EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1455-1463 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Hop Dominating Sets in Graphs Under Binary Operations Sergio R. Canoy Jr.1, Reynaldo V. Mollejon2, John Gabriel E. Canoy∗ 1 Department of Mathematics and Statistics, College of Science and Mathematics, Center for Graph Theory, Algebra, and Analysis, Premier Research Institute in Science and Mathematics, MSU-Iligan Institute of Technology, Tibanga, Iligan City, Philippines 2 Department of Teacher Education, Visayas State University-Villaba, Villaba, Leyte, Philippines Abstract. Let G be a (simple) connected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. The minimum cardinality of a hop dominating set of G, denoted by γh(G), is called the hop domination number of G. In this paper we revisit the concept of hop domination, relate it with other domination concepts, and investigate it in graphs resulting from some binary operations. 2010 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Domination, hop domination, join, corona, and lexicographic product 1. Introduction Domination in graph and several variations of the concept have been widely studied by many researchers. The two books by Haynes et al. [3, 4] give an excellent treatment of the standard domination concept and some of its variants. Recently, Natarajan and Ayyaswamy [6] introduced and studied the concept of hop domination in a graph. In another study, Ayyaswamy et al. [2] investigated the same concept and gave bounds of the hop domination number of some graphs. Henning and Rad [5] also studied the concept and answered a question posed by Ayyaswamy and Natarajan in [6]. They presented probabilistic upper bounds for the hop domination number and showed that the decision problems for the 2-step dominating set and hop dominating set problems are NP-complete for planar bipartite graphs and planar chordal graphs. Pabilona and Rara [7] considered the variant called connected hop domination and studied it in graphs under some binary operations. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3550 Email addresses: serge canoy@yahoo.com (S. Canoy Jr.), reynaldo.mollejon@g.msuiit.edu.ph (R. Mollejon),jgcanoy@mail.com (JG. Canoy) http://www.ejpam.com 1455 c© 2019 EJPAM All rights reserved. S. Canoy Jr., R. Mollejon, JG. Canoy / Eur. J. Pure Appl. Math, 12 (4) (2019), 1455-1463 1456 Let G = (V (G), E(G)) be a simple graph. The open neighbourhood of a vertex v of G is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)} and its closed neighbourhood is the set NG[v] = NG(v) ∪ {v}. The degree of v, denoted by degG(v), is equal to |NG(v)| and the maximum degree of G, denoted by ∆(G), is equal to max{degG(v) : v ∈ V (G)}. The open hop neighbourhood of vertex v is the set NG(v, 2) = {w ∈ V (G) : dG(v, w) = 2}, where dG(v, w) denotes the distance between v and w (the length of a shortest path joining v and w). The open neighbourhood of a subset S of V (G) is the set NG(S) = ∪v∈SNG(v) and its closed neighbourhood is the set NG[S] = NG(S) ∪ S. A set S ⊆ V (G) is a dominating set (resp. total dominating set) of G if NG[S] = V (G) (resp. NG(S) = V (G)). The smallest cardinality of a dominating (resp. total dominating) set of G, denoted by γ(G) (resp. γt(G)), is called the domination number (resp. total domination number) of G. A dominating (resp. total dominating) set S of G with |S| = γ(G) (resp. |S| = γt(G)), is called a γ-set (resp. γt-set) of G. It should be noted that only graphs without isolated vertices admit total dominating sets. A set S ⊆ V (G) is a hop dominating set (total hop dominating set) of G if for each x ∈ V (G) \ S (resp. x ∈ V (G)), there exists z ∈ S such that dG(x, z) = 2. The smallest cardinality of a hop dominating (total hop dominating) set of G, denoted by γh(G) (resp. γth(G)), is called the hop domination number (total hop domination number) of G. A hop dominating (total hop dominating) set S of G with |S| = γh(G) (resp. |S| = γth(G)) is called a γh-set (resp. γth-set) of G. A set S ⊆ V (G) is a (1, 2)∗-dominating set (resp. (1, 2)∗-total dominating set) of G if it is a dominating (resp. total dominating) set of G and for each x ∈ V (G) \ S, there exists z ∈ S such that dG(x, z) = 2. The smallest cardinality of a (1, 2)∗-dominating (resp. (1, 2)∗-total dominating) set of G, denoted by γ∗1,2(G) (resp. γ∗t1,2(G)), is called the (1, 2)∗- domination number (resp. (1, 2)∗-total domination number) of G. A (1, 2)∗-dominating (resp. (1, 2)∗- total dominating) set S with |S| = γ∗1,2(G) (resp. |S| = γ∗t1,2(G)) is called a γ∗1,2-set (resp. γ∗t1,2-set) of G. Clearly, S ⊆ V (G) is a (1, 2)∗-dominating (resp. (1, 2)∗-total dominating) set if and only if it is both a dominating (resp. total dominating) and a hop dominating set. The concept of (1, 2)∗-domination (a variation of (1, 2)-domination) is introduced and investigated in [1]. A set D ⊆ V (G) is a point-wise 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 point-wise non- dominating set of G, denoted by pnd(G), is called the point-wise non-domination number of G. A dominating set S which is also a point-wise non-dominating set of G is called a dominating point-wise non-dominating set of G. The smallest cardinality of a dominating point-wise non-dominating set of G will be denoted by γpnd(G). Any point-wise non- dominating (resp. dominating point-wise non-dominating) set S of G with |S| = pnd(G) (resp. |S| = γpnd(G)), is called a pnd-set (resp. γpnd-set) of G. 2. Results The first result, which will be needed later, is found in [1]. S. Canoy Jr., R. Mollejon, JG. Canoy / Eur. J. Pure Appl. Math, 12 (4) (2019), 1455-1463 1457 Proposition 1. [1] Let G be a graph. Then 1 ≤ pnd(G) ≤ |V (G)|. Moreover, (i) pnd(G) = |V (G)| if and only if G is a complete graph; (ii) pnd(G) = 1 if and only if G has an isolated vertex; and (iii) pnd(G) = 2 if and only if G has no isolated vertex and there exist distinct vertices a and b of G such that NG(a) ∩NG(b) = ∅. The join of graphs G and H is the graph G+H with vertex set V (G+H) = V (G) ∪ V (H) and edge set E(G+H) = E(G) ∪ E(H) ∪ {uv : u ∈ V (G) and v ∈ V (H)}. Theorem 1. Let G and H be any two graphs. A set S ⊆ V (G + H) is hop dominating set of G+H if and only if S = SG∪SH , where SG and SH are point-wise non-dominating sets of G and H, respectively. Proof. Suppose that S is a hop dominating set of G + H. Let SG = S ∩ V (G) and SH = S ∩ V (H). If SG were empty, then S = SH . Since V (G) ⊆ NG(S), it follows that S is not a hop dominating set, a contradiction. Thus, SG 6= ∅. Similarly, SH 6= ∅. Now let v ∈ V (G)\SG. Since S is hop dominating set, there exists z ∈ S such that dG+H(v, z) = 2. Hence, z ∈ SG and v /∈ NG(z). This shows that SG is a point-wise non-dominating set of G. Similarly, SH is a point-wise non-dominating set of H. For the converse, suppose that S = SG ∪ SH , where SG and SH are point-wise non- dominating sets of G and H, respectively. Let v ∈ V (G + H) \ S. If v ∈ V (G), then v ∈ NG+H(SH). Since SG is a point-wise non-dominating set of G, there is a vertex y ∈ SG \ NG(v). It follows that dG+H(v, y) = 2. The same argument can be used if v ∈ V (H). Therefore S is a hop dominating set of G+H. The next result is a consequence of Theorem 1 and Proposition 1 Corollary 1. Let G and H be any two graphs of orders m and n, respectively. Then γh(G+H) = pnd(G) + pnd(H). In particular, (i) γh(G+H) = m+ n if G and H are complete; (ii) γh(G+H) = 2 if G and H have isolated vertices; (iii) γh(G+H) = 1 + pnd(H) if G = K1; (iv) γh(G+H) = 4 if G = Pm and H = Pn (m,n ≥ 2); and (v) γh(G+H) = 4 if G = Cm and H = Cn (m,n ≥ 4). The corona of graphs G and H, denoted by G ◦H, is the graph obtained from G by taking a copy Hv of H and forming the join 〈v〉+Hv = v +Hv for each v ∈ V (G). S. Canoy Jr., R. Mollejon, JG. Canoy / Eur. J. Pure Appl. Math, 12 (4) (2019), 1455-1463 1458 Theorem 2. 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 ∈ V (G) ∩NG(w) with V (Hy) ∩ C 6= ∅, (ii) Sv ⊆ V (Hv) for each v ∈ V (G) ∩NG(A), and (iii) Ew ⊆ V (Hw) is a point-wise non-dominating set of Hw for each w ∈ V (G)\NG(A). Proof. Suppose C is a hop dominating set of G ◦ H and set A = C ∩ V (G). Let w ∈ V (G) \ A. Then there exists x ∈ C such that dG◦H(w, x) = 2. If x ∈ A, then dG(w, x) = 2. Suppose that x /∈ A. Then there exists y ∈ V (G) such that x ∈ V (Hy). Since dG◦H(w, x) = 2, it follows that y ∈ NG(w). Thus, (i) holds. Let v ∈ V (G). Set Sv = C ∩ V (Hv) if v ∈ V (G) ∩ NG(A) and Ew = C ∩ V (Hw) if v ∈ V (G) \ NG(A). Then, clearly, Sv ⊆ V (Hv) and Ew ⊆ V (Hw). Suppose that w ∈ V (G) \ NG(A) and let q ∈ V (Hw) \ Ew. Since C is a hop dominating set of G ◦H, there exists u ∈ C such that dG◦H(q, u) = 2. By assumption, u /∈ A. Thus, u ∈ Ew and qu /∈ E(Hw). Therefore Ew is a point-wise non-dominating set of Hw, showing that (iii) holds. For the converse, suppose that C has the given form and satisfies properties (i), (ii), and (iii). Let z ∈ V (G ◦H) \C and let v ∈ V (G) such that z ∈ V (v +Hv). Consider the following cases: Case 1. z = v Then z /∈ A. From the assumption that (i) holds, it follows that there exists y ∈ C such that dG◦H(z, y) = 2. Case 2. z 6= v Then z ∈ V (Hv). If v ∈ NG(A), say vw ∈ E(G) for some w ∈ A, then dG◦H(z, w) = 2. Suppose that v /∈ NG(A). Then z ∈ V (Hv) \Ev where Ev is a point-wise non-dominating set of Hv by property (iii). Thus, there exists p ∈ Ev ⊂ C such that dG◦H(x, p) = 2. Accordingly, C is a hop dominating set of G ◦H. Corollary 2. Let G be a connected non-trivial graph and let H be any graph. Then: (i) γh(G ◦H) ≤ min{γ∗t1,2(G), [1 + pnd(H)]γ(G)}. (ii) γh(G ◦H) = 2 if γ∗t1,2(G) = 2. (iii) γh(G ◦H) = 2 if γ(G) = 1 and H has an isolated vertex. Let A be a γ∗t1,2-set of G. Since A is a total dominating set of G, V (G) \NG(A) = ∅. Let w ∈ V (G) \ A. Since A is a hop dominating set of G, there exists x ∈ A such that dG(x,w) = 2. Setting Sv = ∅ for each v ∈ A ∩NG(A) = A, we find that C = A satisfies S. Canoy Jr., R. Mollejon, JG. Canoy / Eur. J. Pure Appl. Math, 12 (4) (2019), 1455-1463 1459 conditions (i), (ii), and (iii) of Theorem 2. Thus, C = A is a hop dominating set of G◦H and γh(G ◦H) ≤ |C| = |A| = γ∗t1,2(G). Next, let A0 be a γ-set of G and let D0 be a pnd-set of H. Set Sv = Dv, where Dv ⊆ V (Hv) and 〈Dv〉 ∼= 〈D〉, for each v ∈ A0. Since A0 is a dominating set of G, w ∈ NG(A0) for each w ∈ V (G) \ A0 (hence, [V (G) \ A0] \ NG(A0) = ∅). Thus, by Theorem 2, C0 = A0 ∪ (∪u∈A0Sv) is a hop dominating set of G ◦ H, and γh(G ◦ H) ≤ |C0| = |A0|+ |A0|.pnd(H) = [1 + pnd(H)]γ(G). Therefore, γh(G ◦H) ≤ min{γ∗t1,2(G), [1 + pnd(H)]γ(G)}, showing that (i) holds. Statements (ii) and (iii) are immediate from (i) and the fact that γh(G ◦H) ≥ 2. Observation: The bound given in Corollary 2(i) is attainable (as given in (ii) and (iii)). It can also be verified easily that γh(C5 ◦ P3) = γ∗t1,2(C5) = 3 < 6 = [1 + pnd(P3)]γ(C5) and γh(K4 ◦ P3) = [1 + pnd(P3)]γ(K4) = 3 < 4 = γ∗t1,2(K4). It is worth noting that the inequality is also attainable. As a matter of fact, it can be shown that γh(K5 ◦K4) = 3 < 5 = min{[1 + pnd(K4)]γ(K5), γ ∗t 1,2(K5)}. The lexicographic product of graphs G and H, denoted by G[H], is the graph with vertex set V (G[H]) = V (G)× V (H) such that (v, a)(u, b) ∈ E(G[H]) if and only if either uv ∈ E(G) or u = v and ab ∈ E(H). Note that every non-empty subset C of 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. Theorem 3. 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 of G[H] if and only if the following conditions hold: (i) S is a hop dominating set of G; (ii) Tx is a point-wise non-dominating set of H for each x ∈ S with |NG(x, 2) ∩ S| = 0. Proof. Suppose C is a hop dominating set of G[H]. Let u ∈ V (G) \ S and pick any a ∈ V (H). Since C is a hop dominating set and (u, a) /∈ C, there exists (y, b) ∈ C such that dG[H]((u, a)(y, b)) = 2. This implies that y ∈ S and dG(u, y) = 2. Since u was arbitrarily chosen, it follows that S is a hop dominating set of G. Thus, (i) holds. Now let x ∈ S∗ and let p ∈ V (H) \ Tx. Then (x, p) /∈ C. Again, noting that C is a hop dominating set of G[H], there exists (z, q) ∈ C such that dG[H]((x, p)(z, q)) = 2. By the assumption that x ∈ S∗, we find that x = z. Hence, q ∈ Tx and q /∈ NH(p). Thus, Tx is a point-wise non-dominating set of H, showing that (ii) holds. For the converse, suppose that C satisfies properties (i) and (ii). Let (v, t) ∈ V (G[H])\ C and consider the following cases: Case 1. v /∈ S Since S is a hop dominating set of G, there exists w ∈ S such that dG(v, w) = 2. Pick any d ∈ Tw. Then (w, d) ∈ C and dG[H]((v, t)(w, d)) = 2. Case 2. v ∈ S S. Canoy Jr., R. Mollejon, JG. Canoy / Eur. J. Pure Appl. Math, 12 (4) (2019), 1455-1463 1460 If v /∈ S∗, then there exists z ∈ S such that dG(v, z) = 2. It follows that dG[H]((v, t)(z, a)) = 2 for any a ∈ Tz. Suppose that v ∈ S∗. Then, by property (ii), there exists c ∈ Tv such that tc /∈ E(H). Since G is non-trivial and connected, dG[H]((v, t)(v, c)) = 2. Accordingly, C is a hop dominating set of G[H]. Lemma 1. A non-trivial graph G admits a total hop dominating set if and only if γ(C) 6= 1 for every component C of G. Proof. Suppose G admits a total hop dominating set, say S. Suppose further that there exists a component C of G such that γ(C) = 1. Let v ∈ V (C) be such that {v} is a dominating set of C. Since S is a hop dominating set of G, v ∈ S. This, however, contradicts the fact that S is a total hop dominating set. Thus, γ(C) 6= 1 for every component C of G. For the converse, suppose that γ(C) 6= 1 for every component C of G. Clearly, S = V (G) is a hop dominating set of G. Let w ∈ V (G) and Cw be the component of G with w ∈ V (Cw). Since {w} is not a dominating set of Cw, there exists u ∈ V (C) \ {w} such that dC(u,w) = dG(u,w) = 2. This shows that S = V (G) is a total hop dominating set of G. Theorem 4. Let G be a connected graph with γ(G) 6= 1. If S is a hop dominating set of G, then γth(G) ≤ |S ∩NG(S, 2)|+ 2|S \NG(S, 2)|. Moreover, γth(G) ≤ 2γh(G). Proof. Let S be a hop dominating set of G. If S is a total hop dominating set of G (possible by Lemma 1), then S∩NG(S, 2) = S and S\NG(S, 2) = ∅. Hence, the inequality holds. Suppose now that S is not a total hop dominating set. Then S \NG(S, 2) 6= ∅. Let x ∈ S\NG(S, 2). Then, since γ(G) 6= 1, there exists vx ∈ V (G)\S such that dG(x, vx) = 2. Let DS = {vx : x ∈ S \NG(S, 2)}. Then, clearly, |DS | ≤ |S \NG(S, 2)| and S∗ = S ∪DS is a total hop dominating set of G. Thus, γth(G) ≤ |S∗| ≤ |S ∩NG(S, 2)|+ 2|S \NG(S, 2)|. In particular, γth(G) ≤ 2γh(G). In what follows, ρH(G) = min{|S∩NG(S, 2)|+pnd(H)|S\NG(S, 2)| : S is a hop dominating set ofG}. Corollary 3. Let G and H be non-trivial connected graphs of orders m and n, respectively. Then (i) γh(G[H]) = ρH(G) if γ(G) = 1; (ii) γh(G[H]) = γth(G) if γ(G) 6= 1; and (iii) γh(G[H]) = m[pnd(H)] if G = Km. Proof. (i) Suppose first that γ(G) = 1. Then, by Lemma 1, G does not admit a total hop dominating set (hence, γh(G[H]) 6= γth(G)). Now let S′ be a hop dominating set of G such that ρH(G) = |S′ ∩ NG(S′, 2)| + pnd(H)|S′ \ NG(S′, 2)|, and let D′ be a S. Canoy Jr., R. Mollejon, JG. Canoy / Eur. J. Pure Appl. Math, 12 (4) (2019), 1455-1463 1461 pnd-set of H. Set Qx = D′ for each x ∈ S′ \ NG(S′, 2) and Qy = {q}, where q ∈ V (H), for each y ∈ S′ ∩NG(S′, 2). Then C ′ = ∪x∈S′ [{x} ×Qx] is a hop dominating set of G[H] by Theorem 3. Hence, γh(G[H]) ≤ |C ′| = ∑ x∈S′∩NG(S′,2) |Qx|+ ∑ x∈S′\NG(S′,2) |Qx| = ρH(G). Next, suppose that C0 = ∪x∈S0 [{x} × Tx] is a γh-set of G[H]. By Theorem 3, S0 is a hop dominating set of G and Tx is a pnd-set of H for each x ∈ S0 \ NG(S0, 2). Clearly, |Tx| = 1 for all x ∈ S0 ∩NG(S0, 2). Hence, γh(G[H]) = |C0| = |S0 ∩NG(S0, 2)|+ pnd(H)|S0 \NG(S0, 2)| ≥ ρH(G), showing that equality in (i) holds. (ii) Suppose that γ(G) 6= 1. Then G admits a total hop dominating set by Lemma 1. Let S be a γth-set of G and let D = {a}, where a ∈ V (H). Set Tx = D for each x ∈ S. Then C = ∪x∈S [{x}×Tx] = S×D is a hop dominating set of G[H] by Theorem 3. Hence, γh(G[H]) ≤ |S||D| = γth(G). Next, suppose that C∗ = ∪x∈S∗ [{x} ×Rx] is a γh-set of G[H]. By Theorem 3, S∗ is a hop dominating set of G and Rx is a pnd-set of H for each x ∈ S∗ \NG(S∗, 2). Since C∗ is a γh-set, |Rx| = 1 for all x ∈ S∗ ∩NG(S∗, 2). Moreover, since H is a non-trivial connected graph, |Rx| = pnd(H) ≥ 2 for each x ∈ S∗ \ NG(S∗, 2) by Proposition 1(ii). Thus, by Theorem 4, γh(G[H]) = |C∗| ≥ |S∗ ∩NG(S∗, 2)|+ 2|S∗ \NG(S∗, 2)| ≥ γth(G). This establishes the desired equality in (ii). (iii) Suppose that G = Km. Since γ(G) = 1, γh(G[H]) = ρH(G). Now, since S = V (Km) is the only hop dominating set of G, it follows that γh(G[H]) = ρH(G) = m[pnd(H)]. This proves the assertion in (iii). Corollary 4. Let G be a non-trivial connected graph and let H be any non-trivial graph. If H has an isolated vertex, then γh(G[H]) = γh(G). Proof. Since H has an isolated vertex, pnd(H) = 1 by Proposition 1(ii). Let C = ∪x∈S [{x} × Tx] be a γh-set of G[H]. By Theorem 3, S is a hop dominating set of G and Tx is a pnd-set of H for each x ∈ S \NG(S, 2). Further, since C is γh-set, |Tx| = 1 for all x ∈ S ∩NG(S, 2). Hence, γh(G[H]) = |C| = |S ∩NG(S, 2)|+ |S \NG(S, 2)| = |S| ≥ γh(G). S. Canoy Jr., R. Mollejon, JG. Canoy / Eur. J. Pure Appl. Math, 12 (4) (2019), 1455-1463 1462 Now if S0 is a γh-set of G and D0 is a pnd-set of H, then C0 = S0 ×D0 is a γh-set of G[H] by Theorem 3. Thus, γh(G[H]) ≤ |C0| = |S0||D0| = |S0| = γh(G). This establishes the desired equality. The Cartesian product of graphs G and H, denoted by G�H, is the graph with vertex set V (G�H) = V (G) × V (H) such that (v, p)(u, q) ∈ E(G�H) if and only if uv ∈ E(G) and p = q ∈ E(H)] or u = v and pq ∈ E(H). Theorem 5. 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 of G�H if and only if the following conditions hold: (i) For each x ∈ V (G)\S and for each p ∈ V (H), at least one of the following statements is satisfied: (a) There exists y ∈ S ∩NG(x) such that Ty ∩NH(p) 6= ∅. (b) There exists z ∈ S ∩NG(x, 2) such that p ∈ Tz. (ii) For each v ∈ S and for each p ∈ V (H) \ Tv, at least one of the following statements is satisfied: (c) NH(p, 2) ∩ Tv 6= ∅. (d) There exists y ∈ S ∩NG(v) such that Ty ∩NH(p) 6= ∅. (e) There exists z ∈ S ∩NG(v, 2) such that p ∈ Tz. Proof. Suppose C is a hop dominating set of G�H. Let x ∈ V (G) \ S and let p ∈ V (H). Since C is a hop dominating set and (x, p) /∈ C, there exists (y, q) ∈ C such that dG�H((x, p)(y, q)) = 2. Since y ∈ S, x 6= y. If xy ∈ E(G), then pq ∈ E(H). Hence, q ∈ Ty ∩ NH(p), showing that (a) holds. So suppose that y /∈ NG(x). Since dG�H((x, p)(y, q)) = 2, it follows that y ∈ NG(x, 2) and p = q. Hence, p ∈ Ty, showing that (b) holds. Next, let v ∈ S and let p ∈ V (H)\Tv. Since C is a hop dominating set and (v, p) /∈ C, there exists (w, q) ∈ C such that dG�H((v, p)(w, q)) = 2. Suppose that (d) and (e) do not hold. Then, since dG�H((v, p)(w, q)) = 2, v = w and dH(p, q) = 2. Thus, q ∈ Tv∩NH(p, 2), showing that (c) holds. For the converse, suppose that C satisfies properties (i) and (ii). Let (v, t) ∈ V (G[H])\ C and consider the following cases: Case 1. v /∈ S If (a) of (i) holds, then there exist y ∈ S ∩ NG(v) and h ∈ Ty ∩ NH(p). Hence, (y, h) ∈ C ∩ NG�H((v, t), 2). If (b) of (i) holds, then there exists z ∈ S ∩ NG(v, 2) such that t ∈ Tz. It follows that (z, t) ∈ C ∩NG�H((v, t), 2). Case 2. v ∈ S Then t /∈ Tv. If (c) of (ii) holds, then we may take any q ∈ NH(t, 2) ∩ Tv. Clearly, (v, q) ∈ C∩NG�H((v, t), 2). As in the first case, if (d) or (e) of (ii) holds, then there exists (w, h) ∈ C ∩NG�H((v, t), 2). Accordingly, C is a hop dominating set of G�H. REFERENCES 1463 Corollary 5. Let G and H be non-trivial connected graphs. Then γh(G�H) ≤ min{γ(G)γ∗t1,2(H), γ(H)γ∗t1,2(G)}. Proof. Let S be a γ-set of G and let D be a γ∗t1,2-set of H. Set Tx = D for each x ∈ S and let C = ∪x∈S [{x} × Tx] = S ×D. Let x ∈ V (G) \ S and let p ∈ V (H). Since S is a dominating set of G, there exists y ∈ S ∩ NG(x). Now, since Ty = D is a total dominating set of H, there exists q ∈ Ty ∩ NH(p). Thus, (a) of property (i) of Theorem 5 holds. Next, let v ∈ S and let t ∈ V (H) \ Tv. Since Tv = D is a hop dominating set of H, Tv ∩ NH(t, 2) 6= ∅. Hence, (c) of property (ii) of Theorem 5 holds. Therefore, by Theorem 5, C is a hop dominating set of G�H. Thus, γh(G�H) ≤ |C| = γ(G)γ∗t1,2(H). This proves the assertion. Remark 1. The bound given in Corollary 5 is tight. Moreover, the inequality is also attainable. To see this, consider P3�P4 and P4�P4. It can easily be verified that γh(P3�P4) = 2 = γ(P3)γ ∗t 1,2(P4) and γh(P4�P4) = 4 = γ(P4)γ ∗t 1,2(P4). The inequality is attainable since γh(K4�K4) = 3 < 4 = γ(K4)γ ∗t 1,2(K4). References [1] S. Arriola, and S. Canoy, Jr., (1, 2)∗-domination in graphs, Advances and Applications in Discrete Mathematics, 2017,18, 2, 179− 190. [2] S. Ayyaswamy, B. Krishnakumari, B. Natarjan, and Y. Venkatakrishnan, Bounds on the hop domination number of a tree, Proceedings -Mathematical Sciences, 2015, 125, 4, 449-455. [3] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcell Dekker , 1998, New York. [4] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Domination in Graphs, Advanced Topics, Marcell Dekker, 1998, New York. [5] M. Henning, and N. Rad, On 2-step and hop dominating sets in graphs, Graphs and Combinatorics, 2017, 33, 4, 913-927. [6] C. Natarajan and S. Ayyaswamy, Hop domination in graphs II, Versita, 2015, 23, 2, 187-199. [7] Y. Pabilona and H. Rara, Connected hop domination in graphs under some binary operations, Asian-European Journal of Mathematics, 11(2018), 11, 5, 1850075-1- 1850075-11.