Ratio Mathematica Volume 44, 2022 Independent Restrained k - Rainbow Dominating Function M. Esakki Dharani* A. Nagarajan† K. Palani‡ Abstract Let G be a graph and let f be a function that assigns to each vertex a set of colors chosen from the set {1, 2…, k} that is f: V(G) → P [1, 2, …, k]. If for each vertex v ∈ V(G) such that f(v) = ∅ .we have ⋃ 𝑓(𝑢) = {1, 2, . . , 𝑘} 𝑢∈𝑁[𝑣] then f is called the k – Rainbow Dominating Function (KRDF) of G. A k – Rainbow Dominating Function is said to be Independent Restrained k - Rainbow Dominating function if (i)The set of vertices assigned with non – empty set is independent. (ii)The induced subgraph of G, by the vertices with label ∅ has no isolated vertices. The weight w(f) of a function f is defined as w(f) = ∑ |𝑓(𝑣)|𝑣 ∈ 𝑉(𝐺) .The Independent Restrained k – Rainbow Domination number is the minimum weight of G. In this paper we introduce Independent Restrained k – Rainbow Domination and find for some graphs Keywords: Independent, Restrained, Rainbow domination number, weight. 2010 AMS subject classification: 05C69§ *Research Scholar (Reg.No.21212232092011), PG & Research Department of Mathematics, V.O. Chidambaram College, Thoothukudi-628008, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.dhanushiya1411@gmail.com †Associate Professor (Retd.), V.O. Chidambaram College, Thoothukudi-628008, Tamil Nadu, India Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.nagarajan.voc@gmail.com ‡Associate Professor, A.P.C. Mahalaxmi College for Women, Thoothukudi-628008, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.palani@apcmcollege.ac.in § Received on June 10th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.924. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. 349 M. Esakki Dharani, A. Nagarajan, and K. Palani 1. Introduction Domination in graphs originates from location problems in operations research. As a variation of domination in graphs, rainbow domination was introduced by Bresar et al. [2]. Shao et al. [7] gave bounds for the k – rainbow domination number on an arbitrary graph. Hao et al. [4] studied the k – rainbow domination number of directed graphs. Independent rainbow domination was introduced by Zehui Shao et al. [8]. Amjadi et al. [1] was investigated the rainbow restrained domination number. In this paper we introduce Independent Restrained k – Rainbow Domination and find for some graphs. 2. Preliminaries A graph G consists of pair(V(G), E(G)) where V(G) is a non-empty finite set whose elements are called points or vertices and E(G) is a set of unordered pair of distinct elements of V(G). The elements of E(G) are called lines or edges of the graph G. For any vertex u in G, the open neighbourhood of u, is denoted by N(u) is the set of vertices adjacent to u and the closed neighbourhood of u, is denoted by N[u] = N(u) ∪ {u}. A set of vertices in a graph is said to be an independent set of vertices or simply an independent if no two vertices in the set are adjacent. The splitting graph of G is denoted by S[G]For each vertex v of a graph G, take a new vertex 𝑣′ and join 𝑣′ to all vertices adjacent to v in G.The corona product of two graphs G and H is defined as the graph obtained by taking one copy of G and |𝑉(𝐺)| copies of H and joining the 𝑖𝑡ℎ vertex of G to every vertex in the 𝑖𝑡ℎ copy of H. It is denoted as 𝐺 ∘ 𝐻. Let G be a graph and let f be a function that assigns to each vertex a set of colors chosen from the set {1, 2, …, k} that is f : V(G) → P [1,2,…,k]. If for each vertex v∈ V(G) such that f(v) = ∅ .we have ⋃ 𝑓(𝑢) = {1,2, . . , 𝑘} 𝑢∈𝑁[𝑣] then f is called the k – Rainbow Dominating Function (KRDF) of G. A k – Rainbow dominating function is called independent k – Rainbow Domination if vertices assigned with non – empty sets are pairwise non- adjacent. A k – Rainbow dominating function is called Rainbow Restrained Domination function if vertices assigned with empty sets has no isolated vertex. 3. Main Results Definition 3.1. Let G be a graph and let f be a function that assigns to each vertex a set of colors chosen from the set {1, 2, …, k} that is f : V(G) → P [1,2,…,K]. If for each vertex v∈ V(G) Such that f(v) = ∅ . we have ⋃ 𝑓(𝑢) = {1,2, . . , 𝑘} 𝑢∈𝑁[𝑣] then f is called the k – Rainbow Dominating Function (KRDF) of G. A k – Rainbow Dominating Function is said to be Independent Restrained k- Rainbow Dominating function if i. The set of vertices assigned with non – empty set is independent. ii. The induced subgraph of G, by the vertices with label ∅ has no isolated vertices. 350 Independent Restrained k - Rainbow Dominating Function The weight w(f) of a function f is defined as w(f) = ∑ |𝑓(𝑣)|𝑣∈𝑉(𝐺) . The Independent Restained k – Rainbow Domination number is the minimum weight of G. Observation 3.2. 1. Let u be a pendant vertex to v. If one of the vertex u and v is assigned {1, 2, … k} then the other may be assigned ∅. (i.e) if the end vertex is assigned {1, 2, … k} then the support vertex may be assigned ∅ 2. Always 𝑘 ≤ 𝛾𝑖𝑟𝑘𝑟 ≤ 𝑛𝑘 3. If a graph G has a clique as a subgraph, then one of the vertices of the clique should be labelled {1, 2, … k} and all the remaining vertices are labelled ∅. Theorem 3.3. Let G be a graph and let v be a full degree vertex in G. Suppose G – {v} has no isolated vertex then 𝛾𝑖𝑟𝑘𝑟(𝐺) = 𝑘 Proof: Define f: V(G) → P [1,2,…,k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑣 ∅ otherwise } Then clearly f is an independent restrained rainbow dominating function. Further w(f) = k As 𝛾𝑖𝑟𝑘𝑟(𝐺) ≥ 𝑘, f is a minimum independent restrained k rainbow dominating function. Therefore, 𝛾𝑖𝑟𝑘𝑟(𝐺) = 𝑘 Corollary 3.4: Complete graph, Wheel graph, Fan graph, Flower graph all have independent restrained k rainbow domination number is equal to k. Theorem 3.5: The Independent Restrained k- Rainbow Dominating function exists for path graph 𝑃𝑛 if and only if 𝑛 ≡ 1 𝑚𝑜𝑑 3 and then 𝛾𝑖𝑟𝑘𝑟 (𝑃𝑛) = ⌈ 𝑛 3 ⌉ k, where 𝑛 ≡ 1 𝑚𝑜𝑑 3 Proof: When 𝑛 ≡ 0 𝑜𝑟 2 𝑚𝑜𝑑 3, the graph G does not admit any Independent Restrained k Rainbow dominating function. Since any k Rainbow Dominating function f fails to satisfy either Independent or Restrained condition. let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛 be the vertices of the path graph. Define f: V(G) → P [1, 2, …, k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑣𝑖 𝑤ℎ𝑒𝑟𝑒𝑖 ≡ 1 𝑚𝑜𝑑 3 𝑎𝑛𝑑 0 ≤ 𝑖 ≤ 𝑛 ∅ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 } Then clearly f is an Independent Restrained k Rainbow Dominating function. Therefore, w(f) = ∑|𝑓(𝑣𝑖 )| = |𝑓(𝑣1)| + |𝑓(𝑣4)| + |𝑓(𝑣7)| + ⋯ + |𝑓(𝑣𝑛)| Therefore, 𝛾𝑖𝑟𝑘𝑟(𝑃𝑛) = ⌈ 𝑛 3 ⌉ k 351 M. Esakki Dharani, A. Nagarajan, and K. Palani Theorem 3.6: The Independent Restrained k- Rainbow Dominating function exists for cycle graph 𝐶𝑛 if and only if 𝑛 ≡ 0 𝑚𝑜𝑑 3 and then 𝛾𝑖𝑟𝑘𝑟 (𝐶𝑛) = 𝑛𝑘 3 , 𝑛 ≡ 0 𝑚𝑜𝑑 3 Proof: When 𝑛 ≡ 1 𝑜𝑟 2 𝑚𝑜𝑑 3 , the graph G does not admit any Independent Restrained k Rainbow dominating function. Since any k Rainbow Dominating function f fails to satisfy either Independent or Restrained condition. let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛 be the vertices of the cycle graph. Define f: V(G) → P [1, 2, …, k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑣𝑖 𝑤ℎ𝑒𝑟𝑒𝑖 ≡ 0 𝑚𝑜𝑑 3 𝑎𝑛𝑑 0 ≤ 𝑖 ≤ 𝑛 ∅ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 } Then clearly f is an Independent Restrained k Rainbow Dominating function. Therefore, w(f) = ∑|𝑓(𝑣𝑖 )| = |𝑓(𝑣3)| + |𝑓(𝑣6)| + |𝑓(𝑣9)| + ⋯ + |𝑓(𝑣𝑛)| Therefore, 𝛾𝑖𝑟𝑘𝑟(𝐶𝑛) = ⌈ 𝑛 3 ⌉ k Theorem 3.7: Let 𝐻𝑛 be the helm graph obtained from the wheel by attaching a pendant vertex to each rim vertex. Then 𝛾𝑖𝑟𝑘𝑟(𝐻𝑛) = k + n - 2 for n ≥ 4 Proof: Let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛−1 be the rim vertex. Let u be the apex vertex. Let 𝑤1, 𝑤2, … 𝑤𝑛−1such that 𝑤𝑖 is adjacent to 𝑣𝑖 for 1 ≤ 𝑖 ≤ 𝑛 − 1 . Define f: V(G) → P [1, 2, …, k] by f(x) = { { 1}𝑖𝑓𝑥 = 𝑤𝑖 where 1 ≤ 𝑖 ≤ 𝑛 − 1 { 2,3, … , 𝑘 }𝑖𝑓𝑥 = 𝑢 ∅ 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 } Obviously, every vertex v with labelled ∅ satisfies the condition ⋃ 𝑓(𝑢) =𝑢∈𝑁[𝑣] {1,2, . . , 𝑘} ∴f is a k Rainbow Dominating Function. Let S be the set of vertices assigned ∅ labelled then S = {𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛−1}. Here has no isolated vertices. Let S’ be the set of vertices assigned non empty labelled. Then S’ = {𝑢, 𝑤1, 𝑤2, … 𝑤𝑛−1} is Independent. Then clearly f is an Independent Restrained k Rainbow Dominating function. ∴ w(f) = ∑ (|𝑓(𝑢)| + |𝑓(𝑤𝑖 )| ) where i = 1 to n – 1 Therefore, 𝛾𝑖𝑟𝑘𝑟(𝐻𝑛) = k + n – 2 Theorem 3.8: Let (𝐶𝑛 ∘ 𝐾1)be the crown graph obtained by joining a pendant edge to each vertex of 𝐶𝑛. Then for n ≥ 1, 𝛾𝑖𝑟𝑘𝑟 (𝐶𝑛 ∘ 𝑘1) = nk Proof: Let 𝑣1, 𝑣2, 𝑣3, … 𝑣𝑛 be the vertices of the cycle. Let 𝑤1, 𝑤2, 𝑤3, … 𝑤𝑛 be the set of end vertices of the crown graph, where 1 ≤ 𝑖 ≤ 𝑛 . Define f: V(G) → P [1, 2, …, k] by f(x) = { { 1,2, … , 𝑘}𝑖𝑓𝑥 = 𝑤𝑖 ; 1 ≤ 𝑖 ≤ 𝑛 ∅ otherwise } 352 Independent Restrained k - Rainbow Dominating Function Obviously, every vertex v with labelled ∅ satisfies the condition ⋃ 𝑓(𝑤) =𝑤∈𝑁[𝑣] {1,2, . . , 𝑘} ∴f is a k Rainbow Dominating Function. By Observation 3.2(1), we assigned { 1,2, … , 𝑘} to the end vertices, which is independent and we assigned∅ to all the support vertices. Then the induced subgraph of empty set is also connected. Hence it satisfies both Independent and Restrained condition. 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