EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5770 ISSN 1307-5543 – ejpam.com Published by New York Business Global Weakly Connected k-Rainbow Domination in Graphs Jamil J. Hamja1,2, Seyed Mahmoud Sheikholeslami3,∗, Imelda S. Aniversario2,4, Lyster Rey B. Cabardo2,4 1 Mathematics and Sciences Department, College of Arts and Sciences, MSU - Tawi-Tawi College of Technology and Oceanography, 7500 Tawi-Tawi, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, MSU - Iligan Institute of Technology, 9200 Iligan City, Philippines 3 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran 4 Center for Mathematical and Theoretical Physical Sciences, Premier Research Institute of Science and Mathematics (PRISM), MSU - Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a simple and connected graph, and let f be a function that assigns to each vertex a set of colors chosen from the set {1, 2, 3, . . . , k}, i.e., f : V (G) → P({1, 2, 3, . . . , k}). If for each vertex v ∈ V (G) such that f(v) = ∅, we have ⋃ u∈NG(v) f(u) = {1, 2, 3, . . . , k}, then f is called a k-rainbow dominating function (kRDF) of G. A kRDF f : V (G) → P({1, 2, . . . , k}) is said to be a weakly connected k-rainbow dominating function (WCkRDF) if the set S = {v ∈ V (G) : f(v) ̸= ∅} is weakly connected dominating. The weight w(f) of f is defined as ω(f) = ∑ v∈V (G)|f(v)|. The weakly connected k-rainbow domination number of G, denoted by γwc rk (G) is the minimum weight of WCkRDF. A weakly connected k-rainbow dominating function of G with weight γwc rk (G), i.e., ω(f) = γwc rk (G) is referred to as a γwc rk -function of G. In this paper, we initiate the study of the weakly connected k-rainbow domination parameter. First, we establish fundamental properties and bounds for weakly connected k-rainbow domination. Then, we determine the weakly connected k-rainbow domination number for various classes of graphs. Furthermore, we characterize the weakly connected k-rainbow dominating function under the join of graphs and determine the weakly connected k-rainbow domination number for this binary operation. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Weakly connected dominating set, k-rainbow dominating function, weakly connected k-rainbow dominating function ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5770 Email addresses: jamilhamja@msutawi-tawi.edu.ph (J. J. Hamja), s.m.sheikholeslami@azaruniv.ac.ir (S. M. Sheikholeslami), imelda.aniversario@g.msuiit.edu.ph (I. S. Aniversario), lysterrey.cabardo@g.msuiit.edu.ph (L. B. Cabardo) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 2 of 11 1. Introduction Graph domination provides a foundational framework for various applications, where the vertices in a dominating set represent service providers or essential resources that en- sure accessibility to every vertex in the network, as described by T. W. Haynes et al. [1]. In 1987, Hedetniemi [2] introduced the concept of dominating functions, offering an analyt- ical approach to studying this discrete structure. This framework established connections between domination, graph labelings, and colorings, leading to the development of new domination function parameters. Two decades later, in 2008, Brešar et al. [3] introduced the concept of rainbow domination, which extends the notion of domination by incorpo- rating multiple service types, each represented by a distinct color. The goal of rainbow domination is to assign services so that any vertex not directly receiving a service has access to all service types within its neighborhood. In the following years, this concept gained significant attention from researchers, leading to numerous studies exploring its properties and applications, as discussed in [4–11]. In 1997, J. E. Dunbar et al. [12] introduced the concept of a weakly connected dominat- ing set in a connected graph and examined the weakly connected domination number along with related parameters. Further insights into weakly connected domination can be found in [13–19]. Rainbow domination extends to weakly connected domination by ensuring re- source distribution while maintaining a weakly connected dominating set. This integration is crucial for networks that require both connectivity and service diversity, leading to new parameters and optimization techniques in graph theory. In this paper, we initiate the study of the weakly connected k-rainbow domination parameter. First, we establish fundamental properties and bounds for weakly connected k-rainbow domination. Then, we determine the weakly connected k-rainbow domination number for various classes of graphs. Furthermore, we characterize the weakly connected k- rainbow dominating function under the join of graphs and determine the weakly connected k-rainbow domination number for this binary operation. 2. Terminology and Notation For general graph theory terminology, we adhere to the definitions provided by Harary in [20]. Let G = (V,E) be a simple and connected graph, where V = V (G) repre- sents the vertex set and E = E(G) denotes the edge set of G. The number of edges incident to a vertex v is called its degree, denoted as deg(v). The maximum degree of G, represented by ∆(G), is given by ∆(G) = max{deg(v) : v ∈ V (G)}. The open neighborhood of a vertex u, denoted by NG(u), is the set of all vertices adjacent to u, i.e., NG(u) = {v ∈ V (G) : uv ∈ E(G)}. The closed neighborhood of u is defined as NG[u] = NG(u) ∪ {u}. Similarly, for a subset S ⊆ V (G), the closed neighborhood of S is given by NG[S] = ⋃ v∈S NG[v]. The subgraph weakly induced by a set S ⊆ V (G), as defined by E. P. Sandueta et al. [18], is the graph ⟨S⟩w = (NG[S], Ew(S)), where the edge set is J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 3 of 11 given by Ew(S) = {uv ∈ E(G) : u ∈ S or v ∈ S}. A set S ⊆ V (G) is said to be a dominating set of G if NG[S] = V (G). A dominating set S is called a minimal dominating set if no proper subset S′ ⊂ S is itself a dominating set. The domination number, denoted by γ(G) is the minimum cardinality of a dominating set in G. A dominating set S with |S| = γ(G) is referred to as a γ-set of G. A set S ⊆ V (G) is said to be a weakly connected dominating set in G if S is dominating and the subgraph ⟨S⟩w weakly induced by S is connected. The weakly connected domina- tion number, denoted by γw(G) is the minimum cardinality among all weakly connected dominating sets. A weakly connected dominating set S with |S| = γw(G) is referred to as a γw-set of G, as defined by J. E. Dunbar et al. in [12]. A function f : V (G) → P({1, 2, 3, . . . , k}) assigns to each vertex of a graph G a set of colors chosen from the set {1, 2, 3, . . . , k}. If, for every vertex v ∈ V (G) such that f(v) = ∅, we have ⋃ u∈NG(v) f(u) = {1, 2, 3, . . . , k}, then f is called a k-rainbow domi- nating function (kRDF) of G. The weight ω(f) of f is defined as ω(f) = ∑ v∈V (G)|f(v)|. The k-rainbow domination number of G, denoted by γrk(G), is the minimum weight of a kRDF. A k-rainbow dominating function of G with weight γrk(G), i.e., ω(f) = γrk(G), is referred to as a γrk-function of G, as defined by B. Brešar et al. in [3]. A kRDF f : V (G) → P({1, 2, . . . , k}) is said to be a weakly connected k-rainbow dom- inating function (WCkRDF) if the set S = {v ∈ V (G) : f(v) ̸= ∅} is weakly connected dominating. The weight ω(f) of f is defined as ω(f) = ∑ v∈V (G)|f(v)|. The weakly con- nected k-rainbow domination number of G, denoted by γwc rk (G), is the minimum weight of a WCkRDF. A weakly connected k-rainbow dominating function of G with weight γwc rk (G), i.e., ω(f) = γwc rk (G), is referred to as a γwc rk -function of G. Clearly, when k = 1, the weakly connected 1-rainbow domination number γwc r1 (G) is equivalent to the classical weakly con- nected domination number γw(G) of G. For any graph G and a γwc rk -function f of G, set V f i = {x ∈ V (G) : |f(x)| = i} for i ∈ {1, 2, . . . , k}. 3. Preliminary Results We begin this section by presenting some properties and bounds, and then we determine the weakly connected k-rainbow domination number of G. Remark 1. Every weakly connected k-rainbow dominating function f of G is also a k- rainbow dominating function of G. In particular, γrk(G) ≤ γwc rk (G). Theorem 1. Let G be a connected graph with ∆(G) ≤ k for some positive integer k ≥ 2. Then there exists a γwc rk -function f of G such that |f(v)| < k for every vertex v ∈ V (G). J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 4 of 11 Proof. Let g be a γwc rk -function of G such that |V g k | is as small as possible. We claim that |f(v)| < k for every vertex v ∈ V (G) as desired. Assume, to the contrary, that there exists a vertex v ∈ V (G) such that |g(v)| = k. Since |g(v)| = k, by the definition there exists a vertex x1 ∈ NG(v) such that g(x1) = ∅. Let there exist r vertices x1, x2, . . . , xr ∈ NG(v) such that g(xi) = ∅ for i ∈ {1, 2, . . . , r}, where r ≤ ∆(G) ≤ k. Now, define a new function f : V (G) → P({1, 2, . . . , k}) by f(xi) = {i} for i ∈ {1, 2, . . . , r − 1}, f(xr) = {r, r + 1, . . . , k}, f(v) = ∅, and f(u) = g(u) for all other vertices u ∈ V (G). Clearly, f is a weakly connected k-rainbow dominating function of G of weight ω(g), contradicting the choice of g. Thus |f(v)| < k for every vertex v ∈ V (G), and the proof is complete. Theorem 2. Let G be a connected graph. Then γwc r2 (G) ≥ γw(G). Proof. Let f be a γwc r2 -function of G. By definition the set V (G) \ V f 0 is a weakly connected dominating set of G and thus γwc r2 (G) = ∑ v∈V (G)\V f 0 |f(v)| ≥ ∑ v∈V (G)\V f 0 1 = |V (G) \ V f 0 | ≥ γw(G), as desired. The graph G illustrated in Figure 1 demonstrates that the bound in Theorem 2 is sharp. ... ...... ... x1 xn y1 yn G : Figure 1: A graph G attaining the bound in Theorem 2 The exact values of the k-rainbow domination number for k ∈ {2, 3} of paths and cycles determined in [6] and [21] as follows. Theorem 3. (i) γr2(Pn) = ⌊ n 2 ⌋ + 1 and γr2(Cn) = ⌊ n 2 ⌋ + ⌈ n 4 ⌉ − ⌊ n 4 ⌋ . (ii) For n ≥ 5, γr3(Pn) =  ⌈ 3n 4 ⌉ + 1 if n ≡ 0 (mod 4)⌈ 3n 4 ⌉ if n ≡ 1, 2, 3 (mod 4). (iii) For n ≥ 5, γr3(Cn) = ⌈ 3n 4 ⌉ . J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 5 of 11 Using Theorem 3 and Remark 1, we will determine the weakly connected k-rainbow domination number for k ∈ {2, 3} for paths Pn and cycle Cn. Proposition 1. Let n be a positive integer. Then (i) for all n ≥ 1, γwc r2 (Pn) = ⌊ n 2 ⌋ + 1. (ii) for all n ≥ 1, γwc r3 (Pn) = { ⌈3n4 ⌉+ 1 if n ≡ 0 (mod 4) ⌈3n4 ⌉ if n ≡ 1, 2, 3 (mod 4). (iii) for all n ≥ 3, γwc r2 (Cn) = ⌊ n 2 ⌋ + ⌈ n 4 ⌉ − ⌊ n 4 ⌋ (iv) for all n ≥ 3, γwc r3 (Cn) = ⌈ 3n 4 ⌉ . Proof. Let Pn = [v1, v2, . . . , vn−1, vn] be a path of order n and Cn = [v1, v2, . . . , vn−1, vn, v1] be a cycle of order n. We first establish the result (i) and (ii) for paths. For k ∈ {2, 3}, define the function fk : V (Pn) → P({1, 2, . . . , k}) as follows: (a) If n ≡ 1 (mod 4), then let fk(v4i+1) = {1} for 0 ≤ i ≤ n−1 4 , fk(v4i+3) = {2, k} for 0 ≤ i ≤ n−5 4 , and fk(x) = ∅ otherwise. (b) If n ≡ 2 (mod 4), then let fk(vn) = {1}, fk(v4i+1) = {1} for 0 ≤ i ≤ n−2 4 , fk(v4i+3) = {2, k} for 0 ≤ i ≤ n−6 4 , and fk(x) = ∅ otherwise. (c) If n ≡ 3 (mod 4), then let fk(v4i+1) = {1}, fk(v4i+3) = {2, k} for 0 ≤ i ≤ n−3 4 and fk(x) = ∅ otherwise. (d) If n ≡ 0 (mod 4), then let fk(vn) = {1}, fk(v4i+1) = {1}, fk(v4i+3) = {2, k} for 0 ≤ i ≤ n−4 4 and fk(x) = ∅ otherwise. In all cases, fk is a WCkRDF of Pn of weight γrk(Pn) and thus γwc rk (Pn) ≤ γrk(Pn) for k ∈ {2, 3}. By Remark 1, we obtain γwc rk (Pn) = γrk(Pn) for k ∈ {2, 3}, and Theorem 3-(i,ii) leads to the desired values. Now, we prove (iii) and (iv) for cycles simultaneously. Let k ∈ {2, 3}. If n ≡ 0 (mod 4), then define the function fk : V (Cn) → P({1, 2, . . . , k}) by fk(v4i+1) = {1}, fk(v4i+3) = {2, k} for 0 ≤ i ≤ n−4 4 , and fk(x) = ∅ otherwise. If n ̸≡ 0 (mod 4), then let fk be the function defined in the above item (i) and (ii) depending on n. In all cases, fk is a WC2RDF of Cn of weight γrk(Cn), and thus γwc rk (Cn) ≤ γrk(Cn) for k ∈ {2, 3}. Now, Remark 1 leads to γwc rk (Cn) = γrk(Cn) for k ∈ {2, 3}, and (iii) and (iv) follow from Theorem 3-(i,iii). J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 6 of 11 Proposition 2. Let k be a positive integer and G be a connected graph of order n. Then min{n, k} ≤ γwc rk (G) ≤ n. In particular, if n ≤ k, then γwc rk (G) = n. Proof. Suppose that f is a γwc rk -function of G. If V f 0 = ∅, then we have γwc rk (G) = ∑ v∈V (G) |f(v)| ≥ ∑ v∈V (G) 1 = n. Assume that V f 0 ̸= ∅ and v ∈ V f 0 . Then we have ⋃ u∈NG(v) f(u) = {1, 2, 3, . . . , k}, and thus γwc rk (G) = ∑ u∈V (G) |f(u)| ≥∑ u∈NG(v) |f(u)| ≥ k. Combining the above inequalities, we get min{n, k} ≤ γwc rk (G). For the upper bound, consider the function g : V (G) → P({1, 2, . . . , k}) defined by g(v) = {1} for all v ∈ V (G). Then g is a k-rainbow dominating function. Let S = {v ∈ V (G) : g(v) ̸= ∅}. Observe that S = V (G) since g(v) = {1} for all v ∈ V (G). Thus, S is a weakly connected dominating set of G. It would imply that g is a weakly connected k-rainbow dominating function. Thus, γwc rk (G) ≤ n. Clearly, if n ≤ k, then γwc rk (G) = n. Shao et al. in [21] proved the next result. Theorem 4. For positive integers n and k ≥ 2, let G be a connected graph of order n ≥ k with k > ∆(G)2. Then γrk(G) = n. The next results are direct consequences of Theorem 4 and Remark 1. Corollary 1. For a positive integer n and k ≥ 2, let G be a connected graph of order n ≥ k with k > ∆(G)2. Then γwc rk (G) = n. Corollary 2. For positive integers n and k ≥ 5, γwc rk (Pn) = γwc rk (Cn) = n. The following result shows that the difference γwc r2 (G)−γr2(G) can be arbitrarily large. A caterpillar C(n; d1, . . . , dn) is defined as a tree in which removal of all its leaves yields a path Pn = [x1, x2, . . . , xn] and that di is the number of leaf neighbors of xi for each i. The path Pn = [x1, x2, . . . , xn] is called the backbone of the caterpillar. Theorem 5. For the integer k ≥ 2 and every non-negative integer c, there exists a con- nected graph G such that γwc r2 (G)− γr2(G) = c. Proof. Consider the caterpillar G = C(3c + 1; 2k, 0, 0, 2k, . . . , 0, 0, 2k) with backbone Pn = [x1, x2, . . . x3c+1] in Figure 2. It is easily seen that the function f : V (G) → P({1, 2, . . . , k}) defined by f(x3i+1) = {1, 2, . . . , k} for 0 ≤ i ≤ c and f(x) = ∅ for other vertices, is the unique γrk-function of G of weight k(c + 1), and the function g : V (G) → P({1, 2, . . . , k}) defined by g(x3i+1) = {1, 2, . . . , k} for 0 ≤ i ≤ c, g(x3i) = {1} for 1 ≤ i ≤ c, and g(x) = ∅ for other vertices, is a γwc rk -function of G of weight k(c+1)+ c. Thus, γwc r2 (G)− γr2(G) = c and the proof is complete. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 7 of 11 x1 x2 x3 x4 x3c−1 x3c x3c+1 . . . . . . . . . . . . Figure 2: A caterpillar G = C(3c+ 1; 2k, 0, 0, 2k, . . . , 0, 0, 2k) 4. Graphs with γwc rk (G) = k In this section, we characterize all graphs G with γwc rk (G) = k. Theorem 6. Let k ≥ 1 be an integer, and let G be a connected graph of order n ≥ k. Then γwc rk (G) = k if and only if n = k or n > k and there exists a set X = {x1, x2, . . . , xm} of vertices with 1 ≤ |X| ≤ k such that (V (G) \X) ⊆ ⋂m i=1NG(xi). Proof. Let n = k or n > k and there exists a set X = {x1, x2, . . . , xm} of vertices with 1 ≤ |X| ≤ k such that (V (G) \X) ⊆ ⋂m i=1NG(xi). By Proposition 2, we have γwc rk (G) ≥ k. If n = k, then obviously γwc rk (G) = k. Assume that n > k. Then the function f : V (G) → P({1, 2, . . . , k}) defined by f(xi) = {i} for 1 ≤ i ≤ m − 1, f(xm) = {t, t + 1, . . . , k} and f(x) = ∅ otherwise, is clearly a WCkRDF of G of weight k, and so γwc rk (G) ≤ k. Thus, γwc rk (G) = k. Conversely, assume that γwc rk (G) = k. Let f be a γwc rk -function of G. If V f 0 = ∅, then we have that n = k. Assume that V f 0 ̸= ∅ and let u ∈ V f 0 . By definition, ∪m i=1f(xi) = {1, 2, . . . , k}. Now let x1, x2, . . . , xm be all vertices in NG(u) such that f(xi) ̸= ∅ for 1 ≤ i ≤ m. It follows from the condition γwc rk (G) = k that ∪m i=1|f(xi)| = k, 1 ≤ i ≤ m, and (V (G) \X) ⊆ ⋂m i=1NG(xi). This completes the proof. As a consequence of Proposition 2 and Theorem 6, we have the following. Corollary 3. Let n and k be positive integers. Then γwc rk (Kn) = { k if n ≥ k, n if n < k. Proposition 3. Let m, n, and k be positive integers with k ≥ 1 and m ≤ n. Then γwc rk (Km,n) =  m+ n if m+ n ≤ k, 2k if m ≥ 2k, max{m, k} if m+ n > k and m < 2k. Proof. Suppose Km,n is a complete bipartite graph with m,n vertices. Let X and Y be the two partite sets of Km,n, where X = {x1, x2, . . . , xm} and Y = {y1, y2, . . . , yn}. If m + n ≤ k, then we have γwc rk (Km,n) = m + n by Proposition 2. Hence we assume that m+ n > k. Consider the following cases: J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 8 of 11 Case 1. m ≥ 2k. Let f : V (Km,n) → P ({1, 2, . . . , k}) be a function defined by f(x1) = f(y1) = {1, 2, . . . , k}, and f(v) = ∅ for every vertex v ∈ V (Km,n)\{x1, y1}. Then ⋃ u∈NG(v) f(u) = {1, 2, . . . , k} for every v ∈ V (Km,n) \ {x1, y1}. Observe that the subgraph ⟨{x1, y1}⟩w induced by {x1, y1} is connected and NKm,n [{x1, y1}] = V (Km,n). Thus, by definition, f is a weakly connected k-rainbow dominating function of Km,n of weight ω(f) = ∑ v∈V (Km,n) |f(v)| = 2k. This implies that γwc rk (Km,n) ≤ 2k. Now, let f∗ be any γwc rk -function of Km,n. If for every vertex x ∈ X, f∗(x) ̸= ∅ or for every vertex y ∈ Y , f∗(y) ̸= ∅, then clearly ω(f∗) ≥ m ≥ 2k. Hence, assume that there are two vertices x ∈ X and y ∈ Y such that f∗(x) = ∅ and f∗(y) = ∅. By definition, we have ω(f∗) = ∑ v∈V (Km,n) |f∗(v)| = ∑ v∈X |f∗(v)|+ ∑ v∈Y |f∗(v)| = ∑ v∈NKm,n (y) |f∗(v)|+ ∑ v∈NKm,n (x) |f∗(v)| ≥ 2k. Therefore, γwc rk (Km,n) ≥ 2k, and as a result, we conclude γwc rk (Km,n) = 2k. Case 2. m+ n > k and m < 2k. If m ≤ k, then by Theorem 6, we have γwc rk (Km,n) = k = max{m, k}. Assume that k < m < 2k. Let f : V (Km,n) → P({1, 2, . . . , k}) be a function defined by f(xi) = {i} for 1 ≤ i ≤ k and f(xi) = {1} for each i ∈ {k + 1, . . . ,m} and f(y) = ∅ for each y ∈ Y . As Case 1, we observe that f is a WCkRDF of Km,n, and thus γwc rk (Km,n) ≤ m. Using a similar argument as in Case 1, we can see that γwc rk (Km,n) = m. Thus, γwc rk (Km,n) = max{m, k}. This completes the proof. 5. Join of Graphs Harary [20] defined the join of two graphs G and H, denoted by G+H, as the graph with V (G+H) = V (G)∪V (H) and E(G+H) = E(G)∪E(H)∪{uv : u ∈ V (G), v ∈ V (H)}. Theorem 7. Let m, n, and k be positive integers such that min{m,n} ≥ k, and let G and H be graphs of order m and n, respectively. Then γwc rk (G + H) = k if and only if γwc rk (G) = k or γwc rk (H) = k. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 9 of 11 Proof. If γwc rk (G) = k (the case γwc rk (H) = k is similar), then by Theorem 6, there exists a set X ⊆ V (G) such that |X| ≤ k and each vertex in V (G) \X is adjacent to all vertices in X. By definition of the join of graphs, each vertex in V (H) is adjacent to all vertices in X, and again Theorem 6 leads to γwc rk (G+H) = k. Conversely, assume that γwc rk (G+H) = k. Using Theorem 6, it follows that there is a set X of vertices V (G+H) such that |X| ≤ k and each vertex in V (G+H)\X is adjacent to all vertices in X. Without loss of generality, we may assume that X ∩V (G) ̸= ∅. Since |X ∩ V (G)| ≤ k and each vertex in V (G) \X is adjacent to all vertices in X, we deduce from Theorem 6 that γwc rk (G) = k. This completes the proof. Theorem 8. Let m, n, and k be positive integers such that min{m,n} ≥ k, and let G and H be connected graphs of order m and n, respectively. Then γwc rk (G+H) ≤ min{2k,min{γwc rk (G), γwc rk (H)}}. Proof. Let first x, y be two vertices of G + H such that x ∈ V (G) and y ∈ V (H), and define the function f : V (G + H) → P({1, 2, . . . , k}) by f(x) = f(y) = {1, 2, . . . , k} and f(z) = ∅ or all vertices z ∈ (V (G) ∪ V (H)) \ {x, y}. Clearly, f is a WCkRDF of G+H and thus γwc rk (G+H) ≤ 2k. Now assume, without loss of generality, that γwc rk (G) = min{γwc rk (G), γwc rk (H)} and let f be a γwc rk -function of G. Since m ≥ k, we may assume that ∪x∈V (G)f(x) = {1, 2, . . . , k}. Since each vertex in V (H) is adjacent to all vertices of V (G) in G+H, f is a WCkRDF of G+H and thus γwc rk (G+H) ≤ min{γwc rk (G), γwc rk (H)}. This proves the upper bound. Open questions and problems: We conclude this paper with some open problem and perspective related to our work. Problem 1. For positive integer k ≥ 3, characterize the graphs G of order n such that γwc rk (G) = n. Problem 2. For positive integer k ≥ 2, characterize the graphs G of order n such that γwc rk (G) = k + ℓ for some positive integer ℓ. Problem 3. Determine Nordhaus-Gaddum type results for γwc rk (G). Problem 4. Design an algorithm for computing the value of γwc rk (T ) for any tree T and k ≥ 2. 6. Conclusion In this paper, we have introduced and investigated the weakly connected k-rainbow domination parameter in graphs. We established fundamental properties and derived bounds for the weakly connected k-rainbow domination number γwc rk (G). Moreover, we determined the exact values of γwc rk (G) for various graph classes, providing insights into their structural dependencies. Additionally, we examined the weakly connected k-rainbow domination behavior under the join operation of graphs. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 10 of 11 Acknowledgements The authors sincerely thank the reviewers for their valuable comments and sugges- tions, which helped improve the quality of this paper. They also acknowledge the fi- nancial support from the Department of Science and Technology—Accelerated Science and Technology Human Resource Development Program (DOST-ASTHRDP), Mindanao State University—Tawi-Tawi College of Technology and Oceanography (MSU-TCTO), and Mindanao State University—Iligan Institute of Technology (MSU-IIT), which made this publication possible. References [1] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning. Topics in Domination in Graphs, volume 64 of Developments in Mathematics. Springer, 2020. [2] S. M. Hedetniemi, S. T. Hedetniemi, and T. V. Wimer. Linear time resource allo- cation for trees. Technical Report URI-014, Dept. Mathematical Sciences, Clemson Univ., 1987. Presented at Southeastern Conf. on Combinatorics, Graph Theory and Computing, Boca Raton, FL, 1987. [3] B. Brešar, M. A. Henning, and D. F. Rall. Rainbow domination in graphs. Taiwanese Journal of Mathematics, 12:213–225, 2008. [4] H. Abdollahzadeh Ahangar, J. Amjadi, N. Jafari Rad, and V. D. Samodivkin. Total k-rainbow domination numbers in graphs. Communications in Combinatorics and Optimization, 3:37–50, 2018. [5] J. Amjadi, N. Dehgardi, M. Furuya, and S. M. Sheikholeslami. A sufficient condition for large rainbow domination number. International Journal of Computer Mathemat- ics: Computer Systems Theory, 2:1–17, 2017. [6] B. Brešar and T. K. Šumenjak. Note on the 2-rainbow domination in graphs. Discrete Applied Mathematics, 155:2394–2400, 2007. [7] S. Fujita, M. Furuya, and C. Magnant. General bounds on rainbow domination num- bers. Graphs and Combinatorics, 31:601–613, 2015. [8] B. Kuzman. On k-rainbow domination in regular graphs. Discrete Applied Mathemat- ics, 184:454–464, 2020. [9] D. Meierling, S. M. Sheikholeslami, and L. Volkmann. Nordhaus-gaddum bounds on the k-rainbow domatic number of a graph. Applied Mathematics Letters, 24:1758–1761, 2011. [10] A. Mahmoodi and L. Volkmann. Outer-independent total 2-rainbow dominating func- tions in graphs. Communications in Combinatorics and Optimization, 8:431–444, 2023. [11] R. Y. Salkhori, E. Vatandoost, and A. Behtoei. 2-rainbow domination number of the subdivision of graphs. Communications in Combinatorics and Optimization. In press. [12] J. E. Dunbar, J. W. Grossman, J. H. Hattingh, S. T. Hedetniemi, and A. A. McRae. On weakly connected domination in graphs. Discrete Mathematics, 167:261–269, 1997. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5770 11 of 11 [13] G. S. Domke, J. H. Hattingh, and L. R. Markus. On weakly connected domination in graphs II. Discrete Mathematics, 305:112–122, 2005. [14] J. J. Hamja, I. S. Aniversario, and H. M. Rara. Weakly connected closed geodetic domination in graphs under some binary operations. European Journal of Pure and Applied Mathematics, 15(2):736–752, 2023. [15] J. J. Hamja, I. S. Aniversario, and C. I. Merca. Weakly connected hop domination in graphs resulting from some binary operations. European Journal of Pure and Applied Mathematics, 16(1):454–464, 2023. [16] M. Lemańska and A. Patyk. Weakly connected domination critical graphs. Opuscula Mathematica, 28:325–330, 2008. [17] J. Raczek and J. Cyman. Weakly connected Roma domination in graphs. Discrete Applied Mathematics, 267:151–159, 2019. [18] E. P. Sandueta and S. R. Canoy Jr. Weakly connected domination in graphs resulting from some graph operations. International Mathematical Forum, 6:1031–1035, 2011. [19] V. Swaminathan. Weakly connected domination in graphs. Electronic Notes in Dis- crete Mathematics, 33:67–73, 2009. [20] F. Harary. Graph Theory. Addison-Wesley Publishing Company, Inc., Massachusetts, 1969. [21] Z. Shao, M. Liang, C. Yin, X. Xu, P. Pavlič, and J. Žerovnik. On rainbow domination numbers of graphs. Information Sciences, 254:225–234, 2014.