj) ≤ γ P 1 (vi) ∨ γ P 1 (vj) γN2 (vi, vj) ≥ γ N 1 (vi) ∧ γ N 1 (vj) βP2 (vi, vj) ≤ β P 1 (vi) ∧ β P 1 (vj) βN2 (vi, vj) ≥ β N 1 (vi) ∨ β N 1 (vj) denote the degree of positive, negative membership, degree of positive, neg- ative non membership and degree of positive, negative hesitancy of the edge (vi, vj) ∈ E respectively and 0 ≤ µP2 (vi, vj) + γ P 2 (vi, vj) + β P 2 (vi, vj) ≤ 1 , −1 ≤ µN2 (vi, vj) + γ N 2 (vi, vj) + β N 2 (vi, vj) ≤ 0 for every (vi, vj) ∈ E. Figure 1: Bipolar Hesitancy Fuzzy Graph 161 R. Jahir Hussain, T.C. Mujeeburahman, D. Dhamodharan Example 3.1. From Fig 1, for vertex v1, µP1 (v1) + γ P 1 (v1) + β P 1 (v1) = 0.4 + 0.2 + 0.4 = 1 µN1 (v1) + γ N 1 (v1) + β N 1 (v1) = −0.6 − 0.2 − 0.2 = −1. For edge (v1, v2); µP2 (v1, v2) + γ P 2 (v1, v2) + β P 2 (v1, v2) = 0.8 ≤ 1 µN2 (v1, v2) + γ N 2 (v1, v2) + β N 2 (v1, v2) = −0.7 ≥ −1. Definition 3.6. A Bipolar Hesitancy Fuzzy Graph G = (V, E) is said to be complete when, µP2 (vi, vj) = µ P 1 (vi) ∧ µP1 (vj),µN2 (vi, vj) = µN1 (vi) ∨ µN1 (vj), γP2 (vi, vj) = γP1 (vi) ∨ γP1 (vj), γN2 (vi, vj) = γN1 (vi) ∧ γN1 (vj) , βP2 (vi, vj) = βP1 (vi) ∧ βP1 (vj) , βN2 (vi, vj) = β N 1 (vi) ∨ βN1 (vj) for every vi, vj ∈ V. Definition 3.7. A Bipolar Hesitancy Fuzzy Graph G = (V, E) is said to be strong when, µP2 (vi, vj) = µ P 1 (vi) ∧ µP1 (vj),µN2 (vi, vj) = µN1 (vi) ∨ µN1 (vj) γP2 (vi, vj) = γP1 (vi) ∨ γP1 (vj), γN2 (vi, vj) = γN1 (vi) ∧ γN1 (vj) , βP2 (vi, vj) = βP1 (vi) ∧ βP1 (vj) , βN2 (vi, vj) = β N 1 (vi) ∨ βN1 (vj) for every (vi, vj) ∈ E. Definition 3.8. Let G be a Bipolar hesitancy fuzzy graph. The neighbourhood of a vertex x in G is defined by N(x) = (NPµ (x), N N µ (x), N P γ (x), N N γ (x), N P β (x), N N β (x)) where NPµ (x) = {y ∈ V/µP2 (x, y) ≤ µP1 (x) ∧ µP1 (x)}; NNµ (x) = {y ∈ V/µN2 (x, y) ≥ µN1 (x) ∨ µN1 (x)};NPγ (x) = {y ∈ V/γP2 (x, y) ≤ γP1 (x) ∧ γP1 (x)}; NNγ (x) = {y ∈ V/γN2 (x, y) ≥ γN1 (x) ∨ γN1 (x)};NPβ (x) = {y ∈ V/β P 2 (x, y) ≤ βP1 (x) ∧ βP1 (x)}; NNβ (x) = {y ∈ V/β N 2 (x, y) ≥ βN1 (x) ∨ βN1 (x)}. Definition 3.9. Let G be a Bipolar Hesitancy Fuzzy Graph. The neighborhood degree of a vertex x in G is defined by deg(x) = [deg µP (x), deg µN(x), deg γP (x), deg γN(x), deg βP (x), deg βN(x)] y ∈ V, where deg µP (x) = ∑ y∈N(x) µP1 (y), deg µ N(x) = ∑ y∈N(x) µN1 (y), deg γ P (x) = ∑ y∈N(x) γP1 (y) deg γN(x) = ∑ y∈N(x) γN1 (y), deg β P (x) = ∑ y∈N(x) βP1 (y), deg β N(x) = ∑ y∈N(x) βN1 (y) . Definition 3.10. Let G = (V, E) be a BHFG. The edge cardinality of G is given by, |E| = r = ∑ (u,v)∈E 3 + µP2 (u, v) + µ N 2 (u, v) + γ P 2 (u, v) + γ N 2 (u, v) + β P 2 (u, v) + β N 2 (u, v) 3 . 162 Some Edge Domination Parameters in Bipolar Hesitancy Fuzzy Graph Definition 3.11. An Arc (u,v) is said to be strong edge in BHFG. then, µP2 (u, v) ≥ (µP2 )∞(u, v),µN2 (u, v) ≥ (µN2 )∞(u, v),γP2 (u, v) ≥ (γP2 )∞(u, v), γN2 (u, v) ≥ (γN2 )∞(u, v),βP2 (u, v) ≥ (βP2 )∞(u, v), βN2 (u, v) ≥ (βN2 )∞(u, v) whereas (µP2 ) ∞(u, v) = max{(µP2 )k(u, v)|k = 1, 2, . . . , n}; (µN2 ) ∞(u, v) = min{(µN2 )k(u, v)|k = 1, 2, . . . , n}; (γP2 ) ∞(u, v) = max{(γP2 )k(u, v)|k = 1, 2, . . . , n}; (γN2 ) ∞(u, v) = min{(γN2 )k(u, v)|k = 1, 2, . . . , n}; (βP2 ) ∞(u, v) = max{(βP2 )k(u, v)|k = 1, 2, . . . , n}; (βN2 ) ∞(u, v) = min{(βP2 )k(u, v)|k = 1, 2, . . . , n}. 4 Edge domination in bipolar hesitancy fuzzy graph Definition 4.1. Let G = (V, E) be a Bipolar Hesitancy Fuzzy Graph. A set S ⊆ E is said to be an edge dominating set of G if every edge not in S is incident to some edge in S. Definition 4.2. An edge dominating set S ⊆ E is said to be minimal if no proper subset of S is an edge dominating set. Definition 4.3. The minimum cardinality out of all minimal dominating sets of BHFG G is said to be lower domination number of G and denoted as dbh(G). Definition 4.4. The maximum cardinality out of all minimal dominating sets of BHFG G is said to be upper domination number of G and denoted as Dbh(G). Figure 2: Edge domination in BHFG 163 R. Jahir Hussain, T.C. Mujeeburahman, D. Dhamodharan Example 4.1. In the above figure 2, {e1, e2, e4, e5},{e2, e3, e5},{e1, e3, e4} are edge dominating sets of G.{e1, e4} {e3, e2},{e5} are minimal edge dominating sets of G. Among all the minimal dominating sets, {e5} has minimum cardinality and edge domination number γbh(G) = 1.06. Theorem 4.1. Let S be a minimal edge dominating set of a BHFG G = (V, E). if for any edge e ∈ S, one of the following condition hold a) N(e) ∩ S = ϕ b) ∃e′ ∈ E − S such that N(e) ∩ S = {e}. Proof. Given G = (V, E) is a BHFG and S is a minimal edge dominating set of G, Then for every edge e ∈ S, S −{e} is not an edge dominating set and hence there exists an edge e′ ∈ E − S which is not adjacent to any element of S − {e}. Thus if e′ = e we get (a) and if e′ ̸= e we get (b). 2 Definition 4.5. An edge e of a BHFG G is called an an isolated edge if no effective edges is incident with the vertices of e and hence it doesn’t dominate any other vertex in G. Theorem 4.2. If G = (V, E) is a BHFG without any isolated edges, then for every minimal edge dominating set S, prove that E − S is also an edge dominating set. Proof. Given G = (V, E) a BHFG without any isolated edges. Let S be minimal edge dominating set of G, then there exists an edge e′ ∈ N((e). From theorem 5.4 we get e′ ∈ E − S which implies every edge in E − S is adjacent to an edge in S. Hence E − S is also an edge dominating set. 2 Corolary 4.1. For any graph G without isolated edges γbh(G) ≤ r 3 . Definition 4.6. Let G = (V, E) be a BHFG. Let S be a minimum edge set of G. If E − S contains an edge dominating set S′ of G, then S′ is said to be inverse edge dominating set of G. The minimum cardinality out of all minmal inverse edge dominating sets is said to be inverse edge domination number and is denoted as γ−1bh (G). Proposition 4.1. For any graph G without isolated edges and vertices γbh(G) ≤ γ−1bh (G) . Proposition 4.2. If G is a graph without isolated edges and vertices and if number of vertices are greater than or equal to 3, then γbh(G) + γ −1 bh (G) ≤ r . 164 Some Edge Domination Parameters in Bipolar Hesitancy Fuzzy Graph Definition 4.7. Let G = (V, E) be BHFG without isolated edges. An edge dominat- ing set S is called as total edge dominating set if < S > has no isolated edge.The minimum cardinality of all minimal total edge dominating sets is said to be total edge domination number of G and is denoted as γtbh. A set F ⊆ E is said to be a total edge dominating set of G if for every edge in E is adjacent to at least one edge in F. Theorem 4.3. For any bipolar hesitancy fuzzy graph G, γbh(G) ≤ γtbh(G). 2 Theorem 4.4. For any bipolar fuzzy graph G with r edges then prove that γtbh = r iff every edge of G has a unique neighbor. Proof. Given a BHFG G with r edges.Let us consider every edge of G has a unique neighbor, then S is the only total edge dominating set of G which implies γtbh = r. Conversely, suppose γtbh = r and if there exists an edge with neighbors s and t then S − {s} gives a total edge dominating set of G. Thus γtbh < r which is a contradiction. 2 5 Conclusions We have established edge domination in Bipolar hesitancy fuzzy graph(BHFG). Along with various domination parameters such as inverse and total edge domi- nation were also discussed. We have also given various examples and theorems supporting the main result. Our result can be extended to other domination pa- rameters as well. 6 Acknowledgements The authors thanks the management, Ratio Mathematica for their constant support towards the successful completion of this work. We wish to thank the anonymous reviewers for a careful reading of manuscript and for very useful com- ments and suggestions. References M. Akram. Bipolar fuzzy graphs. Information Science, 181:5548–5564, 2002. M. Akram, M. Sarwar, and W. A. Dudek. Graphs for the Analysis of Bipolar Fuzzy Information. Springer, Singapore, 2021. 165 R. Jahir Hussain, T.C. Mujeeburahman, D. Dhamodharan K. Anantha Kanaga Jothi and K. Balasangu. Irregular and totally irregular bipolar hesitancy fuzzy graphs and some of its properties. Advances and Applications in Mathematical Sciences, 20:1685–1696, 2021. M. 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