EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 440-453 ISSN 1307-5543 – ejpam.com Published by New York Business Global Hop Differentiating Hop Dominating Sets in Graphs Sergio R. Canoy, Jr.1,∗, Chrisley Jade C. Saromines1 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. A subset S of V (G), where G is a simple undirected graph, is hop dominating if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2 and it is hop differentiating if N2 G[u] ∩ S ̸= N2 G[v] ∩ S for any two distinct vertices u, v ∈ V (G). A set S ⊆ V (G) is hop differentiating hop dominating if it is both hop differentiating and hop dominating in G. The minimum cardinality of a hop differentiating hop dominating set in G, denoted by γdh(G), is called the hop differentiating hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the hop differentiating hop dominating sets in graphs under some binary operations. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Hop domination, hop differentiating, join, corona, lexicographic product 1. Introduction Differentiating-domination in a graph, a variation of the standard domination, was defined by Gimbel et al. in [6]. A differentiating set in a given network can be viewed as a set of sensitive monitors used to safeguard a given facility, that is, to identify the exact location of an intruder (e.g. a burglar, a fire, etc.) whenever a problem in a facility arises. The requirement that the set have to be dominating would mean that every vertex where there is no monitor on it is connected to at least one monitoring device. Moreover, finding the differentiating-domination number of a graph is equivalent to finding the least number of monitors that can do the certain task in a given network. In other studies, a differentiating dominating set is also referred to as an identifying code (see [14]). Differentiating-domination and some related concepts had been studied in [3], [4], [9], [10], [13], [15], [17], and [18]. In 2015, Natarajan et al. (see [16]) introduced hop domination and made an initial investigation of the concept. The study has led other researchers to investigate it further ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4673 Email addresses: sergio.canoy@g.msuiit.edu.ph (S. Canoy), chrisley.saromines@g.msuiit.edu.ph (C. Saromines) https://www.ejpam.com 440 © 2023 EJPAM All rights reserved. S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 441 and define some of its variants. In fact, a number of variations of hop domination had already been investigated (see [1], [2], [7], [8], [11], [12], [19], [20], [21], and [22]). In this paper, we define and do an initial study of the concept of hop differentiating hop dominating set in a graph. It must be pointed out that a hop differentiating set is ‘almost’ a hop dominating set because it may allow at most a vertex outside the set to be ‘hop undominated’. A result that deals with the concept for disconnected graphs would show that the condition ‘hop differentiating hop dominating’ cannot always be replaced by ‘hop differentiating’. This makes ‘hop differentiating hop dominating’ an interesting concept to consider. This present study is motivated by the introduction of hop domination and differentiating-domination concepts. The new parameter, just like differentiating-domination, can also be used to model the problem of determining the location of monitoring devices so as to identify the exact location of an intruder in a certain facility. 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 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}. The open neighborhood ofX ⊆ V (G) is the setNG(X) = ⋃ u∈X NG(u). The closed neighborhood of X is the set NG[X] = NG(X)∪X. The minimum degree of G, denoted by δ(G), is given by δ(G) = min{degG(u) : u ∈ V (G)}, where degG(u) = |NG(u)|. A set D ⊆ V (G) is a dominating set (resp. total dominating set) of G if for every v ∈ V (G) \ D (resp. v ∈ V (G)), there exists u ∈ D such that uv ∈ E(G), that is, NG[D] = V (G) (resp. NG(D) = V (G)). The domination number (resp. total domination number) of G, denoted by γ(G) (resp. γt(G)), is the minimum cardinality of a dominating (resp. total dominating) set in G. Any dominating (resp. total dominating) set in G with cardinality γ(G) (resp. γt(G)), is called a γ-set (resp. γt-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 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. 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 differentiating in G if for any two distinct vertices v, w ∈ V (G), NG[v] ∩ S ̸= NG[w] ∩ S. A differentiating set S is differentiating-dominating in G if S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 442 NG(v) ∩ S ̸= ∅ for each v ∈ V (G) \ S. The smallest cardinality of a differentiating (resp. differentiating-dominating) set in G is denoted by dn(G) (resp. γD(G)). Any differentiating (resp. differentiating-dominating) set in G with cardinality dn(G) (resp. γD(G)) is called a dn-set (resp. γD-set). A set S ⊆ V (G) is hop differentiating in G if N2 G[u]∩S ̸= N2 G[v]∩S for every two distinct vertices u and v of V (G). A hop differentiating set in G which is also hop dominating is called a hop differentiating hop dominating set. The minimum cardinality of a hop differentiating (resp. hop differentiating hop dominating) set in G, denoted by hdn(G) (resp. γdh(G)), is called the hop differentiating number (resp. hop differentiating hop domination number) of G. Any hop differentiating (resp. hop differentiating hop dominating) set in G with cardinality hdn(G) (resp. γdh(G)) is called an hdn-set (resp. γdh-set). Suppose G is a non-trivial connected graph and suppose that there exist distinct vertices u and v of G such that N2 G[u] = N2 G[v]. Then N2 G[u] ∩ S = N2 G[v] ∩ S for any set S ⊆ V (G). This implies that G does not admit a hop differentiating set. A connected graph G is point determining if distinct vertices have distinct open neigh- borhoods, that is, NG(a) ̸= NG(b) for distinct vertices a, b ∈ V (G). Graph G is said to be point distinguishing if distinct vertices have distinct closed neighborhoods, that is, NG[a] ̸= NG[b] whenever a, b ∈ V (G) and a ̸= b (see [5] and [23]). Graph G is distance- two point determining (resp. distance-two point distinguishing) if N2 G(x) ̸= N2 G(y) (resp. N2 G[x] ̸= N2 G[y]) for any distinct vertices x, y ∈ V (G). It is totally distance-two point determining if N2 G(x) ̸= N2 G(y) and N2 G[x] ̸= N2 G[y] for any distinct vertices x, y ∈ V (G). G is complement point distinguishing if V (G) \ NG(x) ̸= V (G) \ NG(y) for any distinct vertices x, y ∈ V (G). In other words, G is complement point distinguishing if G is point distinguishing. A set S ⊆ V (G) is pointwise non-dominating if for every v ∈ V (G) \ S, there exists u ∈ S such that v /∈ NG(u), i.e., [V (G) \ NG(v)] ∩ S ̸= ∅. The minimum cardinality of a pointwise non-dominating set in G, denoted by pnd(G), is called a pointwise non- domination number of G. Let G be a complement point distinguishing graph. A set S ⊆ V (G) is complement differentiating in G (or differentiating in G) if for any two distinct vertices v, w ∈ V (G), NG[v]∩S = [V (G)\NG(v)]∩S ̸= [V (G)\NG(w)]∩S = NG[w]∩S. A complement differentiating set S in G is called complement differentiating-dominating (or complement differentiating and pointwise non-dominating or differentiating-dominating in G) if for each v ∈ V (G)\S, [V (G)\NG[v]]∩S = NG(v)∩S ̸= ∅. The smallest cardinality of a complement differentiating (resp. complement differentiating-dominating) set in G is denoted by cdn(G) (resp. cdpnd(G)). Any complement-differentiating (resp. complement differentiating-dominating) set in G with cardinality cdn(G) (resp. cdpnd(G)) is called a cdn-set (resp. a cdpnd-set) in G. Clearly, cdn(G) = dn(G) and cdpnd(G) = γD(G). Let G and H be any two graphs. The join G + H is the graph with vertex set V (G+H) = V (G)∪ V (H) and edge set E(G+H) = E(G)∪E(H)∪ {uv : u ∈ V (G), v ∈ V (H)}. The corona G ◦ H is the graph obtained by taking one copy of G and |V (G)| copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. We denote by Hv the copy of H in G ◦ H corresponding to the vertex v ∈ G and write v + Hv for ⟨{v}⟩ + Hv. The lexicographic product G[H] is the graph with vertex set S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 443 V (G[H]) = V (G) × V (H) and (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). Any non-empty set C ⊆ V (G) × V (H) can be expressed as C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. Specifically, Tx = {a ∈ V (H) : (x, a) ∈ C} for each x ∈ S. 3. Results Throughout, a graph is understood to be distance-two point distinguishing whenever a hop differentiating set is assumed (or mentioned) in it. Lemma 1. Let G be a graph on n vertices. Then γdh(G) ≥ ⌈ lnn+ ln2 ln2 ⌉. Proof. Let S be a hop differentiating hop dominating set of G. Since S is a hop domi- nating set, N2 G[v]∩S ̸= ∅ for every v ∈ V (G). Moreover, because it is hop differentiating, it follows that 2|S| > n. Hence, |S| ≥ ⌈ lnn+ln2 ln2 ⌉. In particular, if S is a γdh-set of G, then γdh(G) ≥ ⌈ lnn+ln2 ln2 ⌉. Theorem 1. Let G1, G2, . . . , Gk be the distinct (distance-two point distinguishing) com- ponents of G, where k ≥ 2. Then S is a hop differentiating hop dominating set in G if and only if Sj = S ∩ V (Gj) is a hop differentiating hop dominating set in Gj for each j ∈ {1, 2, . . . , k}. Proof. Suppose S is a hop differentiating hop dominating set in G and let j ∈ {1, 2, . . . , k}. Let v ∈ V (Gj) \ Sj . Since v /∈ S and S is a hop dominating set, there exists w ∈ S such that v ∈ N2 G(w). This implies that w ∈ Sj and v ∈ N2 Gj (w). This shows that Sj is a hop dominating set in Gj . Next, let a, b ∈ V (Gj) where a ̸= b. Since S is a hop differentiating set N2 Gj [a] ∩ Sj = N2 G[a] ∩ S ̸= N2 G[b] ∩ S = N2 Gj [b] ∩ Sj . Thus, Sj is a hop differentiating hop dominating set in Gj for each j ∈ {1, 2, . . . , k}. For the converse, suppose that Sj = S∩V (Gj) is a hop differentiating hop dominating set in Gj for each j ∈ {1, 2, . . . , k}. Then clearly, S is a hop dominating set in G. Let v, w ∈ V (G) with v ̸= w and let Gi and Gj be the components of G with v ∈ V (Gi) and w ∈ V (Gj). If i ̸= j, then N2 G[v] ∩ S = N2 Gi [v] ∩ Si ̸= N2 Gj [w] ∩ Sj = N2 G[w] ∩ S. If i = j, then N2 G[v] ∩ S = N2 Gi [v] ∩ Si ̸= N2 Gi [w] ∩ Si = N2 G[w] ∩ S S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 444 since Si is a hop differentiating set in Gi. Therefore, S is a hop differentiating hop dominating set in G. It is worth mentioning that Theorem 1 does not hold if ‘hop differentiating hop dom- inating’ is replaced by ‘hop differentiating’. Indeed, if there are two distinct hop differ- entiating sets Sj and Sk which have each a single vertex in V (Gj) \ Sj and V (Gk) \ Sk, respectively, such that these vertices are not hop-dominated in the respective components, then the set S cannot be a hop differentiating set in G. The next result follows from Theorem 1. Corollary 1. Let G1, G2, . . . , Gk be the distinct components of G. Then γdh(G) =∑k j=1 γdh(Gj). Corollary 2. Let G1, G2, . . . , Gk be the distinct components of G. If each of these com- ponents is complete, then γdh(G) = |V (G)|. In particular, γdh(Kn) = γdh(Kn) = n for all n ≥ 1. Proposition 1. Let G be a graph on n ≥ 3 vertices. Then 3 ≤ γdh(G) ≤ n. Moreover, the following hold: (i) If n = 3 and γdh(G) = 3, then G ∈ {K3,K3,K1 ∪K2}. (ii) If n = 4, then γdh(G) = 3 if and only if G is a graph obtained from K3 by attaching a pendant vertex to one of the vertices of K3. Proof. Suppose S is a γdh-set of G. Clearly, γdh(G) ≤ n. Now, by Lemma 1, γdh(G) ≥ ⌈ lnn+ ln2 ln2 ⌉ ≥ ⌈ ln3 + ln2 ln2 ⌉ = 3. Next, suppose n = 3 and γdh(G) = 3. By Corollary 2, G ∈ {K3,K3,K1∪K2}, showing that (i) holds. Suppose now that n = 4 and γdh(G) = 3. Let S = {a, b, c} be a γdh-set of G and let v ∈ V (G) \ S. Since γdh(K1 ∪K3) = γdh(K2 ∪K2) = γdh(K2 ∪K2) = γdh(K4) = γdh(K4) = 4 by Corollary 2, and because G is distance-two point distinguishing, G /∈ {K1 ∪K3,K2 ∪K2,K2 ∪K2,K4,K1 ∪ P3,K4, P4, C4,K1,3, H}, where H is obtained from C4 by adding an edge connecting the non-adjacent vertices of C4. Since there are only eleven (11) non-isomorphic graphs of order four (4), it follows that G is a graph obtained from K3 by attaching a pendant vertex to one of the vertices of K3. For the converse, suppose that G is a graph obtained from K3 by attaching a pendant vertex to one of the vertices of K3. Let V (G) = {a, b, c, v} such that ⟨{a, b, c}⟩ = K3 and S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 445 .................................... .................................... .................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ... ........................................................................................................................................................ ............................................................................................................... v a b c G • • • Figure 1: Graph G on 4 vertices and γdh(G) = 3 va ∈ E(G) (see Figure 1). Let S = {a, b, c}. Since dG(v, b) = 2, S is a hop dominating set in G. Moreover, since N2 G[v] ∩ S = {b, c}, N2 G[a] ∩ S = {a}, N2 G[b] ∩ S = {b}, and N2 G[c] ∩ S = {c} are all distinct, S is a hop differentiating set. Thus, by the first part, γdh(G) = 3. This completes the proof of (ii). The next result is found in [11]. Theorem 2. Let G and H be any two graphs. A set S ⊆ V (G+H) is hop dominating in G+H if and only if S = SG ∪ SH , where SG and SH are pointwise non-dominating in G and H, respectively. Theorem 3. Let G and H be any two (complement distance-two point distinguishing) graphs. Then S ⊆ V (G + H) is hop differentiating hop dominating in G + H if and only if S = SG ∪ SH , where SG and SH are complement differentiating and pointwise non-dominating sets in G and H (differentiating-dominating in G and H), respectively. Proof. Suppose that S is a hop differentiating hop dominating set in G + H. Let SG = V (G) ∩ S and SH = V (H) ∩ S. Since S is a hop dominating set in G+H, SG ̸= ∅ and SH ̸= ∅. By Theorem 2, SG and SH are pointwise non-dominating sets in G and H, respectively. If SG = V (G), then it is complement differentiating in G. Next, let x, y ∈ V (G) where x ̸= y. Since S is a hop differentiating set, [V (G) \NG(x)]∩SG = N2 G+H [x]∩ S ̸= N2 G+H [y] ∩ S = [V (G) \NG(y)] ∩ SG, showing that SG is complement differentiating in G. Thus, SG is a complement differentiating and pointwise non-dominating set in G. Similarly, SH is a complement differentiating and pointwise non-dominating set in H. For the converse, suppose that S = SG ∪ SH where SG and SH are complement differentiating and pointwise non-dominating sets in G and H, respectively. Then S is a hop dominating set in G + H by Theorem 2. Next, let a, b ∈ V (G + H) where a ̸= b. Suppose that a, b ∈ V (G). Since SG is complement differentiating in G, N2 G+H [a] ∩ S = [V (G) \ NG(a)] ∩ SG ̸= [V (G) \ NG(b)] ∩ SG = N2 G+H [b] ∩ S. Similarly, N2 G+H [a] ∩ S = [V (H) \NH(a)]∩SH ̸= [V (H) \NH(b)]∩SH = N2 G+H [b]∩S if a, b ∈ V (H). Suppose now that a ∈ V (G) and b ∈ V (H). If a ∈ SG, then a ∈ (N2 G+H [a]∩S)\(N2 G+H [b]∩S). If a /∈ SG, then there exists d ∈ SG \ NG(a) because SG is pointwise non-dominating in G. Hence, d ∈ (N2 G+H [a] ∩ S) \ (N2 G+H [b] ∩ S). In either case, we have N2 G+H [a] ∩ S ̸= N2 G+H [b] ∩ S. Therefore, S is a hop differentiating hop dominating set in G+H. S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 446 Corollary 3. Let G be a graph and let n be a positive integer. Then S ⊆ V (Kn +G) is a hop differentiating hop dominating set in Kn + G if and only if S = V (Kn) ∪ SG, where SG is complement differentiating and pointwise non-dominating set in G. Proof. The only pointwise non-dominating set in Kn is V (Kn). Thus, by Theorem 3, the result follows. The next results follow directly from Theorem 3 and Corollary 3. Corollary 4. Let G and H be any two graphs. Then γdh(G+H) = cdpnd(G) + cdpnd(H) = γD(G) + γD(H). Corollary 5. Let G be a graph and let n be a positive integer. Then γdh(Kn + G) = n+ cdpnd(G) = n+ γD(G). The next result is a restatement of the one in [11]. Theorem 4. Let G and H be any two graphs. A set C ⊆ V (G) is a hop dominating set in G ◦H if and only if C = A∪ (∪v∈V (G)Cv), where A ⊆ V (G) and Cv ⊆ V (Hv) for each v ∈ V (G), and satisfies the following conditions: (i) For each w ∈ V (G) \ A, there exists x ∈ A with dG(w, x) = 2 or there exists y ∈ NG(w) with Cy ̸= ∅. (ii) Cw is a pointwise non-dominating set in Hw for each w ∈ V (G) \NG(A). Theorem 5. Let G and H be non-trivial connected graphs such that H is complement point distinguishing. Then S ⊆ V (G ◦H) is hop differentiating hop dominating in G ◦H if and only if S = A ∪ [∪v∈V (G)Dv] and satisfies the following conditions: (i) Dw is a pointwise non-dominating set in Hw for each w ∈ V (G) \NG(A). (ii) Dv is complement differentiating in Hv for each v ∈ V (G). (iii) For any two distinct vertices v, w ∈ V (G), NG(v) ̸= NG(w) or N 2 G[v]∩A ̸= N2 G[w]∩A. (iv) Dw is a total dominating set in Hw whenever NG(v) = {w} for some v ∈ V (G). (v) If Dv and Dw, where v ̸= w, are not pointwise non-dominating in Hv and Hw, respectively, then NG(v) ∩A ̸= NG(w) ∩A. Proof. Suppose S is a hop differentiating hop dominating set inG◦H. LetA = S∩V (G) and let Dv = S ∩ V (Hv) for each v ∈ V (G). Then S = A∪ [∪v∈V (G)Dv] and, by Theorem 4, (i) holds. Let v ∈ V (G) and let a, b ∈ V (Hv) with a ̸= b. Since S is a hop differentiating set, ([V (Hv) \NHv(a)] ∩Dv) ∪ [NG(v) ∩A] = N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S = ([V (Hv) \NHv(b)] ∩Dv) ∪ [NG(v) ∩A]. S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 447 Hence, [V (Hv) \NHv(a)] ∩Dv ̸= [V (Hv) \NHv(b)] ∩Dv, showing that Dv is a complement differentiating set in Hv. Thus, (ii) holds. Next, let v, w ∈ V (G) with v ̸= w. Since S is a hop differentiating set, [N2 G[v] ∩A] ∪ [∪x∈NG(v)Dx] = N2 G◦H [v] ∩ S ̸= N2 G◦H [w] ∩ S = [N2 G[w] ∩A] ∪ [∪y∈NG(w)Dy]. This implies that N2 G[v] ∩ A ̸= N2 G[w] ∩ A or NG(v) ̸= NG(w), showing that (iii) holds. To show (iv), let w ∈ V (G) such that NG(v) = {w} for some v ∈ V (G). Suppose Dw is not a total dominating set in Hw. Then there exists p ∈ V (Hw) such that p /∈ NHw(Dw). It follows that N2 G◦H [p]∩S = (NG(w)∩A)∪[(V (Hw)\NHw(p))∩Dw] = (NG(w)∩A)∪Dw = N2 G◦H [v]∩S, a contradiction to the assumption that S is a hop differentiating set. Therefore, Dw is a total dominating set in Hw, showing that (iv) holds. Finally, suppose Dv and Dw, where v ̸= w, are not pointwise non-dominating sets in Hv. Then there exist p ∈ V (Hv)\Dv and q ∈ V (Hw) \Dw such that (V (Hv) \NHv(p))∩Dv = ∅ and (V (Hw) \NHw(q))∩Dw = ∅. Since S is hop differentiating, NG(v) ∩A ̸= NG(w) ∩A. This shows that (v) holds. For the converse, suppose that S is as described and satisfies properties (i)-(v). Let v ∈ V (G) \ A and choose any u ∈ NG(v). By (ii), Du is complement differentiating and so Du ̸= ∅. Thus, S satisfies (i) and (ii) of Theorem 4, showing that it is a hop dominating set in G ◦H. Now let a, b ∈ V (G ◦H) with a ̸= b and let v, w ∈ V (G) such that a ∈ V (v +Hv) and b ∈ V (w +Hw). Consider the following cases: Case 1: v = w Suppose a, b ∈ V (Hv). Since Dv is a complement differentiating set in Hv (by (ii)), N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S. Suppose a = v and b ∈ V (Hv). Pick any z ∈ NG(v). Since Dz ⊆ N2 G◦H [a] \N2 G◦H [b], it follows that N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S. Case 2: v ̸= w Suppose a = v and b = w. Then v, w ∈ V (G). By property (iii), NG(v) ̸= NG(w) or N2 G[v] ∩ A ̸= N2 G[w] ∩ A. If N2 G[v] ∩ A ̸= N2 G[w] ∩ A, then N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S. Suppose NG(v) ̸= NG(w). We may assume that there exists p ∈ NG(v) \ NG(w). Then Dp ⊆ N2 G◦H [a] \N2 G◦H [b]. Hence, N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S. Next, suppose that a = v and b ∈ V (Hw) (or b = w and a ∈ V (Hv)). If |NG(v)| > 1 or vw /∈ E(G), pick any z ∈ NG(v) \ {w}. Then Dz ⊆ N2 G◦H [a] \ N2 G◦H [b]. It follows that N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S. Suppose that NG(v) = {w}. By (iv), Dw is a total dominating set of Hw. Hence, (V (Hw) \ NHw(b)) ∩ Dw ̸= Dw. This would imply that N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S. Finally, suppose that a ∈ V (Hv) and b ∈ V (Hw). If [V (Hv)\NHw(a)]∩Dv ̸= ∅ or [V (Hw)\NHw(b)]∩Dw ̸= ∅, thenN2 G◦H [a]∩S ̸= N2 G◦H [b]∩S. Suppose both sets are empty. Then NG(v) ∩ A ̸= NG(w) ∩ A by (v). It follows that N2 G◦H [a] ∩ S ̸= N2 G◦H [b] ∩ S. S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 448 Accordingly, S is a hop differentiating hop dominating set of G ◦H. Corollary 6. Let G and H be non-trivial connected graphs such that δ(G) ≥ 2 and G and H are point determining and complement point distinguishing, respectively. Then γdh(G ◦H) ≤ cdpnd(H)|V (G)|. Proof. Let A = ∅ and let Dv be a cdpnd-set of H for each v ∈ V (G). Then S = A ∪ [∪v∈V (G)Dv] = ∪v∈V (G)Dv is a hop differentiating hop dominating set in G ◦ H by Theorem 5. Thus, γdh(G ◦H) ≤ |C| = cdpnd(H)|V (G)|. This proves the assertion. We note that the bound given in Corollary 6 is tight. Indeed, if G = H = K2, then cdpnd(H) = 2 and γdh(G ◦H) = 4 = cdpnd(H)|V (G)|. The next result is found in [11]. Theorem 6. Let G and H be connected non-trivial graphs. A subset C = ⋃ x∈S [{x}×Tx] of V (G[H] is a hop dominating set in G[H] if and only if the following conditions hold. (i) S is a hop dominating set in G. (ii) Tx is a pointwise non-dominating set in H for each x ∈ S \N2 G(S). Theorem 7. Let G and H be non-trivial connected graphs such that G and H are, re- spectively, distance-two point distinguishing and complement point distinguishing. Then C = ⋃ x∈S [{x} × Tx], where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S, is a hop differen- tiating hop dominating set in G[H] if and only if the following conditions hold: (i) S = V (G) (ii) Tx is a pointwise non-dominating set in H for each x ∈ S \N2 G(S). (iii) Tx is a complement differentiating set in H for all x ∈ S. (iv) If N2 G(x) = N2 G(y) for distinct vertices x and y, then Tx or Ty is pointwise non- dominating in H. Proof. Suppose C is a hop differentiating hop dominating set in G[H]. Then, by Theorem 6, (ii) holds. Suppose there exists z ∈ V (G) \ S. Pick distinct vertices a, b ∈ V (H). Then (z, a), (z, b) ∈ V (G[H]) \ C and so N2 G[H][(z, a)] ∩ C = ⋃ x∈N2 G(z)∩S [{x} × Tx] = N2 G[H][(z, b)] ∩ C. S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 449 This implies that C is not a hop differentiating set, contrary to our assumption. Thus, S = V (G), showing that (i) holds. Now let x ∈ S and p, q ∈ V (H) with p ̸= q. Then (x, p), (x, q) ∈ V (G[H]) and N2 G[H][(x, p)] ∩ C = [{x} × [(V (H) \NH(p)) ∩ Tx]] ∪ [∪w∈N2 G(x)∩S({w} × Tw)] and N2 G[H][(x, q)] ∩ C = [{x} × [(V (H) \NH(q)) ∩ Tx]] ∪ [∪w∈N2 G(x)∩S({w} × Tw)]. Since C is a hop differentiating set, (V (H) \NH(p)) ∩ Tx ̸= (V (H) \NH(q)) ∩ Tx. Hence, Tx is a complement-differentiating set inH, showing that (iii) holds. Next, suppose that x and y are distinct vertices of G with N2 G(x) = N2 G(y). Suppose Tx and Ty are not pointwise non-dominating sets. Then there exist p ∈ V (H) \ Tx and q ∈ V (H) \ Ty such that [V (H) \ NH(p)] ∩ Tx = ∅ and [V (H) \ NH(q)] ∩ Ty = ∅. Since N2 G(x) = N2 G(y), it follows that N2 G[H][(x, p)] ∩ C = N2 G[H][(y, q)] ∩ C, contradicting the assumption that C is a hop differentiating set in G[H]. Thus, Tx or Ty is pointwise non-dominating in H, showing that (iv) holds. For the converse, suppose that C satisfies properties (i)-(iv). Since (i) and (ii) hold, C is a hop dominating set by Theorem 6. Next, let (v, q), (w, s) ∈ V (G[H]) with (v, q) ̸= (w, s). Then N2 G[H][(v, q)] ∩ C = [{v} × [(V (H) \NH(q)) ∩ Tv] ∪ [ ⋃ z∈N2 G(v) [{z} × Tz], and N2 G[H][(w, s)] ∩ C = [{w} × (V (H) \NH(s)) ∩ Tw] ∪ [ ⋃ y∈N2 G(w) [{y} × Ty]. Consider the following cases: Case 1: v = w Then q, s ∈ V (H) with q ̸= s. By (iii), Tv is a complement-differentiating set; hence, [V (H)\NH(q)]∩Tv ̸= [V (H)\NH(s)]∩Tv. It follows that N 2 G[H][(v, q)]∩C ̸= N2 G[H][(v, s)]∩ C. Case 2: v ̸= w Suppose first that dG(v, w) ̸= 2. If N2 G(v) ̸= N2 G(w), then clearly, N2 G[H][(v, q)] ∩ C ̸= N2 G[H][(w, s)]∩C. If N2 G(v) = N2 G(w), then Tv or Tw is pointwise non-dominating in H by (iv). Hence, N2 G[H][(v, q)]∩C ̸= N2 G[H][(w, s)]∩C. Next, suppose that dG(v, w) = 2. Since G is distance-two point distinguishing, N2 G[v] ̸= N2 G[w]. It follows that N 2 G[H][(v, q)]∩C ̸= N2 G[H][(w, s)] ∩ C. Accordingly, C is a hop differentiating hop dominating set in G[H]. S. Canoy Jr., C. Saromines / Eur. J. Pure Appl. Math, 16 (1) (2023), 440-453 450 Corollary 7. Let G and H be non-trivial connected graphs such that G and H are, respectively, distance-two point distinguishing and complement point distinguishing. Then γdh(G[H]) ≤ |V (G)|cdpnd(H) = |V (G)|γD(H). If, in addition, G is also distance-two point determining and γ(G) ̸= 1, then γdh(G[H]) = |V (G)|cdn(H) = |V (G)|dn(H). Proof. Let S = V (G) and let Tx be a cdpnd-set in H for each x ∈ V (G). By Theorem 7, C = ⋃ x∈S [{x}×Tx] is a hop differentiating hop dominating set in G[H]. It follows that γdh(G[H]) ≤ |C| = |V (G)|cdpnd(H). Next, suppose that γ(G) ̸= 1. Let S′ = V (G) and let Rx be a cdn-set in H for each x ∈ S′. Since γ(G) ̸= 1, x ∈ N2 G(S ′) for each x ∈ S′. Thus, by Theorem 7, C = ⋃ x∈S′ [{x} ×Rx] is a hop differentiating hop dominating set in G[H]. It follows that γdh(G[H]) ≤ |C| = |V (G)|cdn(H). Now, if C0 = ⋃ x∈S0 [{x} × Tx] is a γdh-set in G[H], then S0 = V (G) and Tx is a complement-differentiating set in H for each x ∈ V (G), by Theorem 7. Hence, γdh(G[H]) = |C0| = ∑ x∈S0 |Tx| ≥ |V (G)|cdn(H). Therefore, γlh(G[H]) = |V (G)|cdn(H). Corollary 8. Let G and H be non-trivial connected graphs such that G and H are, respectively, totally distance-two point determining and complement point distinguishing. If γ(G) = 1, then γdh(G[H]) = cdpnd(H) + (|V (G)| − 1)cdn(H). Proof. Let DG = {v ∈ V (G) : {v} is a dominating set of G}. Since G is distance-two point distinguishing, it follows that |DG| = 1. Set S = V (G). Let Tv be a cdpnd-set in H for v ∈ DG and let Tx be a cdn-set in H for each x ∈ V (G) \ {v}. Then, by Theorem 7, C = [ ⋃ x∈S\{v}({x} × Tx)] ∪ ({v} × Tv) is a hop differentiating hop dominating set in G[H]. Hence, γdh(G[H]) ≤ |C| = cdpnd(H) + (|V (G)| − 1)cdn(H). Suppose now that C∗ = [ ⋃ x∈S∗({x}×Rx)] is a γdh-set in G[H] and let DG = {v}. By Theorem 7, S∗ = V (G), Rv is complement-differentiating and pointwise non-dominating and Rx is complement-differentiating in H for each x ∈ V (G) \ {v}. Thus, γdh(G[H]) = |C∗| = |Rv|+ ∑ x∈S∗\{v} |Rx| ≥ cdpnd(H) + (|V (G)| − 1)cdn(H). Therefore, γdh(G[H]) = cdpnd(H) + (|V (G)| − 1)cdn(H) as asserted. Corollary 9. Let G be a non-trivial connected totally distance-two point determining graph and let p ≥ 2 be a positive integer. Then γdh(G[Kp]) = { (p− 1)|V (G)| if γ(G) ̸= 1 (p− 1)|V (G)|+ 1 if γ(G) = 1. REFERENCES 451 Proof. Suppose first that γ(G) ̸= 1. By Corollary 7 and the fact that cdn(Kp) = dn(Kp) = p− 1, it follows that γdh(G[Kp]) = (p− 1)|V (G)|. Next, suppose that γ(G) = 1. By Corollary 8 and the fact that cdpnd(Kp) = γD(Kp) = p, we have γdh(G[Kp]) = p+ (p− 1)(|V (G)| − 1) = (p− 1)|V (G)|+ 1. Corollary 10. Let H be a non-trivial connected complement point distinguishing graph and let p ≥ 2 be a positive integer. Then γdh(Kp[H]) = p[cdpnd(H)]. Proof. Let G = Kp. Then v is a dominating vertex of G for each v ∈ V (G). Thus, if C0 = ⋃ z∈S0 [{z} × Tz] is a γdh-set of G[H], then S0 = V (G) and each Tz is a cdpnd-set of H by Theorem 7. Consequently, γdh(Kp[H]) = p[cdpnd(H)]. 4. Conclusion Hop differentiating hop domination is introduced and studied for some graphs. In particular, characterizations of the hop differentiating hop dominating sets in the join, corona, and lexicographic product of two graphs are given. These characterizations are used to obtain either an upper bound or the exact value of the hop differentiating hop domination number of the graph. The concept can be studied further for other interesting graphs and the complexity of the hop differentiating hop dominating decision problem can likewise be investigated. 5. Acknowledgements The authors would like to thank the referees for the invaluable assistance they gave us through their comments and suggestions which contributed to the improvement of the paper. 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