EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6635 ISSN 1307-5543 – ejpam.com Published by New York Business Global Secure Pointwise Non-Domination and Secure Hop Domination in Graphs Farene Loida M. Alfeche1,∗, Sergio R. Canoy, Jr.1,2 1 Department of Mathematics and Statistics, College of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines 2 Center for Mathematical and Theoretical Physical Sciences- PRISM, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. In this paper, we revisit the concept of secure hop domination in graphs and define a new concept called secure pointwise non-domination. A pointwise non-dominating set S is a secure pointwise non-dominating set if for every u ∈ V (G) \ S, there exists v ∈ S \ NG(u) such that (S \ {v}) ∪ {u} is a pointwise non-dominating set. The secure pointwise non-domination number spnd(G) of G is the smallest cardinality of a secure pointwise non-dominating set in G. In this paper, we give bounds on the secure pointwise non-domination number and characterize those graphs which attain these bounds. We also determine the secure pointwise non-domination number of some classes of graphs. Necessary and sufficient conditions for a subset in the join of graphs to be a secure hop dominating set is given. Moreover, we show that given positive integers a and b with 2 ≤ a ≤ b, there exists a connected graph such that γh(G) = a and γsh(G) = b, where γh(G) and γsh(G) are the hop domination number and secure hop domination number of G, respectively. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Pointwise non-domination number, secure pointwise non-domination number, hop domination, secure hop domination number, join of graphs 1. Introduction The concept of security in graphs could provide a framework for modeling a security system [1] that could guarantee that no area is left unmonitored. Recently, Alfeche et al. [2] studied the secure hop dominating sets in graphs, where they gave bounds on the secure hop domination number and determine the secure hop domination numbers of the shadow graph and complementary prism. In this paper, we introduce and study the concept of secure pointwise non-dominating set in a graph. We give bounds on the parameter called secure pointwise non-domination number and give necessary and sufficient conditions for ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6635 Email addresses: farene.alfeche@g.msuiit.edu.ph (F.L. Alfeche) sergio.canoy@g.msuiit.edu.ph (S. Canoy) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 2 of 11 those graphs that attain these bounds. Furthermore, we use this newly defined concept to characterize the secure hop dominating sets in the join of graphs. Other related studies could be found in [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], and [18]. 2. Terminology and Notation Let G = V (G), E(G)) be an undirected graph. For any two vertices u and v of G, the distance dG(u, v) is the length of a shortest path joining u and v. Any u-v path of length dG(u, v) is called a u-v geodesic. The interval IG [u, v] consists of u, v, and all vertices lying on a u-v geodesic. The interval IG(u, v) = IG [u, v] \ {u, v}. Vertices u and v are adjacent (or neighbors) if uv ∈ E(G). The set of neighbors of a vertex u in G, denoted by NG(u), is called the open neighborhood of u. The closed neighborhood of u is the set NG[u] = NG(u) ∪ {u}. If X ⊆ V (G), the open neighborhood of X is the set NG(X) = ⋃ u∈X NG(u). The closed neighborhood of X is the set NG[X] = NG(X) ∪X. A set D ⊆ V (G) is a dominating set in G if for every v ∈ V (G)\D, there exists u ∈ D such that uv ∈ E(G), that is, NG[D] = V (G). The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set in G. Any dominating set in G with cardinality γ(G), is called a γ-set in G. If γ(G) = 1 and {v} is a dominating set in G, then we call v a dominating vertex in G. A dominating set D ⊆ V (G) is secure dominating in G if for every v ∈ V (G) \D, there exists w ∈ D ∩ NG(v) such that (D \ {w}) ∪ {v} is a dominating set in G. A vertex v in G is a hop neighbor of vertex u in G if dG(u, v) = 2. The set N2 G(u) = {v ∈ V (G) : dG(v, u) = 2} is called the open hop neighborhood of u. The closed hop neighborhood of u is given by N2 G[u] = N2 G(u) ∪ {u}. The open hop neighborhood of X ⊆ V (G) is the set N2 G(X) = ⋃ u∈X N2 G(u). The closed hop neighborhood of X is the set N2 G[X] = N2 G(X) ∪X. If S ⊆ V (G) and v ∈ S, then a vertex w ∈ V (G) \ S is an external private hop neighbor of v if N2 G(w) ∩ S = {v}. The set containing all the external private hop neighbors of v with respect to S is denoted by ephn(v;S). A set S ⊆ V (G) is a hop dominating set in G if N2 G[S] = V (G), that is, for every v ∈ V (G)\S, there exists u ∈ S such that dG(u, v) = 2. The minimum cardinality among all hop dominating sets in G, denoted by γh(G), is called the hop domination number of G. Any hop dominating set with cardinality equal to γh(G) is called a γh-set. A set S ⊆ V (G) is a pointwise non-dominating set of G if for each v ∈ V (G) \S, there exists u ∈ S such that v /∈ NG(u). The smallest cardinality of a pointwise non-dominating set of G, denoted pnd(G), is called the pointwise non-domination number of G. A pointwise non-dominating set S is a secure pointwise non-dominating set if for every u ∈ V (G) \ S, there exists v ∈ S \ NG(u) such that (S \ {v}) ∪ {u} is a pointwise non- dominating set. The secure pointwise non-domination number spnd(G) ofG is the smallest cardinality of a secure pointwise non-dominating set in G. A pointwise (secure pointwise non-dominating) set in G having cardinality equal to pnd(G) (resp. spnd(G)) is called a F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 3 of 11 pnd-set (resp. spnd-set) in G. A hop dominating set S is secure hop dominating if for each v ∈ V (G)\S, there exists w ∈ S ∩ N2 G(v) such that (S \ {w}) ∪ {v} is a hop dominating set in G. The minimum cardinality among all secure hop dominating sets of G, denoted by γsh(G), is called the secure hop domination number of G. Any secure hop dominating set with cardinality equal to γsh(G) is called a γsh-set. The join of graphs G1 and G2, denoted G1 + G2, is the graph with V (G1 + G2) = V (G1) ∪ V (G2) and E(G1 + G2) = E(G1) ∪ E(G2) ∪ {uv : u ∈ V (G1) and v ∈ V (G2)}. For other graph theoretic terms not mentioned here, readers may refer to [19] and [20]. 3. Results We shall need the following results. Theorem 1. [2] γsh(Kn) = γsh(Kn) = n for every positive integer n. Theorem 2. [21] Let G be a graph of order n. Then each of the following holds: (i) pnd(G) = 1 if and only if G has an isolated vertex. (ii) pnd(G) = n if and only if G = Kn. It should be noted that every graph admits a secure pointwise non-dominating sets. Theorem 3. Let G be a graph of order n. Then 1 ≤ pnd(G) ≤ spnd(G) ≤ n. Moreover, each of the following statements holds. (i) spnd(G) = 1 if and only if G = Kn. (ii) spnd(G) = 2 if and only if the following conditions hold: (j1) G ̸= Kn; (j2) there exist distinct vertices p, q such that NG(p) ∩NG(q) = ∅; and (j3) for each x ∈ V (G) \ {p, q}, either xp /∈ E(G) and NG(x) ∩ NG(q) = ∅ or xq /∈ E(G) and NG(x) ∩NG(p) = ∅. (iii) spnd(G) = n if and only if G = Kn. Proof. Since every secure pointwise non-dominating set is pointwise non-dominating and 1 ≤ pnd(G), it follows that 1 ≤ pnd(G) ≤ spnd(G) ≤ n. (i) Suppose spnd(G) = 1. Then pnd(G) = 1. By Theorem 2, G has an isolated vertex. Let S = {v} be an spnd-set in G. Then v is an isolated vertex of G (otherwise S is not a pointwise non-dominating set). Let x ∈ V (G) \ S. Since S is a secure pointwise non-dominating set, (S \{v})∪{x} = {x} is a pointwise non-dominating set. This implies that x is an isolated vertex. Therefore, G = Kn. Conversely, suppose G = Kn. Choose any vertex u of G. Then {u} is a secure pointwise non-dominating set in G. Hence, spnd(G) = 1. F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 4 of 11 (ii) Suppose spnd(G) = 2. Then G ̸= Kn. Let D = {p, q} be an spnd-set in G. Since D = {p, q} is a pointwise non-dominating set, NG(p) ∩ NG(q) = ∅. Let x ∈ V (G) \ D. Since D is secure pointwise non-dominating, xp /∈ E(G) and Dx = {x, q} is pointwise non- dominating or xq /∈ E(G) and Dx = {x, p} is pointwise non-dominating. Thus, xp /∈ E(G) and NG(x) ∩NG(q) = ∅ or xq /∈ E(G) and NG(x) ∩NG(p) = ∅. Hence, conditions (j1), (j2), and (j3) hold. Conversely, suppose conditions (j1), (j2), and (j3) hold. By (j1) and part (i), it follows that spnd(G) ≥ 2. Let S = {p, q}. By (j2), S is a pointwise non-dominating set in G. Next, let x ∈ V (G) \ S. Suppose xp /∈ E(G). Set Sx = (S \ {p}) ∪ {x} = {x, q}. By (j3), NG(x) ∩ NG(q) = ∅. This implies that Sx is pointwise non-dominating. If xp ∈ E(G), then xq /∈ E(G) by condition (j2) and NG(x) ∩ NG(p) = ∅ by (j3). It follows that S′ x = (S \ {q}) ∪ {x} = {x, p} is pointwise non-dominating in G. Therefore, S is a secure pointwise non-dominating set in G and spnd(G) = |S| = 2. (iii) Suppose spnd(G) = n. Suppose further that G ̸= Kn. Then there exist non-adjacent vertices x and y of G. Let S = V (G)\{x}. Since y ∈ S and x /∈ NG(y), it follows that S is a pointwise non-dominating set in G. Moreover, because V (G)\{y} is also pointwise non- dominating, S is a secure pointwise non-dominating set in G. Thus, spnd(G) ≤ |S| = n−1, a contradiction. Therefore, G = Kn. The converse is clear. Corollary 1. Let n be a positive integer. Then spnd(Pn) = { 1 if n = 1 2 if n ≥ 2. Proof. Let Pn = [v1, v2, · · · , vn]. By Theorem 3(iii), spnd(P1) = 1 and spnd(P2) = 2. Suppose n ≥ 3. Then spnd(Pn) ≥ 2 by Theorem 3(i). Let p = v1 and q = v2. Then NPn(p) ∩ NPn(q) = ∅. Let x ∈ V (Pn) \ {p, q}. If x = v3, then xp /∈ E(Pn) and NPn(q) ∩NPn(x) = ∅. If x ̸= v3, then xq /∈ E(Pn) and NPn(p) ∩NPn(x) = ∅. Therefore, spnd(Pn) = 2 by Theorem 3(ii). Corollary 2. Let n be a positive integer and n ≥ 3. Then spnd(Cn) = { 3 if n = 3, 5 2 if n /∈ {3, 5}. Proof. Let Cn = [v1, v2, · · · , vn, v1]. By Theorem 3(iii), spnd(C3) = 3. Suppose n = 5. It is easy to show that any set S ⊂ V (Cn) with |S| = 2 is not secure pointwise non-dominating. Since {v1, v2, v3} is secure pointwise non-dominating, it follows that spnd(C5) = 3. Next, suppose n /∈ {3, 5}. Then {v1, v2} is a secure pointwise non- dominating set. Therefore, spnd(Cn) = 2. Theorem 4. Let G1, G2, · · · , Gk be the components of G where k ≥ 2 and at least one component is non-trivial. Then each of the following statements holds: F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 5 of 11 (i) If spnd(Gj) = 2 for some j ∈ [k] = {1, 2, · · · , k} or G has two trivial components, then spnd(G) = 2. (ii) If k ≥ 3, G has at most one trivial component, and pnd(Gj) ̸= 2 for all j ∈ [k], then spnd(G) = 3. (iii) If k = 2 and Gj is complete for each j ∈ {1, 2}, then spnd(G) =  2 if G = K2 ∪Km where m ≥ 1 3 if G = K3 ∪Km where m ≥ 3 4 if G = Kr ∪Km where r,m ≥ 4 m if G = K1 ∪Km where m ≥ 1. Proof. (i) Suppose spnd(Gj) = 2 for some j ∈ [k]. Let S = {p, q} be an spnd-set in Gj . Then S is a pointwise non-dominating set in G. Let x ∈ V (G) \ S. If x ∈ V (Gj), then there exists v ∈ S \ NG(x), say v = p, such that (S \ {p}) ∪ {x} = {x, q} is pointwise non-dominating in Gj because S is secure point- wise non-dominating in Gj . Hence, {x, q} is pointwise non-dominating in G. Suppose x ∈ V (Gi) for i ̸= j. Then, (S \ {q}) ∪ {x} = {x, p} is pointwise non-dominating in G. Therefore, S is secure pointwise non-dominating in G. This implies that spnd(G) = 2. Next, suppose G has two trivial components, say G1 and G2. Then D = V (G1) ∪ V (G2) is cleary an spnd-set in G. Hence, spnd(G) = 2. (ii) Suppose k ≥ 3, G has at most one trivial component, and pnd(Gj) ̸= 2 for all j ∈ [k]. Then spnd(G) ≥ 2. Suppose spnd(G) = 2, say S = {x, y} is an spnd-set in G. Since pnd(Gj) ̸= 2 for all j ∈ [k], it follows that x ∈ V (Gi) and y ∈ V (Gj) for dis- tinct indices i, j ∈ [k]. Since G has at most one trivial component, we may assume that Gi is a non-trivial graph. Let z ∈ NG(x). Since S is secure pointwise non-dominating, (S \ {y}) ∪ {z} = {x, z} is pointwise non-dominating in G. Hence, {x, z} is pointwise non-dominating in Gi, a contradiction. Thus, spnd(G) ≥ 3. Pick any aj ∈ V (Gj) for j ∈ {1, 2, 3} and setD = {a1, a2, a3}. Then clearly, D is a secure pointwise non-dominating set in G. Therefore, spnd(G) = |D| = 3. (iii) Suppose now that k = 2 and Gj is complete for each j ∈ {1, 2}. Since spnd(K2) = 2, it follows from (i) that spnd(G) = 2 whenever G = K2 ∪Km for m ≥ 1. Next, suppose G = K3 ∪Km where m ≥ 3. Clearly, spnd(G) ≥ 3. Since S = V (K3) is a secure pointwise non-dominating set in G, it follows that spnd(G) = |S| = 3. Next, suppose G = Kr ∪ Km where r,m ≥ 4. Clearly, spnd(G) ≥ 3. Suppose spnd(G) = 3, say Q = {x, y, z} is an spnd-set in G. Since pnd(Kr) = r ≥ 4 and pnd(Km) = m ≥ 4, we may assume that x, y ∈ V (Kr) and z ∈ V (Km). Let p ∈ V (Kr) \ {x, y}. Since Q is secure pointwise non-dominating, it follows that (Q \ {z}) ∪ {p} = {x, y, p} is pointwise non-dominating in G (and in Kr), contrary to the fact that pnd(Kr) ≥ 4. Thus, spnd(G) ≥ 4. Choose any a, b ∈ V (Kr) and c, d ∈ V (Km). Then R = {a, b, c, d} is a secure pointwise non-dominating set in G. Thus, spnd(G) = 4. F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 6 of 11 Finally, suppose G = K1 ∪ Km where m ≥ 1. Clearly, spnd(G) = m if m ∈ {1, 2}. Suppose m ≥ 3. Let S be an spnd-set in G. Suppose |S| < m. Since pnd(Km) = m, it follows that V (K1) ⊆ S. This implies that |V (Km)∩S| ≤ m− 2. Let w ∈ V (Km) \S and let V (K1) = {q}. Since S is secure pointwise non-dominating in G, Sw = (S \ {q}) ∪ {w} is pointwise non-dominating in G (and in Km), a contradiction because pnd(Km) = m > m−1 ≥ |Sw|. Therefore, spnd(G) ≥ m. Since V (Km) is a secure pointwise non-dominating set in G, it follows that spnd(G) = m. Given a graph G, the vertex set V (G) is a secure hop dominating set of G. Thus, every graph admits a secure hop dominating set. Theorem 5. [22] Let G and H be any two graphs. A set S ⊆ V (G+H) is hop dominating set in G+H if and only if S = SG ∪SH , where SG and SH are pointwise non-dominating sets in G and H, respectively. Corollary 3. [22] Let G and H be any two graphs. Then γh(G+H) = pnd(G) + pnd(H). Theorem 6. Let G and H be any graphs. Then S ⊆ V (G+H) is a secure hop dominating set if and only if S = SG ∪ SH and SG and SH are secure pointwise non-dominating sets in G and H, respectively. Proof. Suppose S is a secure hop dominating set in G+H. Since S is a hop dominating set, S = SG ∪ SH where SG and SH are pointwise non-dominating sets in G and H, respectively, by Theorem 5. Let v ∈ V (G) \ SG. Then v ∈ V (G + H) \ S. Since S is a secure hop dominating set, there exists w ∈ S \NG+H(v) such that Sv = (S \ {w}) ∪ {v} is a hop dominating set. Note that since w ∈ S \ NG+H(v), w ∈ SG \ NG(v). Hence, Sv = [SG \ {w}) ∪ {v}] ∪ SH . By Theorem 5, (SG \ {w}) ∪ {v} is a pointwise non- dominating set because Sv is a hop dominating set. This shows that SG is a secure pointwise non-dominating set in G. Similarly, SH is a secure pointwise non-dominating set in H. For the converse, suppose that S = SG ∪SH and SG and SH are secure pointwise non- dominating sets in G and H, respectively. Then SG and SH are pointwise non-dominating sets, S is a hop dominating set in G+H by Theorem 5. Let x ∈ V (G+H) \ S. We may assume that x ∈ V (G). Then x /∈ SG. Since SG is a secure pointwise non-dominating set in G, there exists y ∈ S \NG(x) such that (SG \ {y})∪ {x} is a pointwise non-dominating set in G. It follows from Theorem 5 that (S \ {y}) ∪ {x} = [(SG \ {y}) ∪ {x}] ∪ SH is a hop dominating set in G+H. Hence, S is a secure hop dominating set in G+H. Corollary 4. Let G and H be any graphs. Then γsh(G+H) = spnd(G) + spnd(H). F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 7 of 11 Proof. Suppose DG and DH are spnd-sets in G and H, respectively. Then D = DG ∪DH is a secure hop dominating set in G+H by Theorem 6. Hence, γsh(G+H) ≤ |D| = |DG|+ |DH | = spnd(G) + spnd(H). Next, let S be an spnd-set in G + H. Then SG = S ∩ V (G) and SH = S ∩ V (H) are pointwise non-dominating sets in G and H, respectively, by Theorem 6. It follows that γsh(G+H) = |S| = |SG|+ |SH | ≥ spnd(G) + spnd(H). This establishes the desired equality. Corollary 5. Each of the following statements holds. (i) γsh(K1,n) = spnd(K1) + spnd(Kn) = 2 for all n ≥ 1. (ii) If n is a positive integer, then γsh(Fn) = γsh(K1 + Pn) = { 2 if n = 1 3 if n ≥ 2. (iii) If n is a positive integer and n ≥ 3, then γsh(Wn) = γsh(K1 + Cn) = { 4 if n = 3, 5 3 if n /∈ {3, 5}. (iv) If m and n are positive integers with m ≤ n, then γsh(Pm + Pn) = spnd(Pm) + spnd(Pn) =  2 if n = 1 3 if m = 1 and n = 2 4 if m ≥ 2. (v) If m and n are positive integers with 3 ≤ m ≤ n, then γsh(Cm + Cn) = spnd(Cm) + spnd(Cn) =  6 if n = 3, 5 5 if m = 3, 5 and n /∈ {3, 5} 4 if m,n /∈ {3, 5}. (v) If 1 ≤ m1 ≤ m2 ≤ · · · ≤ mk, where k ≥ 2, then γsh(Km1,m2,··· ,mk ) = k. In particular, γsh(Km,n) = 2 for all m,n ≥ 2. Proof. Note that by Corollary 4, Theorem 3(i), Corollary 1, and Corollary 2, state- ments (i), (ii), (iii), (iv), and (v) follow. By repetitive application of Corollary 4 and by Theorem 4(i), we have γsh(Km1,m2,··· ,mk ) = ∑ j∈[k] spnd(Kmj ) = k. F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 8 of 11 Therefore, the assertions hold. Theorem 7. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a connected graph G such that γh(G) = a and γsh(G) = b. Proof. Suppose a = b. Consider G = Ka. Then γh(G) = a because V (Ka) is the only hop dominating set in Ka. With the same reason, γsh(G) = a (also by Theorem 3(iii)). Next, suppose a < b. Let m = b − a and let G = (K1 ∪Km+1) +Ka−1. By Corollary 3 and Theorem 2, γh(G) = pnd(K1 ∪Km+1) + pnd(Ka−1) = 1 + (a− 1) = a. By Corollary 4, and Theorem 3(iii), we have γsh(G) = spnd(K1 ∪Km+1) + spnd(Ka−1) = 1 + (a− 1) = (m+ 1) + (a− 1) = b. This proves the assertion. The next result is found in [22]. Theorem 8. Let G and H be any two graphs. A set C ⊆ V (G ◦H) is a hop dominating set of G ◦H if and only if C = A ∪ (∪v∈V (G)∩NG(A)Sv)) ∪ (∪w∈V (G)\NG(A)Ew); where (i) A ⊆ V (G) such that for each w ∈ V (G) \A, there exists x ∈ A with dG(w, x) = 2 or there exists y ∈ NG(w) with V (Hy) ∩ C ̸= ∅, (ii) Sv ⊆ V (Hv) for each v ∈ NG(A), and (iii) Ew is a pointwise non-dominating set in Hw for each w ∈ V (G) \NG(A). Theorem 9. Let G and H be any two non-trivial graphs. If C = A∪ (∪v∈V (G)Sv)), where A is a secure hop dominating set in G and Sv is a secure pointwise non-dominating set in Hv for each v ∈ V (G), then C is a secure hop dominating set in G ◦H. Proof. By Theorem 8, C is a hop dominating set in G ◦H. Let x ∈ V (G ◦H) \C and let v ∈ V (G) such that x ∈ V (v +Hv). Consider the following cases: Case 1. x = v. Then x ∈ V (G) \ A. Since A is secure hop dominating in G, (A \ {y}) ∪ {x} is hop dominating for some y ∈ A ∩N2 G(x). Hence, (C \ {y}) ∪ {x} = [(A \ {y}) ∪ {x}] ∪ (∪w∈V (G)Sw) is hop dominating in G ◦H by Theorem 8. Case 2. x ∈ V (Hv). F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 9 of 11 Then x ∈ V (Hv) \ Sv. Since Sv is secure pointwise non-dominating in Hv, there exists p ∈ Sv \NHv(x) such that (Sv \ {p})∪{x} is pointwise non-dominating in Hv. Therefore, by Theorem 8, (C \ {p}) ∪ {x} = A ∪ [(∪w∈V (G)\{v}Sw)] ∪ ((Sv \ {p}) ∪ {x}) is hop dominating in G ◦H. Therefore, C is a secure hop dominating set in G ◦H. Corollary 6. Let G and H be any two non-trivial graphs. Then γsh(G ◦H) ≤ γsh(G) + |V (G)|spnd(H). Proof. Let A be a γsh-set in G and let Sv be an spnd-set in Hv for each v ∈ V (G). Then C = A∪ (∪v∈V (G)Sv)) is a secure hop dominating set in G ◦H by Theorem 9. Thus, γsh(G ◦H) ≤ |C| = |A|+ ∑ v∈V (G) |Sv| = γsh(G) + ∑ v∈V (G) spnd(H) = γsh(G) + |V (G)|spnd(H). This proves the assertion. Remark 1. The bound given in Corollary 6 is sharp. Strict inequality is also attainable. To see this, consider G1 = K2, H1 = K2, G2 = K2, and H2 = K2. Then γsh(G1 ◦H1) = γsh(K3 ∪K3) = 6 = γsh(G1) + 2spnd(H1) and γsh(G2 ◦H2) = 2 < 4 = γsh(G2) + 2spnd(H2). Theorem 10. Let G be a non-trivial connected graph with δ(G) ≥ 2 and let H be any graph. Then γsh(G ◦H) ≤ |V (G)|. Proof. By Theorem 8, C = V (G) is a hop dominating set in G ◦ H. Next, let x ∈ V (G ◦ H) \ C and let v ∈ V (G) such that x ∈ V (v + Hv). Since C = V (G), it follows that x ∈ V (Hv) \ Sv. Pick any w ∈ NG(v) and let Cx = [V (G) \ {w}] ∪ {x}. Let p ∈ V (G ◦ H) \ Cx. Suppose p = w. Then x ∈ Cx ∩ N2 G◦H(p). Suppose p ∈ V (Hz) for some z ∈ V (G). If z = w, then v ∈ Cx ∪ NG◦H(p) and dG◦H(u, p) = 2. Suppose z ̸= w. Since δ(G) ≥ 2, we may choose u ∈ NG(z) \ {w}. This implies that u ∈ Cx and dG◦H(u, p) = 2. Thus, Cx is hop dominating in G ◦ H. Since x was arbitrarily chosen in V (G ◦ H) \ C, it follows that C is a secure hop dominating set in G ◦ H. Therefore, γsh(G ◦H) ≤ |V (G)|. F. Alfeche, S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6635 10 of 11 4. Conclusion Secure pointwise non-domination was introduced and investigated in this study. Bounds on the secure pointwise non-domination number were established, and graphs attaining these bounds were characterized. Necessary and sufficient conditions for a subset in the join of graphs to be a secure pointwise non-dominating set was obtained. Moreover, it was shown that given positive integers a and b with 2 ≤ a ≤ b, there exists a connected graph such that γh(G) = a and γsh(G) = b, where γh(G) and γsh(G) are the hop domination number and secure hop domination number of G, respectively. Secure pointwise non-domination may be used to characterize the secure hop dominat- ing sets in the corona and lexicographic product of graphs. Acknowledgements The authors would like to thank the referees for their comments and suggestions which led to this much improved version of the paper. 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