EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6429 ISSN 1307-5543 – ejpam.com Published by New York Business Global Generous Roman Domination Subdivision Number in Graphs Jamil J. Hamja1,5, Seyed Mahmoud Sheikholeslami2,∗, Mina Esmaeili2, Lutz Volkmann3, Imelda S. Aniversario4,5, Lucille M. Bugo4,5 1 Department of Mathematics, College of Arts and Sciences, MSU-Tawi-Tawi College of Technology and Oceanography, 7500 Tawi-Tawi, Philippines 2 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, I.R. Iran 3 Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany 4 Department of Mathematics and Statistics, College of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 5 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 = (V,E) be a simple graph, and let f : V → {0, 1, 2, 3} be a function. A vertex u is considered an undefended vertex with respect to f if f(u) = 0 and there is no adjacent vertex v satisfying f(v) ≥ 2. A function f is termed a generous Roman dominating function (GRD-function) if, for every vertex u with f(u) = 0, there exists at least one adjacent vertex v such that f(v) ≥ 2 and the modified function f ′ : V → {0, 1, 2, 3}, defined as f ′(u) = α, f ′(v) = f(v)− α, where α ∈ {1, 2}, and f ′(w) = f(w) for all w ∈ V \ {u, v}, ensures that no vertex remains undefended. The weight of a GRD-function f is defined as f(V ) =∑ u∈V f(u). The smallest possible weight of a GRD-function on G is known as the generous Roman domination number of G, denoted by γgR(G). The generous Roman domination subdivision number, denoted sdγgR(G), is the minimum number of edges that must be subdivided (each at most once) to increase γgR(G). In this paper, we establish upper bounds on sdγgR(G), and determine its exact value for certain families of graphs, including paths, cycles, and ladders. Furthermore, we provide sufficient conditions for a graph G to have a small subdivision number. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Generous Roman domination, generous Roman domination subdivi- sion number ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6429 Email addresses: jamilhamja@msutawi-tawi.edu.ph (J. J. Hamja), s.m.sheikholeslami@azaruniv.ac.ir (S. M. Sheikholeslami), minaesmaeeli1999@gmail.com (M. Esmaeili), volkm@math2.rwth-aachen.de (L. Volkmann), imelda.aniversario@g.msuiit.edu.ph (I. S. Aniversario), lucille.bugo@g.msuiit.edu.ph (L. M. Bugo) 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 (4) (2025), 6429 2 of 16 1. Introduction Motivated by resource–allocation strategies for defending the Roman Empire, where lightly defended regions must be able to call in reinforcements from nearby strongholds, as discussed by ReVelle and Rosing [1] and by Stewart [2], Cockayne et al. introduced Roman domination in graphs in 2004 [3]. Since its introduction, Roman domination has attracted wide and sustained interest, spawning a rich family of variants and extensions, e.g., double Roman domination, Roman {2}-domination, independent Roman domination, total Roman domination, bondage and domatic parameters, and directed versions, now documented in well over two hundred publications. For comprehensive accounts of core re- sults, and structural characterizations, we refer to the book chapters and survey papers by Chellali, Jafari Rad, Sheikholeslami, and Volkmann [4, 5]. Focused overviews of specific di- rections include surveys on varieties of Roman domination [6–8], Roman domatic problems in graphs and digraphs [9], and Roman domination parameters for directed graphs [10]. These sources together chart the evolution of Roman domination from its facility-location and defense-strategy origins [1, 2] to a mature theory with broad connections across dom- ination theory and algorithmic graph problems [4–6, 9, 10]. Among the most recent contributions to Roman domination theory is the notion of gen- erous Roman domination, introduced by Benatallah, Blidia, and Ouldrabah in 2024 [11]. This strengthening reflects scenarios where “reinforcements” must be guaranteed from more than one source, thereby increasing robustness compared to classical Roman domination. In their foundational paper, they provided exact values of the generous Roman domination number for paths and cycles, derived an upper bound for general graphs, and characterized cubic graphs of order n with respect to this parameter. Furthermore, they investigated a Nordhaus–Gaddum type inequality for the generous Roman domination number, and they established results concerning its computational complexity, demonstrating that the problem of determining this parameter remains NP-complete for general graphs. Building on this foundation, Sheikholeslami, Chellali, and Kor [12] advanced the study of gener- ous Roman domination in 2025 by determining its exact value for ladder graphs. They also provided an upper bound for trees in terms of the order, the number of leaves, and the number of stems. Their results highlighted structural dependencies of this parameter within tree families, leading to a sharper understanding of its extremal behavior. In par- ticular, they proved that for every tree T on at least three vertices, the generous Roman domination number satisfies γgR(T ) ≥ γ(T ) + 2, where γ(T ) denotes the domination number of T , and they completely characterized the extremal trees attaining this lower bound. These findings not only extended the appli- cability of generous Roman domination to broader classes of graphs, but also established meaningful links between classical domination parameters and this new variant. Together, the works of Benatallah et al. [11] and Sheikholeslami et al. [12, 13] demonstrate the rapid development of generous Roman domination as a promising research direction. In par- ticular, they show how the interplay between structural graph properties and domination J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 3 of 16 parameters yields both exact values and tight bounds, while also raising new complexity and extremal problems. This suggests that generous Roman domination may follow a tra- jectory similar to classical Roman domination, giving rise to a wide array of variants and applications in the years to come. Given that graphs serve as models for numerous real-world systems, it is natural to explore how structural changes such as vertex or edge deletions, edge additions, or edge sub- divisions affect domination-related parameters. In this direction, Velammal [14] introduced the domination subdivision number measuring the minimal number of edge subdivisions needed to increase a graph’s domination number, with each edge subdivided at most once. This line of research has since been extended to various domination parameters [15–22]. In this paper, we extend the study of generous Roman domination by examining its behavior under edge subdivision. To this end, we introduce the generous Roman domi- nation subdivision number, which is defined as the minimum number of edges in a graph G that must be subdivided (each at most once) in order to increase the generous Roman domination number of G. We establish general upper bounds on generous Roman dom- ination subdivision number and determine its exact value for several families of graphs, including paths, cycles, and ladders. In addition, we provide sufficient conditions under which a graph G admits a small generous Roman domination subdivision number. 2. Terminology and Notation We consider finite, undirected, and simple graphs G, where the vertex set is denoted by V = V (G) and the edge set by E = E(G). The order of G is given by |V | = n, while the size of G is represented as |E| = m. The open neighborhood of a vertex v ∈ V is defined as N(v) = NG(v) = {u ∈ V | uv ∈ E}, whereas its closed neighborhood is given by N [v] = N(v) ∪ {v}. The degree of a vertex v, denoted by degG(v) = deg(v), refers to the number of neighbors of v, i.e., degG(v) = |NG(v)|. As usual, a path, cycle, star, and complete graph with n vertices are denoted by Pn, Cn, K1,n−1, and Kn, respectively. Similarly, Kn,m denotes the complete bipartite graph of order m+ n, while DSp,q denotes the double star of order p + q + 2. A vertex with degree one is referred to as a leaf, and its adjacent vertex is known as a support vertex. A vertex connected to at least two leaves is called a strong support vertex. The Cartesian product of graphs G and H, denoted by G□H, is the graph with vertex set V (G□H) = V (G)×V (H) such that two vertices (v, p) and (u, q) are adjacent in G□H, i.e., (v, p)(u, q) ∈ E(G□H), if and only if one of the following holds: v = u and pq ∈ E(H), or p = q and vu ∈ E(G). For any subset A ⊆ V (G) and a function f that maps V (G) to a numerical set, the function sum over A is given by f(A) = ∑ x∈A f(x). The total sum over all vertices, f(V (G)), is referred to as the weight of f , denoted by ω(f). Let f be a function from V (G) to {0, 1, 2, 3}. A vertex u is considered undefended with respect to f if f(u) = 0 and no adjacent vertex v satisfies f(v) ≥ 2. The function f is a generous Roman dominating function (GRD-function) if, for every vertex u with J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 4 of 16 f(u) = 0, there exists at least one adjacent vertex v with f(v) ≥ 2 such that the function g : V → {0, 1, 2, 3}, given by g(u) = α, g(v) = f(v) − α, where α ∈ {1, 2}, and g(w) = f(w), for all w ∈ V \ {u, v}, ensures that no vertex remains undefended. The weight of a GRD-function f is defined as f(V ) = ∑ u∈V f(u). The total sum over all vertices, f(V ), is referred to as the weight of f , denoted by ωg R(f), and the smallest possible weight of a GRD-function on G is referred to as the generous Roman domination number (GRD-number), denoted by γgR(G), as introduced by Benatallah, Blidia, and Ouldrabah [11]. For any GRD-function f of G, let Vi = {v ∈ V | f(v) = i}, where i ∈ {0, 1, 2, 3}. Since these four sets uniquely define f , we can represent f as (V0, V1, V2, V3). Furthermore, a γgR(G)-function is a GRD-function of G if ωg R(f) = γgR(G). The generous Roman domination is a variant of double Roman domination with less restriction. Importantly, if G1, G2, . . . , Gs are the components of G, then γgR(G) = ∑s i=1 γgR(Gi). Furthermore, if G1, G2, . . . , Gs represent the components of G with order at least 2, then sdγgR(G) = min{sdγgR(Gi) | 1 ≤ i ≤ s}. Consequently, we restrict our study to connected graphs of order at least two. 3. Preliminary Results We begin this section with some results that will be utilized later. Proposition 1. Let G be a connected graph of order n ≥ 2, and let F ⊆ E(G) be a subset of edges in G. If G′ is obtained by subdividing each edge in F , then γgR(G ′) ≥ γgR(G). Proof. The proof is carried out using induction on the size of |F |. First, assume that |F | = 1, and let e = uv be an element of F . Construct G′ from G by introducing a new vertex x to subdivide the edge e. Let f be a γgR(G ′)-function. Since f is a GRD-function on G′, it follows that f(u) + f(v) + f(x) ≥ 1. Let g : V (G) → {0, 1, 2, 3} be a function defined by g(u) = min{3, f(u)+f(x)}, and for all y ∈ V (G)\{u}, by g(y) = f(y). It is clear that g is a GRD-function on G, and ωg R(g) ≤ ωg R(f). Thus, we obtain γgR(G ′) ≥ γgR(G), which establishes the base case. Suppose the statement holds for any subset F of edges with 1 ≤ |F | < k. Let F ⊆ E(G) be a set of edges of size k, where F = {e1, e2, . . . , ek}, k ≥ 2. Let G′ be the graph obtained from G by subdividing all edges in F \{ek}. Then, let G′′ be the graph formed by further subdividing the edge ek in G′. By the induction hypothesis, we obtain γgR(G ′′) ≥ γgR(G ′) ≥ γgR(G), which completes the proof. Proposition 2. Let G be a connected graph of order n ≥ 2, and let F ⊆ E(G) be a set of edges in G. Let G′ be the graph obtained from G by subdividing the edges in F . If there exists a γgR(G ′)-function that assigns a value of 1 or 3 to at least one subdivision vertex, then γgR(G ′) > γgR(G). Proof. Let e = uv ∈ F and suppose that e is subdivided by introducing a new vertex x. Consider a function f that is a γgR(G ′)-function such that f(x) ∈ {1, 3}. Construct G′′ from G by subdividing all edges in F − {e}, noting that if F = {e}, then G = G′′. If f(x) = 1, then the restriction of f to G′′ acts as a GRD-function with a weight less J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 5 of 16 than ωg R(f), which leads to the inequality γgR(G ′) > γgR(G ′′) ≥ γgR(G). Next, assume f(x) = 3. Define a function g on G′′ by g(u) = min{3, f(u) + 1}, g(v) = min{3, f(v) + 1}, and g(z) = f(z), for all z ∈ V (G′′)− {u, v}. Then, g is a GRD-function with a weight less than ωg R(f), leading to the conclusion that γgR(G ′) > γgR(G ′′) ≥ γgR(G). Thus, the proof is complete. 4. Exact Values The exact values of the GRD-numbers of paths and cycles are determined in [11]. Proposition 3. [11] For n ≥ 1, γgR(Pn) = ⌈ 6n 7 ⌉ . Proposition 4. [11] For n ≥ 4, γgR(Cn) = ⌈ 6n 7 ⌉ . The following results directly follow from Propositions 3 and 4. Corollary 1. For n ≥ 2, sdγgR(Pn) = { 2 if n ≡ 6 (mod 7) 1 otherwise. Corollary 2. For n ≥ 4, sdγgR(Cn) = { 2 if n ≡ 6 (mod 7) 1 otherwise. Proposition 5. [11] γgR(G) = 2 if and only if G = Kn or G = K2. Proposition 6. [13] Let G be a connected graph of order n ≥ 3 different from Kn. Then γgR(G) = 3 if and only if ∆(G) = n− 1. As a direct consequence of Propositions 5 and 6, we obtain: Corollary 3. If G is a connected graph of order n ≥ 2 with γgR(G) ∈ {2, 3}, then sdγgR(G) = 1. Proof. If γgR(G) = 2, then according to Proposition 5, we have G = Kn. Furthermore, Proposition 5 implies that subdividing each edge of Kn increases the GRD-number, leading to sdγgR(G) = 1. Now, suppose γgR(G) = 3. Subdividing any edge e of G results in a new graph G′ with order n + 1 and maximum degree n − 1. By Proposition 6, it follows that γgR(G) > 3, which implies sdγgR(G) = 1. According to Corollary 3, the following result is obtained. Corollary 4. If G is a connected graph of order n ≥ 3 with sdγgR(G) ≥ 2, then γgR(G) ≥ 4. Next, we examine the ladder graphs of the form G = P2□Pn and demonstrate that sdγgR(P2□Pn) = 1 when n is odd or n = 4. We label the vertices of the i-th copy of P2 in the ladder P2□Pn as ui and vi, where i = 1, 2, . . . , n. The GRD-number of ladder graphs is established by Sheikholeslami et al. in [12]. Theorem 1. For n ≥ 1, γgR(P2□Pn) = ⌈3n+1 2 ⌉. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 6 of 16 Proposition 7. sdγgR(P2□P4) = 1. Proof. Suppose that G′ is obtained from G = P2□P4 by introducing a new vertex x through the subdivision of the edge u2v2. Let f be a γgR(G ′)-function. It is enough to establish that γgR(G′) > γgR(G). If f(x) ∈ {1, 3}, then by Proposition 2, we conclude that γgR(G ′) > γgR(G), as required. We now analyze two cases. Case 1. f(x) = 2. If f(u2) ≥ 1 (a similar argument applies for f(v2) ≥ 1), then updating v2 to the value min{3, f(v2) + 1} yields a GRD-function for G with a weight smaller than ωg R(f). Now, suppose f(u2) = f(v2) = 0. In this case, it is evident that f(u1)+f(v1) ≥ 2. Additionally, the function f restricted to G \ {u1, v1, u2, v2} serves as a GRD-function of P2□P4 with weight at most ωg R(f) − 4. Consequently, we obtain ωg R(f) ≥ 8 > 7 = γgR(G) (refer to Theorem 1). Case 2. f(x) = 0. Without loss of generality, we assume that u2 is a moving neighbor of x, implying that f(u2) ≥ 2. If f(u2) = 3, it is easy to verify that in order to protect the vertices u4, v1, v2, v3, v4, we must have ωg R(f) − f(u2) ≥ 5, which implies ωg R(f) ≥ 8 > γgR(G). Let f(u2) = 2 and define t = f(u1)+ f(u2)+ f(v1)+ f(v2). Since u2 is a moving neighbor of x, we must have either f(u1) ≥ 1 or f(v1) ≥ 2. First, assume that f(v2) ≥ 2. If f(v2) = 3, the result follows as before. If f(v2) = 2, then we have t ≥ 5. If t ≥ 6, in order to protect other vertices, we must have f(u3) + f(u4) + f(v3) + f(v4) ≥ 2, which leads to ωg R(f) ≥ 8 > γgR(G). If t = 5, then f(u1) = 1, f(v1) = 0, and v2 is a moving neighbor only for v1. To protect the vertices u4, v3, v4, we must have f(u3) + f(u4) + f(v3) + f(v4) ≥ 3, so ωg R(f) ≥ 8 > γgR(G). Assume now that f(v2) ≤ 1. In this case, u2 is a moving neighbor only for x. If f(v2) = 1, then to protect the vertices u1 and v1, we must have f(u1) + f(v1) ≥ 2. Reassigning u1 and v2 the values 0 and 2, respectively, yields a GRD-function of G with a weight less than γgR(G1), as desired (note that u2 will be a moving neighbor for u1 in the new assignment). Thus, we assume that f(v2) = 0. To protect u1 and v1, we must have f(u1)+ f(v1) ≥ 2. If f(u1)+ f(v1) ≥ 3, as before, we observe that γgR(G′) ≥ 8 > γgR(G). Let f(u1) + f(v1) = 2. Without loss of generality, assume that f(v1) = 2 and f(u1) = 0. Since u2 is a moving neighbor only for x, v1 becomes a moving neighbor of u1, leading to f(v3) ≥ 2. If f(v3) = 3, the result follows as before. Suppose f(v3) = 2. Now, to protect u4 and v4, we must have f(u4)+f(v4)+f(u3) ≥ 2, which again gives ωg R(f) ≥ 8 > γgR(G). This concludes the proof. Theorem 2. If n ≥ 1 is odd, then sdγgR(P2□Pn) = 1. Proof. Let G = P2□Pn. If n = 1, the result follows from Corollary 1. Assume n ≥ 3, and let G′ be the graph derived from G by subdividing the edge u1v1 with a new vertex x. Let f be a γgR(G ′)-function. If f(x) ∈ {1, 3}, then Proposition 2 implies that γgR(G ′) > γgR(G), so we have sdγgR(P2□Pn) = 1. Next, we consider two cases. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 7 of 16 Case 1. f(x) = 2. If f(u1) ≥ 1 (the case f(v1) ≥ 1 is similar), then by reassigning v1 the value min{3, f(v1)+ 1}, we obtain a GRD-function for G with a weight smaller than ωg R(f), which leads to sdγgR(P2□Pn) = 1. Now, assume that f(u1) = f(v1) = 0. If x is a moving neighbor of neither u1 nor v1, or if x is a moving neighbor of both u1 and v1, then we must have min{f(u2), f(v2)} ≥ 2. Reassigning v1 the value 1, u2 the value 2, v2 the value 0 and v3 the value min{3, f(v3)+2} gives a GRD-function for G with a weight smaller than ωg R(f), so we have sdγgR(P2□Pn) = 1. Next, assume that x is a moving neighbor of u1 but not of v1. In this case, v2 must be a moving neighbor of v1, so f(v2) ≥ 2. It is straightforward to observe that the function f restricted to G = P2□Pn \ {u1, v1} is a GRD-function with weight ωg R(f)− 2. By Theorem 1, we can deduce that ωg R(f) ≥ ωg R(f |V (G)) + 2 > ⌈ 3n+1 2 ⌉ , which results in sdγgR(P2□Pn) = 1. Case 2. f(x) = 0. To protect the vertex x, we may assume that f(u1) ≥ 2. If f(u1) = 3, then changing the value of u1 to 2 yields a GRD-function for P2□Pn with a weight of ωg R(f) − 1, which implies that sdγgR(P2□Pn) = 1. Therefore, we assume that f(u1) = 2. If f(v1) ≥ 2, then the function g, defined on (P2□Pn) \ {u1, v1, u2, v2} by g(u3) = min{3, f(u3) + f(u2)}, g(v3) = min{3, f(v3) + f(v2)}, g(z) = f(z) for the remaining vertices z, is a GRD- function with weight at most ωg R(f)−4. By Theorem 1, it follows that ωg R(f) ≥ ωg R(g)+4 ≥⌈ 3(n−2)+1 2 ⌉ + 4 > ⌈ 3n+1 2 ⌉ . If f(v1) = 1, then assigning the value 0 to v1 gives a GRD- function for P2□Pn with weight ωg R(f) − 1, leading to sdγgR(P2□Pn) = 1. Hence, we assume f(v1) = 0. In this case, v2 becomes the moving neighbor for v1, so we must have f(v2) ≥ 2. If f(v2) = 3 or if f(v2) = 2 and f(u2) ≥ 1, then changing the value of u1 to 1 results in a GRD-function for P2□Pn with weight ωg R(f)−1, and again sdγgR(P2□Pn) = 1. Let f(v2) = 2 and f(u2) = 0. From f(x) = 0, it follows that v2 is only a moving neighbor for v1 and that u3 is the only moving neighbor of u2, meaning f(u3) ≥ 2. If n = 3, then we have ωg R(f) ≥ 6 > ⌈ 3n+1 2 ⌉ . Therefore, let n ≥ 5. If f(u3) = 3 or f(v3) ≥ 1, then the function f restricted to (P2□Pn)−{v1, v2, u1, u2} is a GRD-function with weight ωg R(f)−4, and as before, we conclude that ωg R(f) > ⌈ 3n+1 2 ⌉ , leading to sdγgR(P2□Pn) = 1. Let f(u3) = 2 and f(v3) = 0. If f(u4) ≥ 1 or f(v4) ≥ 2, then the function f restricted to (P2□Pn) \ {v1, v2, u1, u2} is a GRD-function with weight ωg R(f) − 4, and as before, we conclude that ωg R(f) > ⌈ 3n+1 2 ⌉ , showing that sdγgR(P2□Pn) = 1. Hence, we assume f(u4) = 0 and f(v4) ≤ 1. If f(v4) = 1, then the function f restricted to (P2□Pn) \ {vi, ui | 1 ≤ i ≤ 4} is a GRD-function with weight ωg R(f) − 7, and using Theorem 1, we obtain ωg R(f) ≥ 7 + ⌈ 3(n−4)+1 2 ⌉ > ⌈ 3n+1 2 ⌉ , Thus sdγgR(P2□Pn) = 1. Thus, let f(v4) = 0. To pro- tect v4, we must have f(v5) ≥ 2. On the other hand, since u3 is the only moving neighbor of u2, we have f(u5) ≥ 2. If n = 5, then ωg R(f) ≥ 10 > ⌈ 3n+1 2 ⌉ . Hence, let n ≥ 7. By reassigning u7 the value min{3, f(u7) + f(u6)} and v7 the value min{3, f(v7) + f(v6)}, we obtain a GRD-function on (P2□Pn) \ {ui, vi | 1 ≤ i ≤ 6} with weight at most ωg R(f)− 10. By Theorem 1, we have ωg R(f) ≥ ωg R(g) + 10 ≥ ⌈ 3(n−6) 2 ⌉ +10 > ⌈ 3n+1 2 ⌉ . In conclusion, we obtain sdγgR(P2□Pn) = 1. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 8 of 16 5. Sufficient conditions on G having small sdγgR(G) In this section, we present several sufficient conditions for a graph G to have a small value of sdγgR(G). Proposition 8. If a connected graph G has a support vertex with at least three leaf neigh- bors, then sdγgR(G) = 1. Proof. If G is a star, then γgR(G) = 3, and by Corollary 3, we have sdγgR(G) = 1. Assume now that G is not a star. Let u, v and y be three leaves adjacent to a support vertex w, and let z be a non-leaf neighbor of w. Let G′ be the graph obtained by subdividing the edge uw with a new vertex x, and let f be a γgR(G ′)-function such that f(w) + f(z) is maximized. If f(w) = 3, it follows that f(v) = f(x) = f(y) = 0, and thus f(u) = 1. Reassigning the value 0 to u results in a GRD-function for G with weight smaller than ωg R(f); therefore, sdγgR(G) = 1. Now, assume that f(w) = 2. If f(v) ≥ 1 or f(y) ≥ 1, then by reassigning w the value 3 and v and y the value 0, we obtain a γgR(G ′)-function g such that g(w) + g(z) > f(w) + f(z), contradicting the maximality of f(w) + f(z). Therefore, we must have f(v) = f(y) = 0. However, in this case, the vertices v and y cannot be protected by f , which is a contradiction. If f(w) = 1, then f(v) = 1, and by reassigning w and v the values 2 and 0, respectively, we get a γgR(G)-function g such that g(w) + g(z) > f(w) + f(z), contradicting the maximality of f(w) + f(z). Thus, we assume that f(w) = 0. In this case, it follows that f(v) = f(y) = 1 and f(x) + f(u) = 2. By reassigning the values of w, u, v, and y to 3, 1, 0, and 0, respectively, we obtain a γgR(G)-function g such that g(w)+ g(z) > f(w)+ f(z), which contradicts the maximality of f(w) + f(z). Therefore, we conclude that sdγgR(G) = 1. Proposition 9. If G has a support vertex w with at least two leaf neighbors, then sdγgR(G) ≤ 2. Proof. If G is a star, the result follows directly from Corollary 3, and if w has at least three leaf neighbors, the conclusion holds by Proposition 8. Thus, assume that G is not a star and that w has exactly two leaf neighbors u and v. Let z be a non-leaf neighbor of w and let G′ be the graph derived from G by inserting new vertices x and y to subdivide the edges uw and vw, respectively. Let f be a γgR(G ′)-function that maximizes the sum f(w) + f(z). If f(w) = 3, then we have f(u) = f(v) = 1, and by reassigning f(u) = f(v) = 0, we obtain a GRD-function for G with weight less than ωg R(f), thus implying that sdγgR(G) ≤ 2. Assume now that f(w) = 2. To protect the vertices u, v, x and y, we must have f(u) + f(v) + f(x) + f(y) ≥ 3. By reassigning f(w) = 3, f(u) = 0, and f(v) = 0, we obtain a GRD-function for G with weight less than ωg R(f), again showing that sdγgR(G) ≤ 2. Finally, if f(w) ≤ 1, then to protect the vertices u, v, x and y, we must have f(u)+f(v)+f(x)+f(y) ≥ 4, and as before, we can conclude that sdγgR(G) ≤ 2. Proposition 10. For every tree T of order n ≥ 3, sdγgR(T ) ≤ 2. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 9 of 16 Proof. If T is a star or T contains a strong support vertex, then the result follows directly from Corollary 3 or Proposition 9. Now, assume that T is neither a star nor does it have a strong support vertex. Let x1x2 . . . xk represent a diametral path of T , and root T at vertex xk. According to our previous assumption, we have deg(x2) = 2. Next, assume that T ′ is obtained by subdividing the edges x1x2 and x2x3 with new vertices x and y, respectively, and let f be a γgR(T ′)-function. If f(x3) ≤ 1, then we must have f(x1) + f(x2) + f(x) + f(y) ≥ 4. By assigning the values 0 and 2 to x1 and x2, respectively, we obtain a GRD-function for T with weight less than ωg R(f). On the other hand, if f(x3) ≥ 2, then we have f(x1)+ f(x2)+ f(x)+ f(y) ≥ 3, and assigning the values 0 and 2 to x1 and x2, respectively, provides a GRD-function for T with weight less than ωg R(f). Therefore, we conclude that sdγgR(G) ≤ 2. Proposition 11. Let G be a connected graph of order n ≥ 3. If G contains a vertex v that lies in a triangle uvwu such that N(u) ∪N(w) ⊆ N [v], then sdγgR(G) ≤ 3. Proof. Let G1 be the graph obtained from G by subdividing the edges vu, vw, and uw with new vertices x1, x2 and x3, respectively. It suffices to prove that γgR(G1) > γgR(G). Let g represent a minimum GRD-function of G1. By Proposition 2, we know that for each i ∈ {1, 2, 3}, g(xi) /∈ {1, 3}. If two of the subdivision vertices, say x1 and x2, have positive weights under g, then it follows that g(x1) + g(x2) ≥ 4, and by reassigning the value 3 to v, we obtain a GRD-function on G with weight less than ωg R(g). If exactly one of the subdivision vertices, say x1, has a positive weight under g, then we must have g(x2) = g(x3) = 0 and g(u) + g(v) + g(w) ≥ 2. Again, by reassigning the value 3 to v, we get a GRD-function on G with weight less than ωg R(g), as the neighbors of u and z are also neighbors of v. Thus, we assume that g(x1) = g(x2) = g(x3) = 0. To protect the subdivision vertices, we must have g(v) + g(u) + g(w) ≥ 4. The function defined above is a GRD-function on G with weight less than ωg R(g). This completes the proof. 6. Bounds In this section, we derive several bounds for the generous Roman domination subdivi- sion number. Theorem 3. Let G be a connected graph. If x ∈ V (G) has degree at least two, then sdγgR(G) ≤ deg(x). Proof. Let s = deg(x) and N(x) = {x1, x2, . . . , xs}, and let G1 be the graph ob- tained from G by subdividing the edges xx1, xx2, . . . , xxs with new vertices y1, y2, . . . , ys, respectively. To prove the desired result, it is sufficient to show that γgR(G1) > γgR(G). Let f be a γgR(G1)-function. By Proposition 2, we may assume that f(yi) /∈ {1, 3} for all 1 ≤ i ≤ s. If ∑s i=1 f(yi) + f(x) ≥ 4, then assigning the value 3 to x results in a GRD-function on G with a weight smaller than ωg R(f), as desired. Therefore, we as- sume that ∑s i=1 f(yi) + f(x) ≤ 3. If f(x) ∈ {2, 3}, then we must have ∑s i=1 f(yi) = 0 J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 10 of 16 because f(yi) ̸= 1 for each i. In this case, reassigning the value 1 to x results in a GRD- function on G with weight less than ωg R(f), as desired. If f(x) = 1, then since s ≥ 2 and∑s i=1 f(yi)+f(x) ≤ 3, there must exist a vertex yi, say y1, such that f(y1) = 0. Therefore, x1 is a moving neighbor of y1, and thus f(x1) ≥ 2. By reassigning the value 0 to x, we obtain a GRD-function on G with weight less than ωg R(f). Finally, assume that f(x) = 0. In this case, one of the vertices yi, say y1, must be a moving neighbor of x, so f(y1) ≥ 2. From ∑s i=1 f(yi) + f(x) ≤ 3 and the assumption that f(yi) ̸= 1 for each i, we conclude that f(yi) = 0 for each i ∈ {2, . . . , s}. Therefore, f(xi) ≥ 2, and each xi is a moving neighbor of yi for i ∈ {2, . . . , s}. Now, by reassigning the value min{3, f(x1) + 1} to x1, we get a GRD-function on G with weight less than ωg R(f), completing the proof. As a result of Corollary 1 and Theorem 3, sdγgR(G) is well-defined for any connected graph G of order n ≥ 2. Moreover, we derive the following result. Corollary 5. If G is a connected graph with δ ≥ 2, then sdγgR(G) ≤ δ. Corollary 6. If G is a connected graph with δ = 1, then sdγgR(G) ≤ 3. Proof. If G is a star, then the result follows directly from Corollary 3. Suppose that G is not a star. Let v ∈ V (G) be a support vertex, and let v1 be a leaf adjacent to v. If deg(v) = 2, then, as shown in the proof of Proposition 10, we obtain sdγgR(G) ≤ 3. Thus, assume that deg(v) ≥ 3 and consider two neighbors v2, v3 ∈ N(v) \ {v1}. Let G1 be the graph obtained from G by subdividing the edges vv1, vv2, vv3 with new vertices x1, x2, x3, and let f be a γgR(G1)-function such that f(v) is maximized. By Proposition 2, we may assume that f(xi) /∈ {1, 3} for all i ∈ {1, 2, 3}. If f(v) + f(v1) + ∑3 i=1 f(xi) ≥ 4, then assigning the value 3 to v produces a GRD-function of G with a weight smaller than ωg R(f). Now, assume that f(v) + f(v1) + ∑3 i=1 f(xi) ≤ 3. This implies that f(v) ∈ {0, 1, 2}. If f(v) ≤ 1, then we must have f(x1) + f(v1) = 2 and f(x2) = f(x3) = 0. Since we previously assumed that f(yi) ̸= 1 for all i, the function g defined on G by g(v1) = 1, g(vi) = min{3, f(vi) + f(xi)} for i = 2, 3, and g(x) = f(x) for all other vertices x, is a GRD-function of G with a weight smaller than ωg R(f). Note that v2 is a moving neighbor of x2 under f , and hence, it is a moving neighbor of v under g. Next, assume that f(v) = 2. Since f(v)+f(v1)+ ∑3 i=1 f(xi) ≤ 3, it follows that f(v1) = 1 and f(xi) = 0 for all i ∈ {1, 2, 3}. In this case, v is the moving neighbor only of x1, and reassigning v1 the value 0 produces a GRD-function of G with a weight smaller than ωg R(f). Thus, we conclude that sdγgR(G) ≤ 3, completing the proof. Since every planar graph contains at least one vertex of degree at most five, the following result is an immediate consequence of Corollaries 5 and 6. Corollary 7. If G is a planar graph, then sdγgR(G) ≤ 5. In the following, we establish an upper bound on the generous Roman domination number of a graph based on the number of vertices that are at a distance of 2 from a given vertex. For a vertex x ∈ V (G), let N2(x) denote the set of vertices in G that are exactly two edges away from x, and define d2(x) = |N2(x)|. In this setting, we introduce δ2(G) = min{d2(x) | x ∈ V (G) and deg(x) ≥ 2}. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 11 of 16 To establish this result, we begin with a few lemmas. The proof of the following lemmas are essentially similar to the proof of corresponding lemmas in [17]. Lemma 1. Let G be a connected graph of order n ≥ 3. If G contains a vertex v1 that is part of a triangle v1v2v3v1 such that N(v2) ⊆ N [v1] and N(v3)−N [v1] ̸= ∅, then sdγgR(G) ≤ 3 + |N(v3)−N [v1]|. Proof. Let w1, w2, . . . , wk be the neighbors of v3 in V (G)−N [v1], and let G1 be obtained from G by subdividing the edges v1v2, v1v3 and v2v3 with vertices x, y and z, respectively, and the edge v3wi with vertex xi for each 1 ≤ i ≤ k. Assume that f is a γgR(G1)-function. By Proposition 2, we may assume that {1, 3} ∩ {f(x), f(y), f(z), f(x1), . . . , f(xk)} = ∅. Similar to the proof of Proposition 11, we can observe that f(x) + f(y) + f(z) + f(v1) + f(v2) + f(v3) ≥ 4. Define a function g : V (G) → {0, 1, 2, 3} by g(v1) = 3, g(v2) = g(v3) = 0, g(wi) = min{3, f(wi) + f(xi)} for each 1 ≤ i ≤ k, and g(t) = f(t) for all t ∈ V (G) \ {v1, v2, v3, wi | 1 ≤ i ≤ k}. It is straightforward to verify that g is a GRD-function of G with a weight smaller than γgR(G1), thereby completing the proof. Lemma 2. Let G be a connected graph of order n ≥ 3, and let x be a vertex of degree at least 2 in G that satisfies the following conditions: (i) N(y) \N [x] ̸= ∅ for each y ∈ N(x), (ii) there exist vertices a, b ∈ N(x) such that (N(a) ∩N(b)) \N [x] = ∅. Then, sdγgR(G) ≤ 3 + |N2(x)|. Proof. Let deg(x) = t and N(x) = {x1, x2, . . . , xt}. We assume, without loss of generality, that a = x1 and b = x2. Additionally, we assume that the pair a, b is selected first among the adjacent vertices in N(x). Therefore, if ab ∈ E(G), then x must be part of the triangle xx1x2x. Moreover, define S = {x1, x2, . . . , xs} as one of the largest subsets of N(x) containing x1 and x2, where every pair of vertices a, b in S satisfies Condition (ii). According to item (i), for each i ∈ {1, 2, . . . , s}, we define N(xi)\N [x] = {xi1 , xi2 , . . . , xili}. Next, we define G1 as the graph obtained from G by subdividing the edges xx1 and xx2 with new vertices u1 and u2, respectively. For each i ∈ {1, 2, . . . , s}, we subdivide each edge xixij , where 1 ≤ j ≤ li, by introducing a new vertex xij . We define Wi = {xij | 1 ≤ j ≤ li} and W = ⋃ 1≤i≤sWi. Furthermore, if x1 and x2 are adjacent, we also subdivide the edge x1x2 by introducing a new vertex u3. Finally, let f be a γgR(G1)-function. By Proposition 2, we can assume that no subdivision vertex is assigned the values 1 or 3 under f . First let x1x2 ∈ E(G). Similar as in the proof of Proposition 11, we can see that f(x)+ f(x1)+ f(x2)+ f(u1)+ f(u2)+ f(u3) ≥ 4. By reassigning x the value 3, x1, x2 the value 0, and xij the value min{3, f(xij )+f(xij )} for all i and j, we obtain a GRD-function of G of weight less than γgR(G1) as desired. Assume now that x1x2 ̸∈ E(G). By the choice of x1, x2, we deduce that S is indepen- dent. To protect the vertices u1 and u2, we must have f(x)+f(x1)+f(x2)+f(u1)+f(u2) ≥ J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 12 of 16 3. If f(x) + f(u1) + f(u2) + ∑s i=1 f(xi) ≥ 4, then reassigning x the value 3, xi the value 0 for all i ∈ {1, 2, . . . , s}, and xij the value min{3, f(xij ) + f(xij )} for all i, j, pro- vides a GRD-function of G of weight less than γgR(G1). Thus, we may assume that f(x) + f(u1) + f(u2) + ∑s i=1 f(xi) ≤ 3. If f(x) ∈ {0, 1, 2}, then to protect the vertices u1, u2, we must have f(x)+f(u1)+f(u2)+ ∑s i=1 f(xi) ≥ f(x1)+f(x2)+f(u1)+f(u2) ≥ 4 contradicting our assumption. Thus, f(x) = 3 and so f(u1) + f(u2) + ∑s i=1 f(xi) = 0. If ∑li j=1 f(x ij ) ≥ 4 for some 1 ≤ i ≤ s, say i = 1, then reassigning x1 the value 3 and xij the value min{3, f(xij ) + f(xij )} for all i ∈ {2, . . . , s} and all j ∈ {1, 2, . . . , li}, provides a GRD-function of G of weight less than ωg R(f) = γgR(G1). Hence, suppose that∑li j=1 f(x ij ) ≤ 3, for each i ∈ {1, 2, . . . , s}. First, consider the case where there exist some i ∈ {1, 2, . . . , s} and some j ∈ {1, 2, . . . , li} such that f(xij ) = 2. Assume, without loss of generality, that i = j = 1. Then, by updating x11 to take the value min{3, 1+f(x11)} and redefining xij as min{3, f(xij ) + f(xij )}, for ij ̸= 11, we obtain a GRD-function of G with weight strictly less than γgR(G1). Thus, we may assume that f(xij ) = 0 for all i and j. This directly implies that f(xij ) = 2 for every i and j. Clearly, reassigning x the value 2 results in a GRD-function of G with weight strictly less than γgR(G1). All in all, we see that the graph G has a GRD-function of weight less than γgR(G1). Moreover, since G1 is obtained by inserting at most 3 + |W | ≤ 3 + |N2(x)| new vertices, we obtain sdγgR(G) ≤ 3 + |N2(x)|. This completes the proof. Lemma 3. Let G be a connected graph of order n ≥ 3 and v be a vertex of degree at least 2 of G satisfying the following conditions: (i) N(y) \N [v] ̸= ∅ for each y ∈ N(v), (ii) for every pair of vertices a, b in N(v), (N(a) ∩N(b)) \N [v] ̸= ∅. Then sdγgR(G) ≤ 3 + |N2(v)|. Proof. If deg(v) ≤ 3+ |N2(v)|, then the result follows from Theorem 3. Henceforth, we assume that deg(v) ≥ 4 + |N2(v)|. Let N(v) = {v1, v2, . . . , vk} and K = N(v1) \ N [v] = {w1, w2, . . . , wp}. By (ii) each vertex y ∈ N(v) \ {v1} has a neighbor in K. Let S be one of the largest subsets of N(v) \ {v1} such that for every subset S1 ⊆ S, the inequality |N(S1) \ (N [v] ∪ K)| ≥ |S1| holds. By the choice of S, we have |N2(v)| ≥ |K| + |S|. Furthermore, every vertex u in U = N(v)\(S∪{v1}) has at least one neighbor in K, and the set N(u)\N [v] satisfies N(u)\N [v] ⊆ K∪N(S). Additionally, the set K dominates N(v) (as stated in item (ii)). From the inequality 4+|K|+|S| ≤ 4+|N2(v)| ≤ deg(v) = |S|+1+|U |, we conclude that |U | ≥ 4. If S ̸= ∅, then, without loss of generality, we may assume S = {v2, v3, . . . , vs}. Assume that G1 is obtained from G by subdividing the edges v1wj with new vertices yj for all j ∈ {1, . . . , p}, and subdividing the edges vvi with new vertices xi for 1 ≤ i ≤ s+2 when S ̸= ∅, or for 1 ≤ i ≤ 3 when S = ∅. Consequently, the number of subdivided edges is |K| + |S| + 3. It suffices to demonstrate that γgR(G1) > γgR(G). Let f be a γgR(G1)-function. By Proposition 3, we can assume that f(z) /∈ {1, 3} for all subdivision vertices z. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 13 of 16 If f(v)+ f(v1)+ ∑s+2 i=1 f(xi) ≥ 4, then by reassigning v the value 3, v1 the value 0, and wj the value min{3, f(wj) + f(yj)} for all j ∈ {1, . . . , p}, we obtain a GRD-function of G with weight less than γgR(G1). Thus, we assume that f(v) + f(v1) + s+2∑ i=1 f(xi) ≤ 3. (1) We now distinguish four different situations. Case 1. f(v) = 3. From equation (1), we have f(v1) = f(x1) = · · · = f(xs+2) = 0. If ∑p j=1 f(yj) ≥ 4, then by assigning the value 3 to v1, we can obtain a GRD-function for G with weight less than γgR(G1). Therefore, we assume that ∑p j=1 f(yj) ≤ 3. Based on our earlier assumption, we also have ∑p i=1 f(yi) ≤ 2. If there exists some j ∈ {1, 2, . . . , p}, say j = 1, such that f(y1) = 2, then it follows that f(yj) = 0 for all j ∈ {2, . . . , p}. In this case, reassigning w1 the value min{3, 1+f(w1)} provides a GRD-function for G with weight less than γgR(G1). Hence, we assume that ∑p j=1 f(yj) = 0. To protect the vertices yj , the vertex wj must be the moving neighbor of yj , implying that f(wj) = 2 for each j ∈ {1, 2, . . . , p}. Then, by reassigning v the value 2, we obtain a GRD-function for G, where every vertex of U has a neighbor in K, with weight less than γgR(G1). Case 2. f(v) = 2. From equation (1) and our previous assumption, it follows that f(x1) = · · · = f(xs+2) = 0. If f(v1) = 1, then by reassigning the value 0 to v1 and the value min{3, f(wi) + f(yi)} to wi for all i ∈ {1, 2, . . . , p}, we obtain a GRD-function for G with weight less than γgR(G1) (note that v is a moving neighbor of v1). Now, suppose f(v1) = 0. This implies that v is a moving neighbor only for x1, and thus, vi is the moving neighbor of xi in order to protect xi. Consequently, we have f(vi) ≥ 2 for all i ∈ {2, 3, . . . , s + 2}. If ∑p j=1 f(yj) ≥ 4, then reassigning the value 3 to v1 provides a GRD-function for G with weight less than γgR(G1). Hence, we assume that ∑p j=1 f(yj) ≤ 3. From our earlier assumption, it follows that either f(yj) = 0 for all j ∈ {1, 2, . . . , p} or f(yj) = 2 for exactly one j. In the first case, we have f(wj) ≥ 2, and wj is the moving neighbor only for yj if f(wj) = 2. Reassigning the value 1 to v then provides a GRD-function for G with weight less than γgR(G1). In the second case, reassigning the value 1 to wj and the value min{3, f(wi)+1} to wi for i ̸= j provides a GRD-function for G with weight less than γgR(G1). Case 3. f(v) = 1. To protect x1, it is required that f(x1) + f(v1) ≥ 2. From our previous assumption and equation (1), it follows that ∑s+2 i=1 f(xi) = 0. If f(v1) = 2, then f(x1) = 0, which means that v1 is a moving neighbor only for x1. By assigning the value 0 to v and the value min{3, f(wi) + f(yi)} to wi for all i ∈ {1, 2, . . . , p}, we obtain a GRD-function for G with weight less than γgR(G1). If f(x1) = 2, then by reassigning the value 1 to v1 and the value min{3, f(wi) + f(yi)} to wi for all i ∈ {1, 2, . . . , p}, we obtain a GRD-function for G with weight less than γgR(G1). Case 4. f(v) = 0. In order to protect x1, we require that f(x1) + f(v1) ≥ 2, and from equation (1), we know J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 14 of 16 that ∑s+2 i=2 f(xi) = 0. This implies that f(vi) ≥ 2, and that vi is a moving neighbor only for xi for all i ∈ {2, 3, . . . , s + 2}. If ∑p j=1 f(yj) ≥ 2, then by assigning the value 3 to v1, we can obtain a GRD-function for G with weight less than γgR(G1). Therefore, we assume that ∑p j=1 f(yj) = 0. Since v1 is a moving neighbor only for x1, in order to protect yi, we require that f(wi) ≥ 2, and that wi is a moving neighbor of yi. Finally, by assigning the value 1 to v1, we obtain a GRD-function for G with weight less than γgR(G1). This completes the proof. We are now prepared to present our main result. Theorem 4. Let G be a connected graph of order n ≥ 3. Then sdγgR(G) ≤ 3 + min{d2(x) | x ∈ V and deg(x) ≥ 2}. Proof. If G is a star graph K1,n−1, then γgR(G) = 3. Moreover, by Corollary 3, we obtain sdγgR(G) = 1, and thus the result holds. Therefore, from this point onward, we assume that G ̸= K1,n−1. If G contains a leaf, then by Corollary 6, the result also holds. Next, let G be a graph such that δ(G) ≥ 2. Using Proposition 11 and Lemmas 1, 2 and 3, we conclude that sdγgR(G) ≤ 3 + min{d2(x) | x ∈ V and deg(x) ≥ 2}. Let δ2(G) = min{d2(v) | v ∈ V (G) and deg(v) ≥ 2}, and note that for each vertex v with degree ∆, we have δ2(G) ≤ |N2(v)| ≤ n − ∆ − 1. The next two Corollaries follow directly from Theorem 4. Corollary 8. Let G be a connected graph with δ(G) ≥ 2, sdγgR(G) ≤ 3 + δ2(G). Corollary 9. Let G be a connected graph of order n ≥ 3. Then sdγgR(G) ≤ n−∆+ 2. Applying Corollaries 5, 6, and 9, we derive the following result. Proposition 12. Let G be a connected graph of order n ≥ 3. Then sdγgR(G) ≤ n 2 + 1. Conclusion: This study introduced and explored the concept of the generous Roman domination subdivision number in graphs, focusing on its effect on the generous Roman domination number. By analyzing how edge subdivisions influence this parameter, we established upper bounds and exact values for specific graph families. These findings con- tributed to a deeper understanding of domination parameters under graph modifications. Future research may focus on characterizing more graph classes where the exact gener- ous Roman domination subdivision number can be determined. Additionally, algorithmic approaches to compute this parameter efficiently in general graphs can be developed to support applications in network defense and resource allocation. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6429 15 of 16 Acknowledgements We gratefully acknowledge the reviewers for their insightful comments and suggestions, which have greatly enhanced the quality of this paper. 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