Ratio Mathematica Volume 46, 2023 Domination in m− polar soft fuzzy graphs S. Ramkumar * R. Sridevi † Abstract In this article, we have introduced dominating set, minimal dominat- ing set, independent dominating set, maximal independent dominat- ing set in m polar soft fuzzy graphs. We proved theorems and also some properties of dominating set in m polar soft fuzzy graphs Keywords: Dominating set, Independent dominating set, Maxi- mal independent dominating set in m − polar soft fuzzy graphs. 2020 AMS subject classifications: 03E72, 05C72. 1 *PG and Research Department of Mathematics, Sri S. Ramasamy Naidu Memorial College, Sattur-626 203, Tamil Nadu, India. Affiliated to Madurai Kamaraj University, Madurai-625 021, Tamil Nadu, India; ramkumarmat2015@gmail.com. †PG and Research Department of Mathematics, Sri S. Ramasamy Naidu Memorial College, Sattur-626 203, Tamil Nadu, India. Affiliated to Madurai Kamaraj University, Madurai-625 021, Tamil Nadu, India;danushsairam@gmail.com. 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1070. ISSN: 1592-7415. eISSN: 2282-8214. ©S.Ramkumar et al. This paper is published under the CC-BY licence agreement. 138 S.Ramkumar and R.Sridevi 1 Introduction The work of Zadeh [13] in 1975, which interacted with ambiguity and im- precision between absolute true and absolute false, is credited with giving rise to fuzzy set theory. A fuzzy set’s possible values fall between [0,1]. Fuzzy set theory’s astonishing discovery opened up new possibilities for handling uncer- tainty in a variety of scientific and technological fields. Due to its use in engi- neering, communication networks, computer sciences, and artificial intelligence, graph theory is swiftly becoming a mainstream topic in mathematics.Graphs and fuzzy graphs are investigated in [10, 11].The idea of domination in fuzzy graphs was propounded by A. Somasundaram and S. Somasundaram [14] in 1998.Soft set and hybrid models are used to deal with uncertainty based on parametrization tool. Soft set, fuzzy soft set and m-polar fuzzy sets are studied in [1, 4, 6, 7]. The possibilityof domination in m− polar fuzzy graphs was introduced by M. Akram et.al [2] in 2017. Domination in graphs and domination in fuzzy graphs are analysedin [9, 5]. MohintaSumit and Samanta.T.K. [8] proposed the thought of fuzzy soft graph. S. Ramkumar and R. Sridevi [12] presented their perception in m−polar soft fuzzy graphs. These concepts motivate us to introduce domina- tion inm− polar soft fuzzy graphs. 2 Dominating set, Minimal dominating set, Maxi- mal independent set in m− Polar soft fuzzy graphs In this paper m-psf-graph denote m− polar soft fuzzy graph. Definition 2.1. An edge in an m-psf-graph G̃P,V = (G∗, ρ̃, µ̃,P)is defined as an effective edge if µ̃ex1(uv) = (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) = (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) = (ρ̃exm(u) ∧ ρ̃exm(v)) in Nhd(u) = {v ∈ V/v is a neighbor of u} is called the neighbourhood of u. Definition 2.2. A vertex u ∈ V in an m-psf-graph G̃P,V = (G∗, ρ̃, µ̃,P) is said to be an isolated vertex if 139 Domination in m− polar soft fuzzy graphs µ̃ex1(uv) < (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) < (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) < (ρ̃exm(u) ∧ ρ̃exm(v))∀v ∈ V \{u} in H̃P,V (e) ∀ e ∈ P. so that Nhd(u) = φ. Example 2.1. Consider a 3 −psf −graphG̃P,V . t t tt t t t t H̃P,V (e1) H̃P,V (e2) a1(0.8, 0.7, 0.9) (0.2, 0.3, 0.5) a2(0.3, 0.5, 0.6) (0.3, 0.4, 0.6) a3(0.6, 0.4, 0.9) (0.2, 0.4, 0.5) a4(0.2, 0.6, 0.5) (0.1, 0.5, 0.4) a1(0.7, 0.5, 0.8) (0.4, 0.2, 0.6) a2(0.5, 0.9, 0.7) (0.5, 0.5, 0.7) a3(0.7, 0.5, 0.8) (0.3, 0.5, 0.4) a4(0.3, 0.9, 0.4) (0.2, 0.1, 0.3) Figure.1. 3− psf-graph G̃P,V = {H̃P,V (e1),H̃P,V (e2)}. In this example, a2a3 and a3a4 are effective edges. Also Nhd(a1) = {φ},Nhd(a2) = {a3},Nhd(a3) = {a4a2},Nhd(a4) = {a3}. Here a1 is an isolated vertex. Definition 2.3. Let G̃P,V = (G∗, ρ̃, µ̃,P) be an m-psf-graph. For any two vertices u,v ∈ V, we call u dominates v in G̃P,V if µ̃ex1(uv) = (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) = (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) = (ρ̃exm(u) ∧ ρ̃exm(v)) in H̃P,V (e) ∀e ∈ P and ∀u,v ∈ V . Definition 2.4. A subset S̃ of V be an m− psf-graph G̃P,V . Then cardinality of S̃ is defined as, | S̃ |= ∑ e∈P ( ∑ u∈S̃ ρ̃e(u)) in H̃P,V (e)p. 140 S.Ramkumar and R.Sridevi Definition 2.5. A subset S̃ of V is called a dominating set of an m − psf − graphG̃P,V if for every vertex u ∈ V \ S̃ then ∃ v ∈ S̃ such that u dominates v in H̃P,V (e) ∀e ∈ P.The domination number γ(G̃P,V ) means the infimum cardinality of all dominating set in G̃P,V and γ(G̃P,V ) = minS̃∈V ∑ e∈P ( ∑ u∈S̃ ρ̃e(u)). Definition 2.6. A dominating set S̃ is called a minimal dominating set of m − psf − graphG̃P,V = (G∗, ρ̃, µ̃,P) if for any a ∈ S̃, S̃ \ {a} is not a dominating set in H̃P,V (e) ∀ e ∈ P. Definition 2.7. Lower and upper domination number of an m−psf-graph G̃P,V = (G∗, ρ̃, µ̃,P) is denoted by γ(G̃P,V ) and Γ(G̃P,V ), respectively, and defined by infimum cardinality and supremum cardinality of all minimal dominating set of that m−psf-graph, respectively. Example 2.2. Consider a 3−psf-graph G̃P,V . @ @ @ @� � � � q q q q �� � �@ @ @ @ q q qr � � � �� q q q q a1(0.7, 0.8, 0.9) (0.3, 0.2, 0.6) (0.3, 0.2, 0.6) (0.4, 0.5, 0.6) a2(0.4, 0.5, 0.6) a3(0.3, 0.2, 0.6) a4(0.8, 0.6, 0.4) (0.3, 0.2, 0.4) H̃P,V (e1) a1(0.8, 0.6, 0.3) (0.3, 0.6, 0.1) a3(0.3, 0.7, , 0.1)a2(0.2, 0.4, 0.8) (0.2, 0.4, 0.3) a4(0.4, 0.6, 0.3) (0.3, 0.6, 0.1)(0.2, 0.4, 0.1) H̃P,V (e2) a1(0.6, 0.8, 0.4) (0.6, 0.8, 0.4) a3(0.8, 0.9, 0.4) a4(0.6, 0.7, 0.3) (0.6, 0.7, 0.3) (0.3, 0.6, 0.4) a2(0.3, 0.6, 0.7) (0.3, 0.6, 0.4) H̃P,V (e3) Figure.2. Minimal dominating set of 3-psf-graph In Figure.2. Here, the minimal dominating sets in each parameterized graph is S̃1 = {a1,a4}, S̃2 = {a2,a4}, S̃3 = {a3}. Theorem 2.1. A dominating set S̃ is minimal if and only if one of the below mentioned criteria holds. 141 Domination in m− polar soft fuzzy graphs 1. Nhd(a) ∩S̃ = φ. 2. There is a vertex b ∈ V \ S̃ such that Nhd(b) ∩S̃ = {a}, for each a ∈ S̃. Proof. For a minimal dominating set S̃ of a 3−psf-graph G̃P,V , for every vertex a ∈ S̃, S̃ \{a} is not dominating set and so then ∃ b ∈ V \(S̃ \{a}) which is not dominated by any vertex in S̃ \ {a} of H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. If a = b then Nhd(a) ⊆ V \S̃ ⇒ Nhd(a)∩S̃ = φ in H̃P,V (ei) for i = 1, 2, . . . ,n . If a 6= b, then b is not dominated by S̃\{a} but is dominated by S̃, i.e., b is dominated only by a in S̃ in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. ∴ Nhd(b) ∩S̃ = {a} in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Hence Nhd(b) ∩S̃ = {a} in 3−psf-graph G̃P,V . Conversely, let S̃ holds one of the two given criterias. If S̃ is not minimal dominating set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Then ∃ a vertex a ∈ S̃ such that S̃ \ {a} is a dominating set in H̃P,V (ei) for i = 1, 2, . . . ,n. Thus a is dominated by atleast one vertex in S̃ \ {a} in H̃P,V (ei) for i = 1, 2, . . . ,n. Then Nhd(a) * S̃ \ {a} in H̃P,V (ei) for i = 1, 2, . . . ,n. Hence condition (1) does not hold. Also, If S̃ \ {a} is a dominating set in H̃P,V (ei) for i = 1, 2, . . . ,n. Then every vertex b in V \ (S̃ \ {a}) is dominated by at least one vertex in S̃ \ {a} in H̃P,V (ei) for i = 1, 2, . . . ,n. So Nhd(b) ∩ S̃ 6= {a} in H̃P,V (ei) for i = 1, 2, . . . ,n. hence condition (2) does not hold. This leads to a ⇒⇐ . ∴ S̃ must be a minimal dominating set in H̃P,V (ei) for i = 1, 2, . . . ,n. Hence S̃ must be a minimal dominating set in 3− psf-graph G̃P,V . Theorem 2.2. Let G̃P,V = ((P,ρ̃), (P,µ̃)) be a 3−psf-graph without isolated vertices. If S̃ is a minimal dominating set of G̃P,V then V \ S̃ is a dominating set of G̃P,V . Proof. Let S̃ be a minimal dominating set and a ∈ S̃. Since G̃P,V has no isolated vertices ∃ a vertex b ∈ Nhd(a) in H̃P,V (ei) ∀ei ∈ P for i = 1, 2, . . . ,n. Utilization of the same content similar to the proof given for Theorem 2.1, we get that b in V \S̃. Thus every element of S̃ is dominated by some element of V \S̃ in H̃P,V (ei) ∀ei ∈ P for i = 1, 2, . . . ,n. Consequently V \S̃ is a dominating set of G̃P,V . Theorem 2.3. Superset of a dominating set in G̃P,V is a dominating set. Proof. Proof is obvious. Remark 2.1. Subset of a dominating set in G̃P,V need not to be dominating set. Example 2.3. Consider a 3−psf-graph. 142 S.Ramkumar and R.Sridevi @ @ @ @ @ r r r s r r r r r rr r r ra1(0.7, 0.9, 0.9) a4(0.6, 0.7, 0.8) a5(0.6, 0.9, 0.8) a6(0.7, 0.8, 0.9) a7(0.6, 0.7, 0.9) a2(0.4, 0.3, 0.2) (0.6, 0.3, 0.4) a3(0.6, 0.3, 0.4) (0.6, 0.7, 0.8) (0.6, 0.8, 0.8)(0.6, 0.3, 0.4) (0.6, 0.3, 0.4) (0.4, 0.3, 0.2) (0.4, 0.3, 0.2) (0.6, 0.3, 0.4) H̃P,V (e1) a1(0.8, 0.9, 0.6) (0.6, 0.7, 0.6) (0.4, 0.3, 0.5) a2(0.4, 0.3, 0.5) a7(0.7, 0.6, 0.8) a6(0.9, 0.6, 0.8) a5(0.6, 0.7, 0.8) a4(0.3, 0.4, 0.6) a3(0.6, 0.7, 0.8) (0.3, 0.4, 0.6) (0.3, 0.4, 0.6) (0.6, 0.6, 0.8) (0.6, 0.6, 0.8) (0.4, 0.3, 0.5) (0.6, 0.6, 0.8) H̃P,V (e2) Figure.3. 3-psf-graphs. Here each parameterized graph is S̃ = {a2,a3,a5} and S̃∪{b} = {a1,a2,a3,a5} S̃ \{b} = {a2,a5}. Here {a1,a2,a3,a5} is a dominating set. But {a2,a5} is not a dominating set. Definition 2.8. A set S̃ ⊆ V of an m−psf-graph G̃P,V is called an independent set if µ̃ex1(uv) < (ρ̃ex1(u) ∧ ρ̃ex1(v)) µ̃ex2(uv) < (ρ̃ex2(u) ∧ ρ̃ex2(v)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ̃exm(uv) < (ρ̃exm(u) ∧ ρ̃exm(v)) in H̃P,V (e) ∀ e ∈ P and for all u,v ∈ S̃. Definition 2.9. An independent set S̃ of an m− psf-graph G̃P,V = (G∗, ρ̃, µ̃,P)is said to be maximal independent set if for every vertex u ∈ V \ S̃, the set S̃ ∪{u} is not independent in H̃P,V (e) ∀ e ∈ P. 143 Domination in m− polar soft fuzzy graphs Definition 2.10. Lower and upper independence number of an m−psf −graph G̃P,V = (G ∗, ρ̃, µ̃,P) is denoted by i(G̃P,V ) and I(G̃P,V ), respectively, and de- fined by infinimum cardinality and supremum cardinality among all the maximum independent set of that m−psf-graph, respectively. Example 2.4. Consider a 3−psf-graph G̃P,V . r r rr r r r r r ra1(0.6, 0.7, 0.8) (0.6, 0.7, 0.8) a2(0.7, 0.9, 0.6) (0.3, 0.4, 0.5) a3(0.3, 0.4, 0.5) (0.3, 0.4, 0.5) (0.6, 0.7, 0.7) a5(0.4, 0.6, 0.5) (0.4, 0.6, 0.5) a4(0.6, 0.8, 0.7) H̃P,V (e1) H̃P,V (e2) (0.6, 0.4, 0.2) a3(0.8, 0.9, 0.2) (03, 0.4, 0.2) (0.3, 0.4, 0.5) a4(0.3, 0.4, 0.5) a2(0.6, 0.4, 0.2) a1(0.4, 0.6, 0.8) a5(0.7, 0.9, 1.0) (0.3, 0.4, 0.5) (0.4, 0.4, 0.2) r r r r ra1(0.6, 0.7, 0.8) a5(0.7, 0.8, 0.9) (0.6, 0.7, 0.6) (0.7, 0.8, 0.6) a4(0.8, 0.9, 0.6) (0.4, 0.6, 0.6) a3(0.4, 0.6, 0.8) (0.3, 0.5, 0.4) a2(0.3, 0.5, 0.4) (0.3, 0.5, 0.4) H̃P,V (e3) Figure.4. Independent set of a 3-psf-graphs. In this example, the dominating set in each parameterized graph is {{a3,a4}, {a1,a3,a5},{a2,a4}}. In H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, 3. the maximal inde- pendent set is {{a2,a4},{a1,a3,a5}}. Here {a2,a4} is a maximal independent set of G̃P,V with infimum cardinality and i(G̃P,V ) = (3.3, 3.9, 3.0). Also here {a1,a3,a5} is maximal independent set of G̃P,V with supremum cardinality and I(G̃P,V ) = (4.9, 6.2, 6.7). 144 S.Ramkumar and R.Sridevi Proposition 2.1. For any 3 − psf − graph G̃P,V = (G∗, ρ̃, µ̃,P),γ(G̃P,V ) ≤ i(G̃P,V ). Example 2.5. Consider a 3− psf-graph. q q q q q q q q e e e ee qq H̃P,V (e1) H̃P,V (e2) a1(0.9, 0.7, 0.3) a5(0.7, 0.8, 0.9) (0.4, 0.3, 0.6) a4(0.4, 0.3, 0.6) (0.4, 0.3, 0.6) a3(0.8, 0.4, 0.7) (0.5, 0.4, 0.7) a2(0.5, 0.6, 0.9) (0.5, 0.6, 0.3) a1(0.6, 0.9, 0.7) a5(0.3, 0.2, 0.1) (0.3, 0.2, 0.1) a4(0.4, 0.3, 0.8) (0.4, 0.3, 0.6) a3(0.7, 0.9, 0.6) (0.4, 0.6, 0.6) a2(0.4, 0.6, 0.8) (0.4, 0.6, 0.7) Figure.5. Independent set of a 3-psf-graphs In Fig.5. that minimum dominating set of a 3−psf-graph G̃P,V is {a2,a4} and the maximal independent set of G̃P,V is {a1,a3,a5} in H̃P,V (ei) ∀ e ∈ P for i = 1, 2. Also here γ(G̃P,V ) = (1.7, 1.8, 3.1) and i(G̃P,V ) = (4.0, 3.9, 3.3) in H̃P,V (ei) ∀ e ∈ P for i = 1, 2. Clearly, γ(G̃P,V ) ≤ i(G̃P,V ). Theorem 2.4. A set S̃ ⊆ V is a maximal independent set of a 3 −psf − graph G̃P,V if and only if it is independent and dominating set. Proof. Assume that S̃ is a maximal independent set of G̃P,V . Then for each vertex a ∈ V \S̃, the set S̃∪{a} is not independent set in H̃P,V (ei) for i = 1, 2, . . . ,n. In this manner for every vertex a ∈ V \S̃,then ∃ vertex b ∈ S̃ such that b dominates a in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Hence S̃ is a dominating set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Therefore S̃ is both independent and dominating set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Hence S̃ is both independent and dominating set in G̃P,V Conversely if we suppose that S̃ is not maximal independent set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Then ∃ a vertex a ∈ V \ S̃, such that S̃ ∪{a} is in- dependent set. Thus there @ any vertex b in S̃ which dominates a in H̃P,V (ei) for i = 1, 2, . . . ,n. So S̃ is not a dominating set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Which ⇒⇐ to the choice of S̃. Accordingly S̃ is a maximal independent set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Hence S̃ is a maximal independent set of G̃P,V 145 Domination in m− polar soft fuzzy graphs Theorem 2.5. In a 3 − psf − graph G̃P,V = ((P,µ̃), (P,ρ̃)), every maximal independent set is a minimal dominating set Proof. For a maximal independent S̃ of G̃P,V . By Theorem 2.4, S̃ is dominating set in G̃P,V . If we consider S̃ being not minimal dominating set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Then ∃ atleast one vertex a ∈ S̃ so that S̃ \ {a} is a dominating set that means S̃ \ {a} dominates V \ (S̃ \ {a}). Thus, there exists atleast one vertex in S̃ which dominates a. This contradict our assumption. Therefore S̃ is a minimal dominating set in H̃P,V (ei) ∀ ei ∈ P for i = 1, 2, . . . ,n. Hence S̃ is a minimal dominating set in GP,V . 3 Conclusions Due to the large range of applications and domination characteristics that can be defined, domination theory research is interesting. This work introduces the idea of dominating sets, independent sets, domination number etc. for m-polar soft fuzzy graphs and shows some intriguing findings. In a similar situation, future studies can define and examine additional domination parameters. References M. N. W. Akram and B. Davvaz. Certain types of domination in m-polar fuzzy graphs. Journal of Multiple-valued Logic and Soft Computing, 29(6), 2017. S. Ramkumar and R. Sridevi. Proper m - polar soft fuzzy graphs. Adv.Math.,Sci.J., 10(4):1845–1856, 2021. A. Somasundaram and S. Somasundaram. Domination in fuzzy graph-i. Pattern Recognition letter, 19(9):77–95, 1998. M. Sumit and T. Samanta. An introduction to fuzzy soft graph. Mathematica Moravica, 19(3):35–48, 2015. 146