Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 18 https://internationalpubls.com Point Set Neutrosophic Domination in Single Valued Neutrosophic Graph 1R. Poornavalli , 2Dr. P. Solairani, 1Research Scholar, Department of Mathematics, R.V.S Arts and Science College (Affiliated to Bharathiyar University), Sulur, Coimbatore, Tamil Nadu, India. e-mail: poornavallir920@gmail.com. 2Assitant Professor, Department of Mathematics, R.V.S Arts and Science College (Affiliated to Bharathiyar University), Sulur, Coimbatore, Tamil Nadu, India. e-mail: poornavallir920@gmail.com. e-mail: solairani@rvs.com Article History: Received: 11-11-2024 Revised:24-12-2024 Accepted:09-01-2025 Abstract: In this paper, we demonstrate a concept of point set neutro-sophic domination, 2-point set neutrosophic domination, connected point set neutrosophic domination, point set tree neutrosophic domination with appropriate example. Some of their theoretical properties are investigates. Keywords: Neutrosophic graph, Dominance in neutrosophic graph, point set neutrosophic dominance, point set tree neutrosophic dominance, connected point set neutrosophic dominance. 1. Introduction In 1965, L.A. Zadeh [21] gave initial proposal for occurrence of uncertainty in real life situation of mathematical framework. Rosenfeld [13] developed the idea of fuzzy networks with membership value in [0, 1] after noticed Zadeh fuzzy function on fuzzy batches. Idea of expanding fuzzy network to intuition-istic fuzzy networks by K.T. Atanassov [1] and introduced additional level of indeterminacy in intuitionistic fuzzy relationships. Florentine Smarandache et al. [15, 19, 20] gave an idea for neutrosophic network & single valued neutrosophic network or graphs as an extension of K.T. Atanassov concept on the fuzzy network and the intuitionistic fuzzy network. The concept of Single valued neutrosophic graph and its additives was introduced by Said Broumi et al. [3]. Orge [13] and Berge [2] was introduced domination in graphs and In 1977, a study on domination number was begun by Cockayne & Hedetniemi [5]. A. Somasundaram and S. Somasundaram [16] introduced domination in fuzzy network. Domination in fuzzy graph using strong arcs is discussed by A. Nagoorgani V.T. Chandrasekaran [12]. An idea of point set domination in graphs are introduced by Sampathkumar and Pushpalatha [14]. S. Kaspar & B. Gayathri [11] introduced few results on point set tree domination of graphs. V. Swaminathan and R. Poovazhaki [17] introduced the idea of point set domination with reference to degree and also discussed connected point set domination of graph. Idea of connected point set domination of fuzzy graph was discussed by S. Vimala & J.S. Sathya [18]. In this paper Section 2 contains preliminary, section 3 defines point set neutrosophic dominance number, point set tree neutrosophic dominance number, connected point set neutrosophic dominance number in neutrosophic network and their bounds has been formulated and Section 4 concludes the paper. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 19 https://internationalpubls.com 2. Preliminaries Definition 2.1 (8). A Pair 𝐺 = (𝐴, 𝐡) is known as single valued neutrosophic graph with the underlying set 𝑉 . 1. The functions 𝑇𝐴 β†’ [0, 1], 𝐼𝐴 ∢ 𝑉 β†’ [0, 1] and 𝐹𝐴 ∢ 𝑉 β†’ [0, 1] denote the degree of truth- membership, degree of indeterminacy-membership and falsity-membership of the element 𝑣𝑖 ∈ 𝑉 respectively and 0 ≀ 𝑇𝐴(𝑣𝑖) + 𝐼𝐴(𝑣𝑖) + 𝐹𝐴(𝑣𝑖) ≀ 3 for all 𝑣𝑖 ∈ 𝑉 . 2 . The functions 𝑇𝐡 ∢ 𝐸 βŠ† 𝑉 Γ— 𝑉 β†’ [0, 1], 𝐼𝐡 ∢ 𝐸 βŠ† 𝑉 Γ— 𝑉 β†’ [0, 1] and 𝐹𝐡 ∢ 𝐸 βŠ† 𝑉 Γ— 𝑉 β†’ [0, 1] are defined by truth-membership, indeterminacy-membership and falsity-membership of the 𝑇𝐴(𝑣𝑖, 𝑣𝑗) ≀ 𝑇𝐴(𝑣𝑖) ∧ 𝑇𝐴(𝑣𝑗), 𝐼𝐴 (𝑣𝑖, 𝑣𝑗) β‰₯ 𝐼𝐴(𝑣𝑖) ∨ 𝐼𝐴(𝑣𝑗), 𝐹𝐴(𝑣𝑖, 𝑣𝑗) β‰₯ 𝐹𝐴(𝑣𝑖) ∨ 𝐹𝐴(𝑣𝑗), denotes the degree of edge (𝑣𝑖, 𝑣𝑗) ∈ 𝐸 (𝑖, 𝑗 = 1, 2, . . . , 𝑛). Definition 2.2 (4). Let 𝐺 = (𝐴, 𝐡) be a SVNG, G is said to be strong SVNG if 𝑇𝐡(𝑒, 𝑣) = 𝑇𝐴(𝑒) ∧ 𝑇𝐴(𝑣) , 𝐼𝐡(𝑒, 𝑣) = 𝐼𝐴(𝑒) ∨ 𝑇𝐴(𝑣), 𝐹𝐡(𝑒, 𝑣) = 𝐹𝐴(𝑒) ∨ 𝐹𝐴(𝑣) for every (𝑒, 𝑣) ∈ 𝐸. Definition 2.3 (6). Let 𝐺 = (𝐴, 𝐡) be a SVNG, G is said to be complete SVNG if 𝑇𝐡(𝑒, 𝑣) = 𝑇𝐴(𝑒) ∧ 𝑇𝐴(𝑣) , 𝐼𝐡(𝑒, 𝑣) = 𝐼𝐴(𝑒) ∨ 𝑇𝐴(𝑣), 𝐹𝐡(𝑒, 𝑣) = 𝐹𝐴(𝑒) ∨ 𝐹𝐴(𝑣) for every 𝑒, 𝑣 ∈ 𝐸. Definition 2.4 (7). Let 𝐺 = (𝐴, 𝐡) be a SVNG on V , then the neutrosophic vertex cardinality of G is defined by |V | = βˆ‘ 1 + 𝑇𝐡(𝑒,𝑣) + 𝐼𝐡(𝑒,𝑣) βˆ’πΉπ΅(𝑒,𝑣) 2V (𝑒,𝑣)βˆˆπ‘‰ Definition 2.5 (7). Let 𝐺 = (𝐴, 𝐡) be a SVNG on E , then the neutrosophic edge cardinality of G is defined by |E| = βˆ‘ 1 + 𝑇𝐡(𝑒,𝑣) + 𝐼𝐡(𝑒,𝑣) βˆ’πΉπ΅(𝑒,𝑣) 2(𝑒,𝑣)∈𝐸 Definition 2.6 (12). An arc (𝑒, 𝑣) of a SVNG G is called strong arc if 𝑇𝐡(𝑒, 𝑣) = 𝑇𝐴(𝑒) ∧ 𝑇𝐴(𝑣), 𝐼 𝐡(𝑒, 𝑣) = 𝐼𝐴(𝑒) ∨ 𝐼𝐴(𝑣), 𝐹𝐡(𝑒, 𝑣) = 𝐹𝐴(𝑒) ∨ 𝐹𝐴(𝑣). Definition 2.7 (7). Let 𝐺 = (𝐴, 𝐡) be a SVNG on. Let (𝑒, 𝑣) ∈ 𝑉 , we say that 𝑒 dominates 𝑣 in 𝐺, if there exist a strong arc between them. Definition 2.8 (7). Given 𝑆 βŠ‚ 𝑉 is dominating set in G if for every vertex 𝑣 ∈ 𝑉 βˆ’ 𝑆 there exist a vertex 𝑒 ∈ 𝑆 such that u dominates 𝑣, for all 𝑒 ∈ 𝐴, 𝑒, 𝑣 ∈ 𝑉 . Definition 2.9 (10). Let 𝐺 = (𝐴, 𝐡) be a fuzzy graph. Let 𝑒, 𝑣 ∈ 𝑉 and we say that 𝑒 dominates 𝑣 in 𝐺 if Β΅(𝑒, 𝑣) = 𝜎(𝑒) ∨ 𝜎(𝑒). A subset 𝑆 of 𝑉 is called dominance set in 𝐺 if for every 𝑣 ∈ 𝑉 βˆ’ 𝑆 there exist u ∈S such that 𝑒 dominates 𝑣. The minimum fuzzy cardinality of a dominating set in 𝐺 is called the dominance number of 𝐺 and is denoted by 𝛾(𝐺) Definition 2.10 (18). A dominating set 𝐷 βŠ† 𝑉 of a fuzzy graph G is said to be a point set dominating set of 𝐺 if for every 𝑆 βŠ† 𝑉 βˆ’ 𝐷 there exist a node 𝑑 ∈ 𝐷 such that < 𝑆 βˆͺ {𝑑} > is a connected fuzzy graph. The minimum cardinality taken over all minimal connected point set is called the point set domination number of the fuzzy graph 𝐺 and it is denoted by 𝛾𝑝(𝐺) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 20 https://internationalpubls.com Definition 2.11 (18). A point set dominating set 𝐷 βŠ† 𝑉 of any fuzzy graph G is a connected point set dominating set of 𝐺 if the subgraph < 𝐷 > induced by 𝐷 is a connected fuzzy graph. The minimum cardinality taken over all minimal connected point set dominating set is called the connected point set domination number 𝛾𝑐𝑝(𝐺). Definition 2.12 (16). Let 𝐺 = (𝑋, π‘Œ) be a single valued neutrosophic network.consider a subset 𝑆 of 𝑉 such that 𝑒 ∈ 𝑆 dominating v for every 𝑣 ∈ 𝑉 βˆ’ 𝑆, then that subset is known to be a neutrosophic dominance set in G is given by 𝛾𝑛𝑑(𝐺). 3. Point set domination in neutrosophic graph In this paper we use some basic notation, 𝐺 = (𝑋, π‘Œ ) is neutrosophic network or graph, 𝑋 be a Vertex set, π‘Œ be a edge set, 𝑇𝑋(𝑣), 𝐼𝑋(𝑣) 𝐹𝑋(𝑣) be truth, indeterminacy and falsity membership values of vertices in graph 𝐺. π‘‡π‘Œ (u, v), , πΌπ‘Œ (u, v), , πΉπ‘Œ (u, v) is truth, indeterminacy and falsity membership value of the edge (𝑒, 𝑣) of 𝐺. Definition 3.1. Let 𝐺 = (𝑋, π‘Œ) be a single valued neutrosophic network, if for every 𝐴 βŠ† 𝑋 βˆ’ 𝐡 there exist a vertices 𝑏 ∈ 𝐡 such that < 𝐴 βˆͺ {𝑏} > is connected neutrosophic set in G then the subset 𝐡 βŠ† 𝑋 is called point set neutrosophic dominance set (𝐷𝑝𝑠𝑛). Point set domination number of 𝐺 is the number with the minimum vertex cardinality in all point set domination set of 𝐺 and it is denoted by 𝛾𝑝𝑛𝑑(𝐺). Figure 1: 𝐷𝑝𝑠𝑛 = {𝑏, 𝑒}, {π‘Ž, 𝑑, 𝑐}, {𝑑, 𝑐}, {𝑏, 𝑑, 𝑒}, {π‘Ž, 𝑏, 𝑐, 𝑑} are few points set neutrosophic domination & 𝛾𝑝𝑛𝑑(𝐺) = 1.65 Definition 3.2. Let 𝐺 = (𝑋, π‘Œ ) be a single valued neutrosophic network, if the subset < 𝐡 > induced by B is a connected neutrosophic graph then the point set domination 𝐡 βŠ† 𝑋(𝐺) of any neutrosophic graph 𝐺 is connected point set domination set (𝐷𝑐𝑝𝑠𝑛). Connected point set domination number of 𝐺 is the number with the minimum vertex cardinality in all connected point set domination set of 𝐺 and it is denoted by 𝛾𝑐𝑝𝑛𝑑(𝐺). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 21 https://internationalpubls.com Figure 2: 𝐷𝑐𝑝𝑠𝑛 = {𝑏, 𝑐, 𝑑, 𝑒, 𝑓}, {𝑏, 𝑓, 𝑔, 𝑖, 𝑑, 𝑐}, {𝑏, 𝑖, 𝑑, 𝑒, 𝑓} are few connected point set neutrosophic domination & 𝛾𝑐𝑝𝑛𝑑(𝐺) = 3.65 Definition 3.3. Let 𝐺 = (𝑋, π‘Œ) be single valued neutrosophic network, a set 𝐡 βŠ† 𝑋(𝐺) is called 2-point set neutrosophic domination set (𝐷2𝑝𝑠𝑛) of 𝐺 if for every set 𝑇 βŠ† 𝑋 βˆ’ 𝐡 there exists a non-empty set 𝑆 βŠ† 𝐡 containing at-most two vertices such that the induced subgraph < 𝑆 βˆͺ 𝑇 > is connected. 2-point set neutrosophic domination number of 𝐺 is the number with theminimum vertex cardinality in all 2-point set domination set of G and it is denoted by 𝛾2𝑝𝑠𝑛𝑑(𝐺). Figure 3: 𝐷2𝑝𝑠𝑛= {a, f, g, h}, {b, f, g, h}, {b, f, d, h}, {b, f, g, i}, {b, c, e, g, h} are few 2-point set neutrosophic domination & 𝛾2𝑝𝑠𝑛𝑑(𝐺) = 2.95. Definition 3.4. Let G = (X, Y) be single valued neutrosophic network, if for each subset A βŠ† X βˆ’B there exists a vertex b ∈ B such that subgraph < Aβˆͺ {b} > induced by the vertices of Aβˆͺ {b} is tree then a subset B βŠ† X(G) of any graph G is a point set tree neutrosophic domination (𝐷𝑝𝑠𝑑𝑛). point set tree neutrosophic domination number of G is the number with the minimum vertex cardinality in all point set tree domination set of G and it is denoted by 𝛾𝑝𝑠𝑑𝑛𝑑(𝐺). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 22 https://internationalpubls.com Figure 4: 𝐷𝑝𝑠𝑑𝑛= {b, c}, {b, a, c}, {b, a, e, c}, {b, d, c}, {a, b, c, d}, {a, b, d} are point set tree neutrosophic domination & 𝛾𝑝𝑠𝑑𝑛𝑑(𝐺). = 1.8. Theorem 3.5. In Single valued neutrosophic network G = (X, Y), Ξ΄(G) ≀ O(𝐷𝑝𝑠𝑛) & βˆ†(G) ≀ O(𝐷𝑝𝑠𝑛, Where O(𝐷𝑝𝑠𝑛) is point set neutrosophic domination. Proof. From fig:1 The Maximum degree of G : βˆ†T(G) = max dT(vi/vi ∈V ) = 1.1, βˆ†I(G) = max dI(vi/vi ∈V ) = 1.2, βˆ†F(G) = max dF(vi/vi ∈V ) = 1.5. The maximum degree of G is βˆ†(G) = max{dT(vi), dI(vi), dF(vi)} = (1.1, 1.2, 1.5) The Minimum degree of G : Ξ΄T(G) = min dT(vi/vi ∈V ) = 0.8, Ξ΄I(G) = min dI(vi/vi ∈V ) = 0.9, Ξ΄F(G) = min dF(vi/vi ∈V ) = 0.5. The minimum degree of G is Ξ΄(G) = max{dT(vi), dI(vi), dF(vi)} = (0.8, 0.9, 0.5) OT(D) = OT(b, c, d, i) = 2.6, OI(D) = OI(b, c, d, i) = 1.4, OF(D) = OF(b, c, d, i) = 1.9, O(𝐷𝑝𝑠𝑛) = (2.6, 1.4, 1.9) Therefore Ξ΄(G) ≀ O(𝐷𝑝𝑠𝑛) & βˆ†(G) ≀ O(𝐷𝑝𝑠𝑛) Theoren 3.6. Let G be a complete neutrosophic graph and 𝐷𝐾 𝑝𝑠𝑛 is a point set neutrosophic domination set then V βˆ’π·πΎ 𝑝𝑠𝑛 has a point set neutrosophic domination set. Proof. Given G is a complete neutrosophic graph then every edge 𝑒 ∈ π‘Œ (𝐺) is an effective edge and each vertex 𝑣 ∈ 𝑋(𝐺) is dominating all others. Thus a minimum point set neutrosophic dominance set 𝐷𝐾 𝑝𝑠𝑛 contains only one vertex, then 𝑉 βˆ’π·πΎ 𝑝𝑠𝑛 set domination set. is point set domination set. Consequently 𝑉 βˆ’π·πΎ 𝑝𝑠𝑛 has point set domination set. Theorem 3.7. In Single valued neutrosophic network 𝐺 = (𝑋, π‘Œ), complement of the complete graph can not point set neutrosophic domination. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 23 https://internationalpubls.com Theorem 3.8. For single valued Neutrosophic network 𝐺 = (𝑋, π‘Œ), any point set domination set is 2-point set domination set of G Proof. Consider 𝐺 = (𝑋, π‘Œ) is neutrosophic network, Let G be a 2-point set neutrosophic dominance set of 𝐺. If for every set 𝑇 βŠ† 𝑋 βˆ’ 𝐷, there exist a non empty set 𝑆 βŠ† 𝐷 containing at most two vertices such that the induced subgraph < 𝑆 βˆͺ 𝑇 > is connected. Since it is 2-point set neutrosophic dominance set of 𝐺. If for each 𝑇 βŠ† 𝑋 βˆ’ 𝐷 there exist 𝑒 ∈ 𝑆, then < {𝑒} βˆͺ 𝑇 > is connected. Therefore 𝐺 is point set neutrosophic dominance. Theorem 3.9. For SVNG, 𝛾2𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑑𝑛𝑑(𝐺) ≀ 𝛾𝑐𝑝𝑠𝑛𝑑(𝐺). Proof. Let A be a least point set neutrosophic dominance set of neutrosophic network G and 𝛾𝑝𝑠𝑛𝑑(𝐺) = 𝑠. Every point set domination is a 2-point set domination in neutrosophic network so, 𝛾2𝑝𝑠𝑛𝑑(𝐺)) = 𝑠. (𝑖. 𝑒) 𝛾2𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑛𝑑(𝐺) Suppose A is not a least point set neutrosophic domination set and if 𝐴′ is least point set neutrosophic domination set then 𝛾𝑝𝑠𝑛𝑑(𝐺) > 𝑠. Then 𝛾2𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑛𝑑(𝐺) Let 𝐡 be a least point set tree neutrosophic domination set of neutrosophic graph 𝐺 then 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺) = 𝑑. If B be a least connected point set neutrosophic domination set of neutrosophic graph G then 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺) = 𝑑. (𝑖. 𝑒) 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺) ≀ 𝛾𝑐𝑝𝑠𝑛𝑑(𝐺) Suppose 𝐡 is not a least connected point set neutrosophic domination and if 𝐡′ is a least connected point set neutrosophic dominance set then 𝛾𝑐𝑝𝑠𝑛𝑑(𝐺) > 𝑑. Thus 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺)≀ 𝛾𝑐𝑝𝑠𝑛𝑑(𝐺). Let 𝐢 be a least point set neutrosophic domination set of neutrosophic graph and 𝛾𝑝𝑠𝑛𝑑(𝐺) = 𝑒 if 𝐢 is also least point set tree neutrosophic domination of neutrosophic graph 𝐺 then 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺) = 𝑒. (i.e) 𝛾𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺) Suppose 𝐢 is not a least point set tree neutrosophic dominance set and if 𝐢′ is a least point set tree neutrosophic dominance set then 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺) > u. Then 𝛾𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺). From (1), (2), (3) we get, 𝛾2𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑛𝑑(𝐺) ≀ 𝛾𝑝𝑠𝑛𝑑𝑑(𝐺) ≀ 𝛾𝑐𝑝𝑠𝑛𝑑(𝐺). Theorem 3.10. A domination set neutrosophic graph 𝐺 = (𝑋, π‘Œ ) is point set domination if for each vertex 𝑒 ∈ 𝑋 βˆ’ 𝐷 satisfies one of the following conditions, 1. < 𝑋 βˆ’ 𝐷 > is connected 2. If there does not exist 𝑒 βˆ’ 𝑣 path between at most any two vertices of 𝑋 βˆ’ 𝐷 then there exist 𝑑 ∈ 𝐷 such that 𝑁(𝑒) ∩ 𝑁(𝑣) = 𝑑. ∴𝐷 is point set neutrosophic domination set, which satisfies one of the above conditions Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 24 https://internationalpubls.com Definition 3.12. If no proper subset of point set domination set 𝐷 is point set domination set, then 𝐷 is said to be minimal point set domination in neutrosophic graph 𝐺 Theorem 3.13. In a neutrosophic graph 𝐺, a point set domination set is minimal if and only if for each vertex in dominating set 𝑏 ∈ 𝐷 one of the following conditions holds. 1. 𝑏 𝑖s independent vertices in 𝐷 2. There is a vertex 𝑒 ∈ 𝑉 βˆ’ 𝐷 such that 𝑁(𝑒) ∩ 𝐷 = 𝑏 Proof. Assume that D is a minimal point set domination of G. Then for every vertex 𝑏 ∈ 𝐷, 𝐷 βˆ’ 𝑏 is not a point set dominating set and hence there exists 𝑀 ∈ 𝑉 βˆ’ (𝐷 βˆ’ 𝑏) which is not dominated by the vertex in 𝐷 βˆ’ 𝑏. If 𝑀 = 𝑏, w is not a strong neighbour of any vertex in 𝐷. If 𝑀 β‰  𝑏, w is not dominated by 𝐷 βˆ’ 𝑀, but is dominated by 𝐷, then the vertex 𝑀 is a strong neighbor only to 𝑏 in 𝐷. That is 𝑁(𝑀) ∩ 𝐷 = 𝑏. conversely, Assume that 𝐷 is a point set neutrosophic domination set for each vertex 𝑏 ∈ 𝐷, one of the two condition holds Suppose 𝐷 is not a minimal point set neutrosophic domination set, then there exist a vertex 𝑏 ∈ 𝐷, 𝐷 βˆ’ 𝑏 is a point set neutrosophic dominating set. Hence 𝑏 is a strong neighbor to at least one vertex in 𝐷 βˆ’ 𝑏, the condition one does not hold. If 𝐷 βˆ’ 𝑏, the condition one does not hold. If 𝐷 βˆ’ 𝑏 is a point set then every vertex in 𝑉 βˆ’ 𝐷 is a strong neighbor to at least one vertex in 𝐷 βˆ’ 𝑏, the second condition does not hold which is contradiction to our assumption that at least one of the conditions. Definition 3.14. Lower point set dominating number (𝑑𝑝𝑠𝑛) of neutrosophic graph G is minimum cardinality of all minimal point set dominating number. 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