EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6075 ISSN 1307-5543 – ejpam.com Published by New York Business Global Secure Hop Dominating Sets in Graphs Farene Loida M. Alfeche1,∗, Gina A. Malacas1,2, 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. Let G be an undirected (simple) graph with vertex and edge sets V (G) and E(G), respectively. A hop dominating set S in G is secure hop dominating if for each v ∈ V (G)\S, there exists w ∈ S ∩N2 G(v) such that (S \ {w})∪{v} is hop dominating in G. The minimum cardinality of a secure hop dominating in G, denoted by γsh(G), is called the secure hop domination number of G. In this paper, we show that the difference γsh(G) − γh(G) can be made arbitrarily large, where γh(G) is the hop domination number of G. We give bounds on the secure hop domination number and characterize those graphs which attain these bounds. The value of the newly defined parameter is determined for some classes of graphs. Moreover, we characterize the secure hop dominating sets in the shadow graph and complementary prism and determine the value of the parameter for each of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Hop domination, secure hop domination number, shadow graph, complementary prism 1. Introduction In 2003, Cockayne et al. [1] introduced and studied secure domination, a variant of the standard domination concept. As used to model a protection strategy in a given network, a secure dominating set may be viewed as one consisting of guards that protect the network from possible attacks. It is ensured that a guard can respond to a certain attack in some nearby vertex and as the guard moves to this location to defend the attack, the protection or security of the whole network is not compromised. The concept and some of its variants have been considered and studied in [2], [3], [4], [5], [6], [7], [8], [9], and [10]. Another domination-related concept was introduced by Natarajan et al. in [11]. This parameter, called hop domination parameter, and some of its variants have been the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6075 Email addresses: farene.alfeche@g.msuiit.edu.ph (F. L. Alfeche) gina.malacas@g.msuiit.edu.ph (G. A. Malacas) sergio.canoy@g.msuiit.edu.ph (S. R. Canoy) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 2 of 14 subject of interest in a number of recent studies (see [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], and [23]). In this paper, we introduce and study initially the new variant secure hop domination. The motivation stems from the fact that domination and hop domination have many similar applications in networks. It is easily observed from its definition that every graph admits a secure hop dominating set; in fact, the vertex set of a graph is such a set. We give bounds on the parameter and give necessary and sufficient conditions for a hop dominating set to be secure hop dominating. We also study the newly defined parameter in the shadow graph and complementary prism. 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 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 F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 3 of 14 equal to γsh(G) is called a γsh-set. The complement of graph G, denoted by G, is the graph with V (G) = V (G) such that vw ∈ E(G) if and only if vw /∈ E(G). The shadow graph D2(G) of graph G is constructed by taking two copies of G, say G1 and G2, and then joining each vertex u ∈ V (G1) to the neighbors of its corresponding vertex u′ ∈ V (G2). For a graph G, the complementary prism GG is the graph formed from the disjoint union of G and its complement G by adding a perfect matching between corresponding vertices of G and G. In simple terms, the graph GG is formed from G ∪ G by adding the edge vv for every vertex v ∈ V (G), where v is the vertex of G corresponding to vertex v of G. For other graph theoretic terms not mentioned here, readers may refer to [24] and [25]. 3. Results 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. Remark 1. Let G1, G2, . . . , Gk be the components of a graph G. Then S is a hop dominating set in G if and only if Sj = S ∩ V (Gj) is a hop dominating set in Gj for each j ∈ [k] = {1, 2, · · · , k}. Moreover, γh(G) = ∑k j=1 γh(Gj). Theorem 1. Let G1, G2, . . . , Gk be the components of G. Then γsh(G) = ∑k j=1 γsh(Gj). Proof. Suppose S is a secure hop dominating set in G. Then, by Remark 1, S = ∪j∈[k]Sj and Sj = S ∩ V (Gj) is a hop dominating set in Gj for each j ∈ [k]. For j ∈ [k], let x ∈ V (Gj)\Sj . Then x ∈ V (G)\S. Since S is a secure hop dominating set in G, there exists y ∈ S ∩N2 G(x) such that (S \ {y}) ∪ {x} = [(Sj \ {y}) ∪ {x}] ∪ [∪i∈[k]\{j}Si] is a hop dominating dominating set in G. Thus, (Sj \ {y}) ∪ {x} is a hop dominating dominating set in Gj . Since j was arbitrarily chosen, it follows that Sj is a secure hop dominating set in Gj for each j ∈ [k]. Therefore, γsh(G) = |S| = k∑ j=1 |Sj | ≥ k∑ j=1 γsh(Gj). Next, suppose that Dj is a γsh-set in Gj for each j ∈ [k]. Since each Dj is a hop dominating set of Gj , D = ∪j∈[k]Dj is a hop dominating set in G by Remark 1. Let v ∈ V (G) \ D. Then v ∈ V (Gt) \ Dt for a unique t ∈ [k]. Since Dt is a secure hop dominating set in Gt, there exists w ∈ Dt ∩ N2 Gt (v) such that (Dt \ {w}) ∪ {v} is a hop dominating set in Gt. By Remark 1, (D \ {w}) ∪ {v} = [(Dt \ {w}) ∪ {v}] ∪ [∪i∈[k]\{t}Di] F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 4 of 14 is a hop dominating dominating set in G. Hence, D is a secure hop dominating set in G and γsh(G) ≤ |D| = k∑ j=1 |Dj | = k∑ j=1 γsh(Gj). Therefore, the assertion holds. Theorem 2. Let G be any graph. Then γh(G) ≤ γsh(G). Moreover, for each positive integer n, there exists a connected graph G such that γsh(G) − γh(G) = n. In particular, the difference γsh(G)− γh(G) can be made arbitrarily large. Since every secure hop dominating set is hop dominating, it follows that γh(G) ≤ γsh(G). Next, let n be a positive integer. Consider the graph G in Figure 1 obtained from the complete graph Kn+2, where V (Kn+2) = {z1, z2, · · · , zn+2}, by adding the edges vw and wz1. The set {v, w} is a γh-set in G. Thus, γh(G) = 2. Let D be a γsh-set in G. If w /∈ D, then D = {v, z2, · · · , zn+2} or D = {z1, z2, · · · , zn+2} because D is a hop dominating set. Hence, |D| = n + 2. Suppose w ∈ D and let zj ∈ V (G) \ D for some j ∈ {2, 3, · · · , n + 2}. Since D is secure hop dominating, (D \ {w}) ∪ {zj} is hop dominating. Note that N2 G(zj) ∩ [{z2, z3, · · · , zn+2} \ {zj}] = ∅. This implies that {z2, z3, · · · , zn+2} \ {zj} ⊆ D. Hence, D = {v, w} ∪ [{z2, · · · , zn+2} \ {zj}] or D = {w} ∪ [{z1, z2, · · · , zn+2} \ {zj}]. It follows that |D| = n + 2. Therefore, γsh(G) = n + 2 and γsh(G)− γh(G) = n. ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... .................................... ............................................................................................................................................ ............................................................................................................... .................................... .................................................................................................................................................... .................................................................................................................................................... .................................... ............................................................................................................... .................................... .................................... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... .................................... .................................... ..................... .................... .................... .................... .................... .................... .................... .................... .................... .................... ........ .................................... .................................... ......................................................................................................................................................................................................................................................................................................................... ..................................................................................................................................................................................................................................................... .................................... .................................... ............................................................................................................................................ ..................................................................................................................................................................................................................................................... .................................... .................................... ..................................................................................................................................................................................................... .................................... ................................................................................................................................................................................ .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ . .................................... .................................... ........... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ...... .................................... .................................... ..................... .................... .................... .................... .................... .................... .................... .................... .................... .................... ........ .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ . .................................... .................................... . . . z4 z3z2 z1 zn+2 z5 v w Figure 1: Graph G with γsh(G)− γh(G) = n Theorem 3. Let G be any graph and let S be a hop dominating set in G. Then S is a secure hop dominating set in G if and only if for each v ∈ V (G) \ S there exists w ∈ S ∩N2 G(v) such that ephn(w;S) ⊆ N2 G[v]. Proof. Suppose S is a secure hop dominating set in G. Let v ∈ V (G) \ S. Since S is secure hop dominating, there exists w ∈ S ∩N2 G(v) such that Sv = (S \ {w}) ∪ {v} is hop dominating. Let z ∈ ephn(w;S). Then N2 G(z) ∩ S = {w}. Since Sv is a hop dominating set, it follows that z ∈ N2 G[v]. Thus, ephn(w;S) ⊆ N2 G[v]. For the converse, suppose that the given property holds. Let p ∈ V (G) \ S. Then by assumption, there exists q ∈ S ∩ N2 G(p) such that ephn(q;S) ⊆ N2 G[p]. Let Sp = F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 5 of 14 (S \ {q}) ∪ {p} and let x ∈ V (G) \ S. If x = q, then q ∈ N2 G(p) ⊆ N2 G[Sp]. Suppose x ̸= q. If x /∈ ephn(q;S), then there exists y ∈ (S \ {q}) ∩ N2 G(x) since S is a hop dominating set in G. Hence, x ∈ N2 G(y) ⊆ N2 G[Sp]. Next, suppose that x ∈ ephn(q;S). Then by assumption, x ∈ N2 G[p] ⊆ N2 G[Sp]. Therefore, Sp is a hop dominating set in G. Since p was arbitrarily chosen, it follows that S is a secure hop dominating in G. Corollary 1. Let G be a non-trivial graph and let S be a hop dominating set in G. If for each v ∈ V (G) \ S there exists w ∈ S ∩N2 G(v) with |ephn(w;S)| = 0 or |ephn(w;S)| ≥ 1 such that dG(v, p) = 2 for all p ∈ ephn(w;S) \ {v}, then S is a secure hop dominating set in G. Proof. Suppose S satisfies the given property. Let v ∈ V (G)\S. By assumption, there exists w ∈ S ∩ N2 G(v) satisfying the condition. If |ephn(w;S)| = 0, then ephn(w;S) = ∅ ⊆ N2 G[v]. Suppose |ephn(w;S)| ≥ 1. Then dG(v, p) = 2 for all p ∈ ephn(w;S) \ {v} by assumption. Thus, ephn(w;S) ⊆ N2 G[v]. Therefore, S is a secure hop dominating set by Theorem 3. Theorem 4. γsh(Kn) = γsh(Kn) = n for every positive integer n. Proof. Let G ∈ {Kn,Kn}. Since the only hop dominating set in G is V (G), it follows that V (G) is the only secure hop dominating. Therefore, γsh(G) = n. Lemma 1. Let G be a non-trivial graph and let S = {p, q} be a hop dominating set in G. Then ephn(p;S) = V (G) \ (N2 G[q] ∪ {p}) and ephn(q;S) = V (G) \ (N2 G[p] ∪ {q}). Proof. Note that since S is hop dominating, dG(p, q) ̸= 2. Let x ∈ ephn(p;S). Then x ∈ V (G) \ S and N2 G(x) ∩ S = {p}. It follows that x ∈ V (G) \ (N2 G[q] ∪ {p}). Hence, ephn(p;S) ⊆ V (G) \ (N2 G[q] ∪ {p}). Now, let z ∈ V (G) \ (N2 G[q] ∪ {p}). Then z ̸= p and z /∈ N2 G[q]. Since S is hop dominating, it follows that z ∈ N2 G(p). This implies that z ∈ ephn(p;S). Thus, V (G) \ (N2 G[q] ∪ {p}) ⊆ ephn(p;S), showing the desired equality. Similarly, the second equality also holds. Theorem 5. Let G be any graph of order n. Then 1 ≤ γsh(G) ≤ n. Moreover, each of the following statements holds: (i) γsh(G) = 1 if and only if G = K1. (ii) γsh(G) = 2 if and only if there exist two distinct vertices v, w ∈ V (G) satisfying the following conditions: (p1) N2 G[{v, w}] = V (G) and N2 G(v) ∩N2 G(w) = ∅. (p2) For each x /∈ {v, w} such that x ∈ N2 G(v) (or x ∈ N2 G(w)), it holds that V (G) \ (N2 G[w] ∪ {v}) ⊆ N2 G[x] (resp. V (G)\(N2 G[v] ∪ {w}) ⊆ N2 G[x]). (iii) γsh(G) = n if and only if every component of G is complete. F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 6 of 14 Proof. Clearly, 1 ≤ γsh(G) ≤ n. (i) Suppose γsh(G) = 1, say S = {v} is a γsh-set in G. Since S cannot be a hop dominating set if G is non-trivial, it follows that G = K1. Conversely, if G = K1, then γsh(G) = 1. (ii) Suppose γsh(G) = 2. Let D = {v, w} be a γsh-set of G. Since D is hop dominating, V (G) = N2 G[{v, w}]. Suppose p ∈ N2 G(v)∩N2 G(w). Since D is a secure hop dominating set, we may assume that Sp = (D \ {v}) ∪ {p} = {p, w} is hop dominating (otherwise, {p, v} is hop dominating). Let q ∈ NG(p) ∩ NG(w). Then q /∈ N2 G[Sp]. This implies that D is not hop dominating, a contradiction. Thus, (p1) holds. Next, let x ∈ V (G) \D. Assume without loss of generality that x ∈ N2 G(v) (hence, x /∈ N2 G(w)). Since D is a secure hop dominating set in G, it follows that Dx = (D \ {v}) ∪ {x} = {w, x} is a hop dominating set in G. Let z ∈ ephn(v;D). Then N2 G(z) ∩D = {v}. Since Dx is hop dominating, we must have z ∈ N2 G[x]. Hence, ephn(v;D) ⊆ N2 G[x]. By Lemma 1, (p2) holds. For the converse, suppose that there exist distinct vertices v, w ∈ V (G) satisfy- ing properties (p1) and (p2). Set S = {v, w}. Then S is a hop dominating set by (p1). Let x ∈ V (G) \ S. By (p1), x ∈ N2 G(v) \ N2 G(w) or x ∈ N2 G(w) \ N2 G(v). If x ∈ N2 G(v) \ N2 G(w) (x ∈ N2 G(w) \ N2 G(v)), then V (G) \ (N2 G[w] ∪ {v}) ⊆ N2 G[x] (resp. V (G) \ (N2 G[v] ∪ {w}) ⊆ N2 G[x]) by (p2). By Lemma 1 and Theorem 3, S is a secure hop dominating set in G. Since G is non-trivial, γsh(G) = |S| = 2. (iii) Suppose γsh(G) = n. Suppose there exists a component of H of G that is not complete. Then there exists v ∈ V (H) ⊆ V (G) such that N2 H(v) = N2 G(v) ̸= ∅, say w ∈ N2 H(v). Set S = V (G) \ {w}. Then clearly, S is a hop dominating set in G. Since Sw = (S \ {v}) ∪ {w} = V (G) \ {v} is also hop dominating, it follows that S is a secure hop dominating set. This implies that γsh(G) ≤ |S| = n − 1, a contradiction. Therefore, every component of G is complete. For the converse, suppose that every component of G is complete. Let G1, G2, . . . , Gk be the components of G. By assumption and Theorem 4, γsh(Gj) = |V (Gj)| for each j ∈ [k] = {1, 2, · · · , k}. Thus, γsh(G) = ∑k j=1 γsh(Gj) = n by Theorem 1. Lemma 2. Let G be a graph of order n ≥ 3 such that γ(G) = 1. If |D(G)| ≥ 2, where D(G) = {v ∈ V (G) : NG[v] = V (G)}, then γsh(G) ≥ 3. Proof. If G = Kn, then γsh(G) = n ≥ 3. So suppose G ̸= Kn and let S be a γsh-set in G. Since S is a hop dominating set, D(G) ⊆ S. Hence, if |D(G)| ≥ 3, then γsh(G) = |S| ≥ 3. Suppose |D(G)| = 2. Since |N2 G(v)| = 0 for every v ∈ D(G) and S is a hop dominating set, it follows that S ̸= D(G). This implies that 2 = |D(G)| < |S| = γsh(G). This proves the assertion. Theorem 6. Let G be a non-trivial graph of order n such that γ(G) = 1. Then γsh(G) = 2 if and if G = K1,n−1, that is, G has a unique dominating vertex v and |NG(x)| = 1 for x ∈ V (G) \ {v}. F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 7 of 14 Proof. Suppose γsh(G) = 2. If n = 2, then G = K2 = K1,1. Suppose n ≥ 3 and let S be a γsh-set in G. By the contrapositive of Lemma 2, |D(G)| = 1, where D(G) = {u ∈ V (G) : NG[u] = V (G)} that is, G has a unique dominating vertex, say v. This implies that S = {v, w} for some w ∈ V (G) \ {v}. Note that since S is a hop dominating set, |NG(w)| = 1. Suppose there exists x ∈ V (G) \ S with |NG(x)| ≥ 2. Since S is a secure hop dominating set and v is a dominating vertex, it follows that x ∈ N2 G(w) and Sx = (S \ {w}) ∪ {x} = {v, x} is a hop dominating set in G. This, however, is not possible because a vertex y ∈ NG(x) \ {v} is not in N2 G[Sx]. Therefore, |NG(x)| = 1 for every x ∈ V (G) \ {v}. Accordingly, G = K1,n−1. For the converse, suppose that G = K1,n−1. Then γsh(G) ≥ 2 by Theorem 5(i). Let v0 ∈ V (G) be such that |NG(v0)| = n− 1 and let q ∈ V (G) \ {v0}. Then v0 and q satisfy the properties (p1) and (p2) of Theorem 5(ii). Therefore, γsh(G) = 2. Remark 2. There are graphs G with γsh(G) = 2 such that γ(G) ̸= 1. To see this, consider G ∈ {P4, C4, H} in Figure 2. Clearly, γ(G) = 2 ̸= 1. It can be verified easily that the blackened vertices form a γsh-set in G. ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... ................................................................................................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... ................................................................................................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... ............................................................................ ........................................................................ ................................................................................................................ ................................................................................................................ .................................... ............................................................................................. .................................... .................................... ........... .......... .......... .......... .......... .......... .......... .......... .......... .. .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... .................................... .................................... P4 C4 H • • • • • • Figure 2: Graph G with γsh(G) = 2 and γ(G) = 2 The next result is a consequence of Theorem 1, Theorem 6, and Theorem 5(ii). Corollary 2. Let G be a graph of order n where 3 ≤ n ≤ 4. If γsh(G) = 2, then G ∈ {P3, P4, C4,K1,3}. Proof. By Theorem 1 and Theorem 5(ii), none of the disconnected graphs G satisfies γsh(G) = 2. From Theorem 6, it follows that γsh(P3) = γsh(K1,3) = 2. Suppose n = 4. As seen in Remark 2, γsh(P4) = γsh(C4) = 2. If G is connected and G /∈ {P4, C4,K1,3}, then γ(G) = 1 and either G has more that two dominating vertices or contains a single dominating vertex and another vertex with two neighbors. Thus, γsh(G) ≥ 3 by Theorem 6. Therefore, G ∈ {P3, P4, C4,K1,3}. Proposition 1. Let n be any positive integer. Then γsh(Pn) =  n if n ∈ {1, 2} 2 if n = 3 2t if n = 4t, t ≥ 1 2t+ 1 if n = 4t+ 1, t ≥ 1 2t+ 2 if n = 4t+ 2, t ≥ 1 or n = 4t+ 3, t ≥ 1. F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 8 of 14 Proof. Let Pn = [v1, v2, . . . , vn]. Then γsh(Pn) = n for n ∈ {1, 2} by Corollary 4, and γsh(P3) = 2 by Theorem 6. Suppose n ≥ 4 and consider the following cases: Case 1. n = 4t. Let Sj = {v4j−3, v4j−2} for each j ∈ {1, 2, . . . , t} and set S = ∪t j=1Sj . Then S is a secure hop dominating set and |S| = ∑t j=1 |Sj | = 2t. Let S′ be a hop dominating set such that |S′| < |S|. Then one can find a vertex z ∈ V (Pn) \ S′ such that for each v ∈ S′ ∩N2 Pn (z), either |ephn(v;S′)| = 1 and v /∈ ephn(v; s) or |ephn(v;S′)| = 2. Hence, S′ is not a secure hop dominating set. Thus, S is a γsh-set in Pn and γsh(Pn) = |S| = 2t. Case 2. n = 4t+ 1(t ≥ 1). Let Sj = {v4j , v4j+1} for each j ∈ {1, 2, . . . , t}. Then S = {v1}∪[∪t j=1Sj ] is a hop dominat- ing set in Pn. Since |ephn(v;S)| ≤ 1 for every v ∈ S, S is a secure hop dominating set by Corollary 1. Following an argument in the preceding case, any hop dominating set S′ with |S′| < |S| is not secure hop dominating. Therefore, γsh(Pn) = |S| = 1+ ∑t j=1 |Sj | = 2t+1. Case 3. n = 4t+ 2 or 4t+ 3. Consider the set Sj = {v4j−3, v4j−2} for each j ∈ {1, 2, . . . , t + 1}. Then S = ∪t+1 j=1Sj is a γsh-set in Pn. Therefore, γsh(Pn) = |S| = ∑t+1 j=1 |Sj | = 2(t+ 1) = 2t+ 2. This proves the assertion. Proposition 2. Let n be any positive integer such that n ≥ 3. Then γsh(Cn) =  3 if n ∈ {3, 5} 2t if n = 4t, t ≥ 1 or n = 4t+ 1, t ≥ 2 or n = 4t+ 2, t ≥ 1 2t+ 1 n = 4t+ 3, t ≥ 1. Proof. Let Cn = [v1, v2, . . . , vn, v1]. If n = 3, then γsh(C3) = γsh(K3) = 3 by Corollary 4. If n = 5, then D = {v1, v2, v4} is a γsh- set in C5. Hence, γsh(C5) = 3. Now, suppose n /∈ {3, 5}. Consider the following cases: Case 1. n = 4t or n = 4t+ 1. Let Sj = {v4j−3, v4j−2} for each j ∈ {1, 2, . . . , t}. Then S = ∪t j=1Sj is γsh-set in Cn. It follows that γsh(Cn) = |S| = ∑t j=1 |Sj | = 2t. F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 9 of 14 Case 2. n = 4t+ 3. Set Sj = {v4j+3, v4j+4}, j = 1, 2, . . . , t− 1. Then S = {v1, v2, vn} ∪ [∪t−1 j=1Sj ] is γsh-set in Cn. Hence, γsh(Cn) = |S| = 3 + ∑t−1 j=1 |Sj | = 3 + 2(t− 1) = 2t+ 1. This proves the theorem. If G1 and G2 are the copies of graph G in the definition of the shadow graph D2(G) and if SG1 ⊆ V (G1) and SG2 ⊆ V (G2), then the sets S′ G1 and S′ G2 are the sets given by S′ G1 = {a′ ∈ V (G2) : a ∈ SG1} and S′ G2 = {a ∈ V (G1) : a ′ ∈ SG2}. The next result is obtained by Hassan et al. in [26]. Theorem 7. Let G be a non-trivial connected graph. Then S is a hop dominating set in D2(G) if and only if one of the following conditions holds: (i) S is a hop dominating set in G1. (ii) S is a hop dominating set in G2. (iii) S = SG1 ∪ SG2 such that SG1 ∪ S′ G2 and S′ G1 ∪ SG2 are hop dominating sets in G1 and G2, respectively. Theorem 8. Let G be a non-trivial connected graph. Then a set S ⊆ V (D2(G)) is secure hop dominating in D2(G) if and only if one of the following conditions holds: (i) S is a secure hop dominating set in G1. (ii) S is a secure hop dominating set in G2. (iii) S = SG1 ∪ SG2 such that SG1 ∪ S′ G2 and S′ G1 ∪ SG2 are secure hop dominating sets in G1 and G2, respectively. Proof. Let S be a secure hop dominating set in D2(G). Set SG1 = S ∩ V (G1) and SG2 = S ∩ V (G2). If SG2 = ∅, then S = SG1 is a hop dominating set in G1 by Theorem 7(i). Let x ∈ V (G1) \ SG1 . Since S is a secure hop dominating set in D2(G), there exists w ∈ S ∩N2 D2(G)(x) such that (S \ {w})∪{x} is hop dominating in D2(G). Since SG2 = ∅, it follows that w ∈ SG1 . Thus, (S \ {w}) ∪ {x} = (SG1 \ {w}) ∪ {x} is hop dominating in G1 by Theorem 7(i). Therefore, S = SG1 is secure hop dominating in G1. Similarly, S = SG2 is secure hop dominating in G2 whenever SG1 = ∅. Finally, suppose SG1 ̸= ∅ and SG2 ̸= ∅. By Theorem 7(iii), L = SG1 ∪S′ G2 and M = S′ G1 ∪SG2 are hop dominating sets in G1 and G2, respectively. Let p ∈ V (G1) \ L. Since S secure hop dominating in D2(G), there exists q ∈ S ∩N2 D2(G)(p) such that Sp = (S \ {q}) ∪ {p} is hop dominating in D2(G). Suppose q ∈ SG1 . Then Sp = (S \ {q})∪ {p} = [(SG1 \ {q})∪ {p}]∪ SG2 . Since Sp is hop dominating in D2(G), it follows from Theorem 7(iii) that [(SG1 \ {q}) ∪ {p}] ∪ S′ G2 = [(SG1 ∪ S′ G2 ) \ {q}]) ∪ {p} F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 10 of 14 is hop dominating in G1 Theorem 7(iii). Suppose q = t′ ∈ SG2 . Then t ∈ S′ G2 ∩ N2 G1 (p) and Sp = (SG1 ∪ {p}) ∪ [(SG2 \ {t′}]. Since Sp is hop dominating in D2(G), (SG1 ∪ {p}) ∪ [(S′ G2 \ {t}] = [(SG1 \ {t}) ∪ {p}] ∪ S′ G2 = [(SG1 ∪ S′ G2 ) \ {t}] ∪ {p} is hop dominating in G1 by Theorem 7(iii). Thus, L is secure hop dominating in G1. Similarly, M is secure hop dominating in G2. Therefore, one of (i), (ii), and (iii) holds. For the converse, suppose (i) holds. Then S is hop dominating in D2(G) by Theorem 7(i). Let z ∈ V (D2(G)) \ S. Suppose z ∈ V (G1). Since S is secure hop dominating in G1, there exists q ∈ S ∩ N2 G(z) such that (S \ {q}) ∪ {z} is hop dominating in G1. By Theorem 7, (S \ {q}) ∪ {z} is hop dominating in D2(G). Next suppose z = t′ ∈ V (G2). If t ∈ S, then dD2(G)(t, t ′) = 2. If a ∈ ephn(t;S), then dD2(G)(a, t) = dD2(G)(a, t ′) = 2. This implies that ephn(t;S) ⊆ N2 D2(G)[t ′]. Hence, (S \ {t}) ∪ {t′} is a hop dominating set in D2(G). Suppose t /∈ S. Since S is a secure hop dominating set in G1, there exists s ∈ S ∩ N2 G1 (t) such that St = (S \ {s}) ∪ {t} is hop dominating in G1. Thus, St is hop dominating in D2(G). Set St′ = (S \ {s}) ∪ {t′} and let p ∈ V (D2(G)) \ St′ . Then p /∈ S \ {s} and p ̸= t′. Suppose first that p ∈ V (G1). If p ∈ {s, t}, then p ∈ N2 D2(G)(t ′). Suppose p /∈ {s, t}. Since St is hop dominating, there exists r ∈ (St \ {s}) ∩ N2 G1 (p). It follows that r ∈ St′ ∩N2 D2(G)(p). Suppose p = b′ ∈ V (G2). By considering b and following the preceding arguments, it can be shown that there exists d ∈ St′ ∩N2 D2(G)(p). Hence, St′ is hop dominating in D2(G). Therefore, S is a secure hop dominating set in D2(G). The same conclusion holds if (ii) holds. Finally, suppose (iii) holds. Then, by Theorem 7(iii), S = SG1 ∪ SG2 is hop dominating. Let x ∈ V (D2(G)) \ S. Then x /∈ SG1 ∪ SG2 . We may assume that x ∈ V (G1) \ SG1 . If x′ ∈ SG2 , then x′ ∈ S ∩N2 D2(G)(x) and (S \ {x′}) ∪ {x} is hop dominating in D2(G). Suppose x′ /∈ SG2 . Then x /∈ S′ G2 . This implies that x /∈ V (G1) \ (SG1 ∪ S′ G2 . Since SG1 ∪ S′ G2 is secure hop dominating in G1, there exists y ∈ (SG1 ∪ S′ G2 ) ∩N2 G1 (x) such that [(SG1 ∪ S′ G2 ) \ {y}] ∪ {x} = [(SG1 \ {y}) ∪ {x}] ∪ S′ G2 is hop dominating in G1. By Theorem 7(iii), [(SG1 \ {y}) ∪ {x}] ∪ SG2 is hop dominating in G1. Thus, (S \ {y}) ∪ {x} = [(SG1 \ {y}) ∪ {x}] ∪ SG2 is hop dominating in D2(G). Therefore, S is secure hop dominating in D2(G). The next result is a direct consequence of Theorem 8. Corollary 3. Let G be a non-trivial connected graph. Then γsh(D2(G)) = γsh(G). Proof. Let S be a γsh-set of G = G1. Then S is a secure hop dominating set of D2(G) by Theorem 8. Hence, γsh(D2(G)) ≤ |S| = γsh(G). Next, suppose S′ is a γsh-set of D2(G). If S′ ⊆ V (G1) or S′ ⊆ V (G2), then S′ is a secure hop dominating set of G by (i) and (ii) of Theorem 8. It follows that γsh(G) ≤ |S′| = γsh(D2(G)). If S′ = SG1 ∪ SG2 , then SG1 ∪ S′ G2 is secure hop dominating in G by F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 11 of 14 Theorem 8(iii). Hence, γsh(G) ≤ |SG1 ∪ S′ G2 | = |SG1 ∪ SG2 | = |S′| = γsh(D2(G)). This establishes the desired equality. Lemma 3. Let G be a graph. Then S = {x, y}, where x, y ∈ V (G), is a hop dominating set in GG if and only if x = y. Proof. Suppose S is a hop dominating set in GG. Suppose x ̸= y. If xy ∈ E(G), then y /∈ N2 GG (S) because yy ∈ E(GG). If xy /∈ E(G), then x y ∈ E(G). Thus, x /∈ N2 GG (S). In both cases, we obtain a contradiction. Thus, x = y. For the converse, suppose x = y. Then clearly, S = {x, x} is a hop dominating set in GG. Theorem 9. Let G be a graph. Then 2 ≤ γsh(GG) ≤ 4. Moreover, each of the following statements hold: (i) γsh(GG) = 2 if and only if G ∈ {K1,K2,K2}. (ii) γsh(GG) = 3 if and only if G /∈ {K2,K2} and one of the following conditions holds: (i1) γh(G) = 2 or γh(G) = 2. (i2) There exists a secure hop dominating set S of G with |S| = 3 such that ephn(v;S) = 0 for some v ∈ S or a secure hop dominating set S of G with |S| = 3 such that ephn(v;S) = 0 for some v ∈ S. (i3) There exist vertices x, y, z,∈ V (G) such z ∈ N2 G[{x, y}], and dG(v, w) = 2 for all v, w ∈ V (G) \N2 G({x, y}), where v ̸= w. (i4) There exist vertices x, y, z ∈ V (G) such z ∈ N2 G [{x, y}], and dG(v, w) = 2 for all v, w ∈ V (G) \N2 G [{x, y}], where v ̸= w (iii) γsh(GG) = 4 if and only if G does not satisfy any of the properties in (i) and (ii). Proof. Since GG is non-trivial, it follows that 2 ≤ γsh(GG). Since {v, v} is a hop dominating set in GG for each v ∈ V (G), {v, w, v, w} is a secure hop dominating set in GG for each pair of distinct vertices v and w of G. Therefore, 2 ≤ γsh(GG) ≤ 4. (i) Suppose γsh(GG) = 2, say S = {p, q} is a γsh-set inGG. SupposeG /∈ {K1,K2,K2}. Then |V (G)| ≥ 3. Suppose p, q ∈ V (G). Choose any s ∈ V (G) \ S. Then s ∈ V (GG) \ S. Since S is secure hop dominating in GG, {p, s} or {q, s} is hop dominating in GG. Accord- ing to Lemma 3, this is impossible. Similarly, we arrived at a contradiction if p, q ∈ V (G). Suppose now that p ∈ V (G) and q = c ∈ V (G). By Lemma 3, p = c, i.e., S = {p, p}. Let z ∈ V (G) \ {p}. Then, by Lemma 3, (S \ {p}) ∪ {z} = {p, z} is a hop dominating set in G. Suppose G is disconnected. Pick w ∈ V (H) where H is a component of G such F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 12 of 14 that p /∈ V (H). Then (S \ {p}) ∪ {w} = {p, w} is not a hop dominating set, a contradic- tion. Hence, G is connected. Since G /∈ {K1,K2}, we may choose a vertex d such that NG(p) ∩NG(d) ̸= ∅. This implies that (S \ {p}) ∪ {d} = {p, d} is not a hop dominating set in G, a contradiction. Therefore, G ∈ {K1,K2,K2}. For the converse, suppose first that G = K1. Then GG = K2 and γsh(GG) = 2. If G ∈ {K2,K2}, then GG = P4. By Theorem 1, γsh(GG) = 2. (ii) Suppose γsh(GG) = 3. Then G /∈ {K2,K2} by (i). If γh(G) = 2 or γh(G) = 2, then (i1) is satisfied. Suppose γh(G) > 2 and γh(G) > 2. Let D = {x, y, s} be a γsh-set in GG. We may assume that D ⊆ V (G). Then D is a secure hop dominating set in G. Suppose ephn(v;S) ̸= 0 for all v ∈ D. Let u′ ∈ V (G) where u /∈ {x, y, s}. Since D is secure hop dominating in GG, there exists w ∈ D, say w = x, such that (D \ {x}) ∪ {u} is hop dominating in GG. This, however, is not possible because ephn(x;S) ̸= 0. Thus, there exists v ∈ D such that ephn(v;S) = 0. This shows that (i2) holds. Next, suppose that x, y ∈ V (G) and s = z ∈ V (G). If {x, y} is a hop dominating set in G, then γh(G) = 2 and we find that (i1) holds. Suppose {x, y} is not a hop dominating set in G. Since zz ∈ E(GG) and D is a hop dominating set in GG, it follows that z ∈ N2 G({x, y}). Let v ∈ V (G) \N2 G({x, y}). Since D is a secure hop dominating set in GG, Dv = (D \ {z}) ∪ {v}) = {x, y, v} is a hop dominating set in GG. Let w ∈ V (G) \ [N2 G({x, y})∪ {v}]. Since Dv is hop dominating in GG, w ∈ N2 G(v). Hence, dG(v, w) = 2 for any pair of distinct vertices v, w ∈ V (G) \N2 G({x, y}). This shows that (i3) holds. Similarly, (i4) holds. For the converse, suppose (i1) holds. Note that since G /∈ {K1,K2,K2}, it follows that γsh(GG) ≥ 3. Let {x, y} be a γh-set of G and let Q = {x, y, x}. Then clearly, Q is a hop dominating set in GG. Let z ∈ V (GG) \ Q. Suppose z ∈ V (G). Since {x, y} is hop dominating in G, z ∈ N2 G({x, y}). Both sets (Q \ {y}) ∪ {z} and (Q \ {x}) ∪ {z} are hop dominating in GG. Suppose z = s ∈ V (G). If s = y, then s ∈ N2 GG (x) and (Q\{x})∪{s} = {y, s, x} is hop dominating in GG. Suppose s ̸= y. Then s ∈ N2 GG (y) and (Q \ {y}) ∪ {s} = {x, s, x} is hop dominating in GG. Hence, Q is secure hop dominating in GG and γsh(GG) = |Q| = 3. The same conclusion holds if {x, y} is a γh-set in G. Suppose now that (i2) holds. We may assume that there exists a secure hop dominating set S = {a, b, c} of G with |S| = 3 and ephn(a;S) = 0. Clearly, S is hop dominating in GG. Moreover, because of the conditions that S is secure hop dominating in G and ephn(a;S) = 0, S is hop dominating in GG. Hence, γsh(GG) = |S| = 3. Suppose (i3) holds, i.e., there exist vertices x, y, z,∈ V (G) such z ∈ N2 G[{x, y}], and dG(v, w) = 2 for all v, w ∈ V (G) \N2 G({x, y}), where v ̸= w. Let R = {x, y, z}. Then R is a hop dominating set in GG. Let u ∈ V (GG) \ R. Clearly, if u = k′ ∈ V (G), then (R \ {x}) ∪ {u} is hop dominating if k ̸= x and (R \ {y}) ∪ {u} is hop dominating if k ̸= y. Suppose u ∈ V (G). If u ∈ N2 G({x, y}), say u ∈ N2 G(x), then (R \ {x}) ∪ {u} is hop dominating in GG. If u /∈ N2 G({x, y}), then (R \ {z}) ∪ {u} is hop dominating in GG because of the additional assumption that dG(v, w) = 2 for all v, w ∈ V (G) \N2 G({x, y}), where v ̸= w. Hence, R is a secure hop dominating set in GG and γsh(GG) = |R| = 3. The same conclusion holds if (i4) is assumed. (iii) This follows from (i) and (ii). F. L. Alfeche, G. A. Malacas , S. Canoy Jr. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6075 13 of 14 The next result follows from Theorem 9. Corollary 4. Let n be a positive integer and n ≥ 2. Then γsh(KnKn) =  2, if n = 2 3, if n = 3 4, if n ≥ 4. 4. Conclusion Secure hop domination was introduced and initially investigated in this study. Bounds on the parameter were given and graphs which attain these bounds were characterized. It was shown that the difference of the secure hop domination number and the hop dom- ination number can be made arbitrarily large. A necessary and sufficient condition for a hop dominating set to be secure hop dominating was obtained. Moreover, the secure hop dominating sets in the shadow graph and complementary prism were characterized. These characterizations were used to determine the values of the parameter for these graphs. The newly defined parameter can be studied further for trees and even for graphs resulting from some binary operations. Moreover, it may be interesting to consider and investigate the complexity of the secure hop dominating set problem. 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