EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5917 ISSN 1307-5543 – ejpam.com Published by New York Business Global Total Safe Domination on Some Known Families of Graphs Wendel Glenn C. Jumalon1,∗, Isagani S. Cabahug, Jr.1 1 Department of Mathematics, College of Arts and Sciences, Central Mindanao University, Musuan, Maramag, Bukidnon, 8714 Philippines Abstract. A total dominating set in a graph G is a nonempty set S ⊆ V (G) such that every vertex v ∈ V (G), including those in S, is adjacent to at least one vertex in S. A safe dominating set in G is a nonempty set S ⊆ V (G) that is a dominating set, and for every component A of the induced subgraph G[S] and every component B of the induced subgraph G[V (G) ∖ S], with A adjacent to B, it holds that |V (A)| ≥ |V (B)|. This study introduces the concept of total safe domination in graphs which combines total domination and safe domination. Total safe domination ensures total accessibility and structural resilience. This paper provides characterization of total safe dominating sets for some well-known graph families, including: path, cycle, complete, complete bipartite, friendship, sunlet and helm graphs. It also presents the total safe domination number for each of these graph families. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Total domination, safe domination, safe set, total safe domination 1. Introduction The study of domination has grown to be one of the most rapidly expanding areas in graph theory. Domination in graphs has wide-ranging applications in network design, resource allocation, and security management. Traditional domination focuses on identify- ing subsets of vertices that can control or influence the entire graph. However, as systems grow more complex, there is a growing need for more refined approaches. This has led to the emergence of more variations of domination, such as total domination and safe domination, which offer enhanced perspectives and more effective solutions to real-world problems. The formal study of dominating sets in graph theory began in the 1960s. In 1977, Cockayne published a comprehensive survey of the results on dominating sets known at ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5917 Email addresses: wendeljumalon@gmail.com (W. G. Jumalon), isaganicabahugjr@cmu.edu.ph (I. Cabahug, Jr.) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 2 of 12 that time. This survey introduced the notation γ(G) for the domination number of a graph G, which has since become widely adopted [1]. In 1980, Cockayne, Dawes and Hedetniemi published the first paper on total domina- tion in graphs. They obtained results concerning the total domination number of a graph and the total domatic number of a graph (the largest order of a partition of a graph into total dominating sets) [2]. In recent decades, a lot of variations in total domination have been studied. Some of the researches related to this study are given as follows: In 2007, Lam and Wei published their paper entitled “On the Total Domination Num- ber of Graphs”, where they developed theorems for the bounds of the total domination number of connected graphs of order at least 3, and for connected graphs with degree at least 2 [3]. In 2011, Go and Canoy determined the domination, total domination, and secure total domination numbers in the corona and join of graphs [4]. The study was continued by Eballe and Miranda in 2021, where they presented the domination defect for the join and corona of graphs [5]. In 2021, Jose Sigarreta published his study entitled “Total Domination on Some Graph Operators”. In his paper, he introduced bounds for the exact value of the total domination number of some graph operators using some parameters in the original graph [6]. The study published by Klostermeyer in 2008 with the title “Secure Domination and Secure Total Domination in Graphs” has a related title but has a totally different concept. The paper states that a secure (total) dominating set of a graph G = (V,E) is as a (total) dominating set X ⊆ V with the property that for each u ∈ V − X, there exists x ∈ X adjacent to u such that (X − {x}) ∪ {u} is a (total) dominating set [7]. The study entitled “Defensive Alliances in Graphs” by Gaikwad and Maity which was published in 2022 also seems similar to this study but the concept is also different. As defined, a set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbours are in S [8]. In 2024, Chatterjee, Jent, Osborn and Zhang published their paper “Proper Total Domination in Graphs”. The paper states that a total dominating set S in a graph G is called a proper total dominating set if σs(u) ̸= σs(v) for every two adjacent vertices u and v of G [9]. From available resources and online publications, the authors found none that is of exact same concept as this study. In particular, there is no published study about total domination which incorporates the concept of safe set or safe domination. The idea of a safe set in graphs was introduced by Fujita, MacGillivray and Sakuma in 2016 [10]. Their work was motivated by applications related to facility location problems, where the goal is to find a “safe” subset of nodes for placing facilities in a network. The paper states that a safe set is a set in which every component of the set has order at least as large as the order of any adjacent component of the complement set. The concept of safe domination in graphs was introduced by Griño, Maceren, and Cabahug in 2023 [11]. In their paper, they defined a safe dominating set as a set that is both dominating and safe. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 3 of 12 This study introduces the concept of total safe dominating set in a graph. This guar- antees total coverage and structural resilience. The concept may have unique potential applications in areas such as network coverage, facility location and defense infrastructure. This study is limited to undirected graphs that are nontrivial, connected and simple. 2. Terminology and Notation A graph G = (V (G), E(G)) is a finite nonempty set V (G) of objects called vertices together with a possibly empty set E(G) of 2-element subsets of V (G) called edges. A graphH is a subgraph of a graph G if V (H) ⊆ V (G) and E(H) ⊆ E(G). IfH is a subgraph of a graph G where H ≇ G, then H is a proper subgraph of G. If S is a nonempty subset of V (G), then the induced subgraph G[S] of S in G, is the graph whose vertex set is S and whose edge set consists of all of the edges in V (E) that have both endpoints in S [12]. A component of a graph G is defined as a maximal subgraph in which every pair of vertices is connected by a path. The connected graph and the trivial graph both have one component. A subgraph induced by a subset of V (G) may have more than one component. A component A of the induced subgraph G[S] is said to be adjacent to a component B of the induced subgraph G[V (G)∖S], if there is at least one edge joining a vertex u ∈ V (A) to a vertex v ∈ V (B) [12]. For an integer n ≥ 1, the path Pn is a graph of order n and size n− 1 whose vertices can be labeled by v1, v2, . . . , vn and whose edges are vivi+1 for i = 1, 2, . . . , n− 1 [13]. For n ≥ 3, the cycle Cn is a graph of order n and size n whose vertices can be labeled by v1, v2, . . . , vn and whose edges are v1v2, v2v3, ..., vn−1vn, vnv1 [13]. For n ≥ 2, the complete graph Kn is a graph of order n and size n(n−1) 2 whose vertices can be labeled by v1, v2, . . . , vn and whose edges are represented as vivj for all pairs of vertices where 1 ≤ i < j ≤ n [12]. A graph G is a complete bipartite graph if V (G) can be partitioned into two sets U and W (called partite sets) such that uw is an edge of G if and only if u ∈ U and w ∈ W . If |U | = m and |W | = n, then the complete bipartite graph is denoted by Km,n [12]. The friendship graph Fn, also called The Dutch windmill graph D3 (n), is the graph with 2n + 1 vertices obtained by taking n copies of the cycle graph C3 with a common vertex, called the central vertex [14]. The sunlet graph Sn, also called as n-sunlet graph, is the graph with 2n vertices ob- tained by attaching a pendant edge at each vertex of a cycle Cn [15]. The helm graph Hn is the graph with 2n+1 vertices obtained by adjoining a pendant edge at each node of the cycle, whose vertices are adjacent to a common vertex, called the hub [15]. For the main concepts included in this study, consider the following definitions: Let G be a simple connected nontrivial graph. A nonempty set S ⊆ V (G) is a domi- nating set if every vertex in V (G)∖S is adjacent to at least one vertex in S. The minimum cardinality of a dominating set in G is called domination number of G, denoted by γ(G) [16]. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 4 of 12 A nonempty set S ⊆ V (G) of vertices is a safe set if for every component A of G[S] and every component B of G[V (G) ∖ S] adjacent to A, it holds that |V (A)| ≥ |V (B)|. The minimum cardinality of a safe set in G is called safe number of G, denoted by s(G) [17]. A nonempty subset S of V (G) is a safe dominating set if and only if it is both a dominating set and a safe set. The minimum cardinality of a safe dominating set in G is called safe domination number of G, denoted by γs(G) [11]. A total dominating set S in a graph G is a nonempty subset of V (G) such that every vertex in V (G), including those in S, is adjacent to at least one vertex in S. The minimum cardinality of a total dominating set in G is called total domination number of G, denoted by γt(G) [2]. 3. Results Definition 1. A nonempty subset S of V (G) is a total safe dominating set in G if it is both a total dominating set and a safe dominating set in G. The minimum cardinality of a total safe dominating set in G is called total safe domination number of G, denoted by γts(G). Example 1. Consider Figure 1 that shows a graph G together with different dominating sets as in G1, G2, and G3. In G1, S1 = {v3, v4} shows a total safe dominating set in G since it is a total dominating set and the order of the component of G[S1] is at least as large as the order of any adjacent component of G[V (G) ∖ S1]. In G2, S2 = {v3, v5} is not a total safe dominating set. It is a total dominating set but not a safe dominating set since G[V (G) ∖ S2] has a component with order 3 which is greater than the order of the component of G[S2]. Finally, in G3, S3 = {v1, v2, v4, v5} is not a total safe dominating set. It is a safe dominating set but not a total dominating set since v2 is isolated. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 5 of 12 v1 v2 v3 v4 v5 v6 v1 v2 v3 v4 v5 v6 v1 v2 v3 v4 v5 v6 v1 v2 v3 v4 v5 v6 G : G1 : G2 : G3 : Figure 1: A graph G with examples of dominating sets, where G1 shows a total safe dominating set. 3.1. Some Realization Results on Total Safe Domination By assumption, G is a nontrivial connected graph. Hence, the following remark follows. Remark 1. Let G be a nontrivial connected graph. Then V (G) is a total safe dominating set. By definition, the following remarks also follow. Remark 2. Every total safe dominating set of a graph G is a safe dominating set. Thus, γs(G) ≤ γts(G). Remark 3. Every total safe dominating set of a graph G is a total dominating set. Hence, γt(G) ≤ γts(G). Theorem 1. Let G be a nontrivial connected graph and S ⊊ V (G) be a total dominating set in G. Then γts(G) = 2 if and only if γt(G) = 2 and for every component B of G[V (G)∖ S], |V (B)| ≤ 2. Proof. Assume that G is a nontrivial connected graph and S ⊊ V (G) is a total dominating set in G. Suppose that γts(G) = 2. Clearly, γt(G) = 2. Also, for every component B of G[V (G)∖ S], |V (G[S])| ≥ |V (B)|. Thus, |V (B)| ≤ 2. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 6 of 12 For the converse, suppose that γt(G) = 2 and for every component B of G[V (G)∖ S], |V (B)| ≤ 2. Then, γts(G) ≥ 2. Since γt(G) = 2, there exists a total dominating set S with |S| = 2. Since |V (B)| ≤ 2 for every component B of G[V (G) ∖ S], and since |S| = 2, it follows that S is a safe dominating set in G. So, S is a total safe dominating set in G. This implies that γts(G) ≤ |S| = 2. Hence, γts(G) = 2. In the following results, we show by construction the existence of a graph where the parameters γts(G) and γt(G) are equal, strictly unequal and whose difference can be made arbitrarily large. Theorem 2. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a connected graph G such that γt(G) = a and γts(G) = b. Proof. Let a, b ∈ Z+ and consider the following cases: Case 1: a = b Consider graph G in Figure 2. Take a ≥ m. Clearly, P = {xi : i = 1, 2, ..., a} is both a minimum total dominating set and a minimum total safe dominating set. Thus, 2 ≤ γt(G) = |P | = a = b = γts(G). x1 x2 x3 xa−1 xa y1 y2 y3 ym−1 ym Figure 2: A graph G with γt(G) = a = b = γts(G), where a ≥ m. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 7 of 12 Case 2: a < b Consider graph G in Figure 3. Take a < m. Let b = |P ∪ Q|, where P = {xi : i = 1, 2, ..., a} and Q = {yj : j = 1, 2, ..., y⌈m−a 2 ⌉}. Then, γt(G) = |P | = a and γts(G) = b = |P ∪Q| = a+ ⌈ m−a 2 ⌉ = ⌈ a+m 2 ⌉ . Hence, γt(G) = a < b = γts(G). x1 x2 x3 xa−1 xa y1 y2 y3 y⌈m−a 2 ⌉ y⌈m−a 2 ⌉+1 ym−1 ym Figure 3: A graph G with γt(G) = a < b = γts(G), where b = ⌈ a+m 2 ⌉ . This completes the proof. From Theorem 2, the following Corollary immediately follows. Corollary 1. For each positive integer n, there exists a connected graph G such that γts(G)−γt(G) = n, that is, the difference between γts(G) and γt(G) can be made arbitrarily large. 3.2. Results on Total Safe Domination on Some Known Graph Families The following results give the characterization of total safe dominating sets on some well-known graph families. Theorem 3. Let G be a nontrivial connected graph of order n such that ∆(G) = 2. Then a nonempty set S ⊊ V (G) is a total safe dominating set in G if and only if every component of G[S] is a Pk, 2 ≤ k ≤ n− 1, and every component of G[V (G)∖ S] is a trivial graph or a P2 with no end vertex in G. Proof. Assume that S is a total safe dominating set in G. Suppose that there exists a component of G[S] that is a P1, or there exists a component of G[V (G)∖ S] that is not a trivial graph and a P2 with an end vertex in G. The first part implies that G[S] contains an isolated vertex and the latter implies that S is not a dominating set. Both implications contradict the assumption of S. Thus, every component of G[S] is a Pk, 2 ≤ k ≤ n − 1, and every component of G[V (G)∖ S] is a trivial graph or a P2 with no end vertex in G. For the converse, suppose that S ⊊ V (G) such that every component of G[S] is a Pk, 2 ≤ k ≤ n− 1, and every component of G[V (G)∖S] is a trivial graph or a P2 with no end vertex in G. Clearyly, G[S] has no isolated vertex and S is a dominating set. If A and B are components of G[S] and G[V (G)∖ S], respectively, then |V (A)| ≥ |V (B)|. Therefore, S is a total safe dominating set in G. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 8 of 12 Corollary 2. For n ≥ 3, γts(Pn) = γts(Cn) =  n 2 , if n ≡ 0(mod 4) n+1 2 , if n ≡ 1 or 3(mod 4) n+2 2 , if n ≡ 2(mod 4). Proof. Let Pn = [v1, v2, ..., vn] and S ⊊ V (Pn). Consider the following cases: Case 1: n ≡ 0(mod 4) Choose S = {v2, v3, v6, v7, ..., vn−2, vn−1}. Then, |S| = n 2 . By Theorem 3, S is a total safe dominating set in Pn. Thus, γts(Pn) ≤ |S| = n 2 . By vertex selection and labeling, we cannot find a total safe dominating set in Pn with cardinality less than the cardinality of S. Therefore, γts(Pn) = |S| = n 2 . Case 2: n ≡ 1(mod 4) Choose S = {v2, v3, v4, v7, v8, ..., vn−2, vn−1}. Then, |S| = n+1 2 . By Theorem 3, S is a total safe dominating set in Pn. By the same argument as in Case 1, we have γts(Pn) = n+1 2 . Case 3: n ≡ 2(mod 4) Choose S = {v2, v3, v5, v6, ..., vn−1, vn}. By similar argument as in the previous cases, we have γts(Pn) = n+2 2 . Case 4: n ≡ 3(mod 4) Choose S = {v2, v3, v6, v7, ..., vn−1, vn}. So, |S| = n+1 2 . Again, by Theorem 3, S is a total safe dominating set in Pn. Hence, γts(Pn) ≤ |S| = n+1 2 . By vertex selection and labeling, we cannot find a total safe dominating set in Pn with cardinality less than the cardinality of S. Thus, γts(Pn) = |S| = n+1 2 . Similarly, for Cn = [v1, v2, ..., vn, v1] and S ⊊ V (Cn), choose the same vertex labeling for S in each case above. By the same arguments as above, we have γts(Cn) =  n 2 , if n ≡ 0(mod 4) n+1 2 , if n ≡ 1 or 3(mod 4) n+2 2 , if n ≡ 2(mod 4). Theorem 4. Let Kn be a complete graph of order n ≥ 3. Then a nonempty set S ⊊ V (Kn) is a total safe dominating set in Kn if and only if |S| ≥ ⌈ n 2 ⌉ . Proof. Assume that S is a total safe dominating set in Kn. Now, suppose that |S| < ⌈ n 2 ⌉ . Then we have |V (Kn) ∖ S| ≥ ⌈ n 2 ⌉ . Since Kn[S] and Kn[V (Kn) ∖ S] both consist of a single component, this implies that S is not a safe set in Kn. This contradicts our assumption that S is a total safe dominating set. Thus, |S| ≥ ⌈ n 2 ⌉ . W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 9 of 12 Conversely, suppose that |S| ≥ ⌈ n 2 ⌉ . Since any vertex in V (Kn) is adjacent to a vertex in S, S is a total dominating set. Additionally, since |S| ≥ ⌈ n 2 ⌉ , we have enough vertices in S to satisfy the safe set condition. Hence, S is a total safe dominating set in Kn. Corollary 3. For Kn, n ≥ 3, γts(Kn) = ⌈n 2 ⌉ . Proof. By Theorem 4, the lower bound for a total safe dominating set in Kn is ⌈ n 2 ⌉ . Hence, γts(Kn) = ⌈ n 2 ⌉ . Theorem 5. Let Km,n be a complete bipartite graph with partite sets U and W such that |U | = m, |W | = n, and m + n ≥ 3. Then a nonempty set S ⊊ V (Km,n) is a total safe dominating set in Km,n if and only if the following hold: (i) S = S1 ∪ S2 such that ∅ ≠ S1 ⊆ U and ∅ ≠ S2 ⊆ W ; (ii) |S| ≥ ⌈ m+n 2 ⌉ . Proof. Assume that S is a total safe dominating set in Km,n. Suppose that S ⊆ U or S ⊆ W . In either case, Km,n[S] is an empty graph and so S is not a total safe dominating set in Km,n. This is a contradiction to the assumption of S. So, S cannot be contained entirely in either U or W . Thus, S = S1 ∪ S2 such that ∅ ≠ S1 ⊆ U and ∅ ≠ S2 ⊆ W . Now, suppose that |S| < ⌈ m+n 2 ⌉ . Then, |V (Km,n) ∖ S| ≥ ⌈ m+n 2 ⌉ . Since Km,n[S] and Km,n[V (Km,n) ∖ S] both consist of a single component, this means that S is not a safe set in Km,n, a contradiction to the assumption of S. Therefore, |S| ≥ ⌈ m+n 2 ⌉ . For the converse, suppose that S = S1 ∪ S2 such that ∅ ≠ S1 ⊆ U and ∅ ≠ S2 ⊆ W , and |S| ≥ ⌈ m+n 2 ⌉ . The first part guarantees total domination of S in Km,n, and together with the latter, it follows that S is a safe dominating set in Km,n. Thus, S is a total safe dominating set in Km,n. Corollary 4. For a complete bipartite graph Km,n such that m+ n ≥ 3, γts(Km,n) = ⌈ m+n 2 ⌉ . Proof. By Theorem 5 (ii), the lower bound for a total safe dominating set in Km,n is⌈ m+n 2 ⌉ . Thus, γts(Km,n) = ⌈ m+n 2 ⌉ . Remark 4. Let Km,n be a complete bipartite graph with partite sets U and W such that |U | = m, |W | = n. If S ⊆ U or S ⊆ W , then S is not a total safe dominating set in Km,n. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 10 of 12 Theorem 6. Let Fn be a friendship graph of order 2n + 1 with central vertex x. Then a nonempty set S ⊊ V (Fn) is a total safe dominating set in Fn if and only if one of the following holds: (i) |S| ≥ 2, if x ∈ S (ii) |S| = 2n, otherwise Proof. Let S be a total safe dominating set in Fn. Suppose that x ∈ S and |S| < 2. Then |S| = 1, that is, S = {x}. So, S is not a total dominating set in Fn, a contradiction to the assumption of S. Therefore, |S| ≥ 2 if x ∈ S. Now, suppose that x /∈ S and |S| ̸= 2n. Then |S| < 2n. Without loss of generality, suppose |S| = 2n− 1. Let v ̸∈ S with v ̸= x. Since S is a dominating set in Fn, then there exists a vertex u ∈ S such that v ∈ N(u). This implies that Fn[S] has an isolated vertex, a contradiction to the assumption of S. Thus, |S| = 2n if x /∈ S. For the converse, suppose that x ∈ S and |S| ≥ 2. Clearly, S is a total safe dominating set in Fn since x is adjacent to all other vertices of Fn. Now, suppose x /∈ S and |S| = 2n. Again, clearly, S is a total safe dominating set in Fn. Corollary 5. For any friendship graph Fn of order 2n+ 1, γts(Fn) = 2. Proof. By Theorem 6 (i), clearly, γts(Fn) = |S| = 2. Theorem 7. Let Sn be a sunlet graph of order 2n with vertex set V (Cn) ∪ P , where P is the set of pendant vertices in Sn. Then a nonempty set S ⊊ V (Sn) is a total safe dominating set in Sn if and only if V (Cn) ⊆ S. Proof. Assume that S is a total safe dominating set in Sn. Suppose that V (Cn) ̸⊆ S. Then there exists a vertex, say v ∈ V (Cn), that is not in S. Let u ∈ V (Sn) be a pendant vertex such that u ∈ N(v). Then u must be in S since v ̸∈ S and S is a dominating set in Sn. Thus, u is an isolated vertex in Sn[S], and so S is not a total dominating set in Sn, a contradiction to the assumption of S. Hence, V (Cn) ⊆ S. For the converse, let V (Cn) ⊆ S. Clearly, V (Cn) is a total safe dominating set in Sn. Therefore, S is a total safe dominating set in Sn. Corollary 6. For any sunlet graph Sn of order 2n, γts(Sn) = n. Proof. By Theorem 7, V (Cn) is the minimum total safe dominating set in Sn. There- fore, γts(Sn) = |V (Cn)| = n. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 11 of 12 Theorem 8. Let Hn be a helm graph of order 2n + 1 with central vertex x and vertex set V (Cn) ∪ {x} ∪ P , where P is the set of pendant vertices in Hn. Then a nonempty set S ⊊ V (Hn) is a total safe dominating set in Hn if and only if V (Cn) ⊆ S. Proof. Let S be a total safe dominating set in Hn. Suppose that V (Cn) ̸⊆ S. Then there exists a vertex v ∈ V (Cn) with v ̸∈ S. Let u ∈ V (Hn) be a pendant vertex such that u ∈ N(v). Then u ∈ S since S is a dominating set in Hn. So, u is an isolated vertex in Hn[S], and so S is not a total dominating set in Sn, a clear contradiction to the assumption of S. Hence, V (Cn) ⊆ S. Conversely, let V (Cn) ⊆ S. Clearly, V (Cn) is a total safe dominating set in Hn. Thus, S is a total safe dominating set in Hn. Corollary 7. For any helm graph Hn of order 2n+ 1, γts(Hn) = n. Proof. By Theorem 8, V (Cn) is the minimum total safe dominating set in Hn. Hence, γts(Hn) = |V (Cn)| = n. Conclusions The concept of total safe domination has been introduced and explored in this study. Some realizations on how the total safe domination number relate with the total domi- nation number and the safe domination number are presented. Characterizations of total safe dominating sets in several well-known families of graphs are provided and used to determine the exact values of total safe domination number of those graphs. For those interested in further study on this topic, it would be valuable to explore other variations of total safe dominating sets, such as restrained, forcing, and locating sets. The study of domination in graphs resulting from unary operations is relatively underexplored and less studied. Exploring total safe domination in such graphs is worthwhile. With existing results on safe domination in ladder graphs, a special type of grid graph (denoted as (Pm × Pn)), exploration of total safe domination within these graphs is also interest- ing. Finally, the authors encourage other researchers to investigate inequalities related to Vizing’s conjecture and Nordhaus-Gaddum-type inequalities. Acknowledgements The authors would like to express their sincere thanks to the anonymous referees for their helpful and valuable comments. Heartfelt gratitude is also extended to the Depart- ment of Science and Technology, Philippines, for the financial support given to the authors through the DOST-SEI STRAND scholarship program. References [1] E Cockayne and S Hedetniemi. Towards a theory of domination in graphs. Networks, 7(4):247–261, 1977. W. G. Jumalon, I. Cabahug / Eur. J. Pure Appl. Math, 18 (2) (2025), 5917 12 of 12 [2] E Cockayne, R Dawes, and S Hedetniemi. Total domination in graphs. Networks, 10(1):85–95, 1980. [3] P Lam and B Wei. 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