EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1848-1861 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Properties and Realization Problems Involving Connected Outer-Hop Independent Hop Domination in Graphs Javier A. Hassan1,∗, Abdurajan B. Lintasan1, Nurijam Hanna M. Mohammad1 1 Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines Abstract. In this paper, we construct a realization problems involving connected outer-hop in- dependent hop domination and we determine its connections with other known parameters in graph theory. In particular, given two positive integers a and b with 2 ≤ a ≤ b are realizable as the connected hop domination, connected outer-hop independent hop domination, and connected outer-independent hop domination numbers, respectively, of a connected graph. In addition, we characterize the connected outer-hop independent hop dominating sets in some families of graphs, join and corona of two graphs, and we use these results to derive formulas for the parameters of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Hop independent set, connected outer-hop independent hop domi- nating set, connected outer-hop independent hop domination number 1. Introduction Hop domination has been one of the widely studied topics of research in graph theory. Several mathematicians have investigated this concept and introduced variants because of its nice application to different fields and in networks. Some newly defined variations are studied in many classes of graphs (see [3–5, 7–12, 14]). In 2021, Nanding et al. [12] introduced and studied the concept called connected outer- independent hop domination in a graph. They characterized this newly defined sets on graphs under some binary operations and obtained some nice formulas and bounds. Recently, Hassan et al. [6] introduced the concept of hop independent set in a graph and defined the parameter called hop independence number. The authors have shown that the hop independence number is incomparable with the standard independence number ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4684 Email addresses: javierhassan@msutawi-tawi.edu.ph (J. Hassan), abdurajanlintasan@msutawi-tawi.edu.ph (A. Lintasan), hannamohammad@msutawi-tawi.edu.ph (N.H. Mohammad) https://www.ejpam.com 1848 © 2023 EJPAM All rights reserved. J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1849 of a graph. In fact, the authors have shown that the absolute difference between the hop independence number and the independence number of a graph can be made arbitrarily large. Motivated by the aforementioned studies, the concept of connected outer-hop indepen- dent hop domination in a graph will be introduced and investigated in this study. Some properties and realization results involving this parameter will be formulated. Moreover, exact values or bounds for the parameter will be given for some families of graphs, join, and corona of two graphs. Just like hop domination, we believe that this new parameter will yield significant results in the topic of domination and can lead to other interesting research directions in the future. 2. Terminology and Notation Let G be a simple graph. Then S ⊆ V (G) is a clique if the subgraph ⟨S⟩ induced by S is complete. The maximum cardinality of a clique set in G, denoted by ω(G), is called a clique number of G. Any clique set with cardinality equal to ω(G) is called a ω-set. A subset D of V (G) is called a pointwise non-dominating set of G if for each v ∈ V (G) \D, there exists u ∈ D such that v /∈ NG(u). A subset D of V (G) is independent if for every pair of distinct vertices v, w ∈ D, we have dG(v, w) ̸= 1. The maximum cardinality of an independent set in G, denoted by α(G), is called the independence number of G. Any independent set with cardinality equal to α(G) is called an α-set. 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 in G 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 in G is the set N2 G[X] = N2 G(X) ∪X. A subset S of V (G) is a hop dominating of 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 of 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 D ⊆ V (G) is called a connected hop dominating if the subgraph ⟨D⟩ induced by D is connected. The minimum cardinality among all connected hop dominating sets of G, denoted by γch(G), is called the connected hop domination number of G. Any connected hop dominating set with cardinality equal to γch(G) is called a γch-set. A connected hop dominating set C ⊆ V (G) is called a connected outer-independent hop dominating if V (G)\C is an independent set inG. The minimum cardinality of a connected outer-independent hop dominating set in G, denoted by γoich(G), is called the connected outer-independent hop domination number of G. Any connected outer-independent hop dominating set with cardinality equal to γoich(G) is called a γoich-set. A subset D of V (G) is hop independent if for every pair of distinct vertices v, w ∈ D, J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1850 we have dG(v, w) ̸= 2. The maximum cardinality of a hop independent set in G, denoted by αh(G), is called the hop independence number of G. Any hop independent set with cardinality equal to αh(G) is called an αh-set. Let G and H be two graphs. The join of G and H, denoted by 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 and H, denoted by G ◦H, 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⟩. 3. Results We begin this section by introducing the concept called connected outer-hop indepen- dent hop domination in a graph. Definition 1. LetG be a connected graph. A subset C of V (G) is called a connected outer- hop independent hop dominating if C is a connected hop dominating set and V (G)\C is a hop independent set in G. The minimum cardinality of a connected outer-hop independent hop dominating set in G, denoted by, γohich (G) is called the connected outer-hop independent hop domination number of G. Any connected outer-hop independent hop dominating set with cardinality equal to γohich (G) is called a γohich -set of G. Proposition 1. Let G be a connected graph. Then γch(G) ≤ γohich (G), and this bound is sharp. Proof. Let C be a γohich -set of G. Then C is a connected hop dominating set in G (by definition). Since γch(G) is the minimum cardinality among all connected hop dominating sets in G, it follows that γohich (G) = |C| ≥ γch(G). To see that the bound is sharp, consider G = P7 = [v1, v2, . . . , v7]. Let S = {v3, v4, v5}. Observe that ⟨S⟩ is connected and N2 G[S] = V (G). This means that S is a connected hop dominating in G. Since any connected hop dominating set in G contains S, S is the minimum connected hop dominating set in G. Hence, γch(P7) = 3. Moreover, since dG(a, b) ̸= 2 for every a, b ∈ V (G)\S, it follows that S is the minimum connected outer-hop independent hop dominating set in G. Consequently, γch(P7) = 3 = γohich (P7). Theorem 1. Let G be a connected graph with |V (G)| = n ≥ 1. Then each of the following is true. (i) 1 ≤ γohich (G) ≤ n. (ii) γohich (G) = 1 if and only if G is trivial. (iii) γohich (G) = n if and only if G is complete. J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1851 Proof. (i) Since an empty set cannot be a connected outer-hop independent hop dom- inating set in G, we have γohich (G) ≥ 1. Moreover, since any connected outer-hop indepen- dent hop dominating set in G is contained in V (G), we have γohich (G) ≤ n. Consequently, 1 ≤ γohich (G) ≤ n. (ii) Suppose γohich (G) = 1. Suppose that G is non-trivial. Then G is either connected or disconnected. If G is connected, then there exist a, b ∈ V (G) such that dG(a, b) = 1. This means that a /∈ N2 G[b] and b /∈ N2 G[a], showing that a singleton set is not the minimum connected outer-hop independent hop dominating set in G. Thus, γohich (G) > 1, a contradiction. Suppose that G1, . . . , Gk, k ≥ 2 are the components of G and let S be a connected outer-hop independent hop dominating set of G. Then S = S1∪ . . .∪Sk, where Si is a connected outer-hop independent hop dominating set of Gi for each i ∈ {1, . . . , k}. Since k ≥ 2, we have γohich (G) ≥ 2, a contradiction. Therefore, G must be a trivial graph. The converse is clear. (iii) Let γohich (G) = n. Suppose further that G is non-complete. Let x, y ∈ V (G) such that dG(x, y) = ∆(G), ∆(G) is the maximum degree of G. This means that x, y are non-cut vertices of G. Moreover, since G is non-complete, dG(x, y) ≥ 2. Now, let S′ = {V (G)\{x}. Then S′ is a connected outer-hop independent hop dominating set inG. Thus, γohich (G) ≤ n− 1, a contradiction. Therefore, G is complete. Conversely, suppose G is complete. Then N2 G[a] = {a} for every a ∈ V (G). Let S = V (G) = {a1, a2, . . . , an}. Then S is the minimum connected outer-hop independent hop dominating set of G. Thus, γohich (G) = n. The next result follows from Theorem 1. Corollary 1. Let G be a connected graph on n ≥ 2 vertices. Then (i) γohich (G) = n if and only if G = Kn . (ii) 2 ≤ γohich (G) ≤ n− 1 if and only if G is non-complete graph. (iii) 4 ≤ γohich (G)+γohich (G′) ≤ 2n−2 if and only if G and G′ are two non-complete graphs. (iv) 4 ≤ γohich (H) · γohich (J) ≤ n2 − 2n + 1 if and only if H and J are two non-complete graphs. Theorem 2. Let G be a connected graph. Then γohich (G) = γch(G) if and only if G has a γch-set D such that V (G) \D is a hop independent set in G. Proof. Suppose γohich (G) = γch(G). Let D be a γohich -set of G. Then V (G) \D is a hop independent set in G. Since D is a connected hop dominating set and γohich (G) = γch(G) = |D|, it follows that D is a γch-set of G. J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1852 Conversely, suppose G has a γch-set D such that V (G) \ D is a hop independent set in G. Then D is a connected outer-hop independent hop dominating set in G. Hence, γohich (G) ≤ |D| = γch(G). By Proposition 1, γohich (G) = |D| = γch(G). The next result is a realization problem involving connected outer-hop independent hop domination and connected hop domination. Theorem 3. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a connected graph G such that γch(G) = a and γohich (G) = b. Proof. For a = b, consider the following two cases: Case 1: a = 2 Consider Ka. Then by Corollary 1, γohich (Ka) = a. Since γch(Ka) = a, we have γohich (Ka) = a = γch(Ka). Case 2: a ≥ 3 Consider the graph G in Figure 1. Let D = {d1, d2, . . . , da}. Then D is both connected hop dominating and connected outer-hop independent hop dominating of G. Observe that every connected hop dominating (resp. connected outer-hop independent hop dominating) set of G contains D. This follows that D is both a γch-set and a γohich -set of G. Thus, γch(G) = a = γohich (G). G : d2d1 . . . dada−1d3 Figure 1: A graph G with γch(G) = γohi ch (G) Suppose a < b. Let m = b − a and consider the graph G′ given in Figure 2. Let D1 = {x1, x2, . . . , xa} and D2 = {x1, x2, . . . , xa, y1, y2, . . . , ym}. Then N ′2 G [D1] = V (G′) and N ′2 G [D2] = V (G′). Since ⟨D1⟩ and ⟨D2⟩ are connected, it follows that D1 and D2 are both connected hop dominating sets of G′. Moreover, since any connected hop dominating (resp. connected outer-hop independent hop dominating) set D contains D1 (resp. D2), D1 and D2 are γch-set and γohich -set of G, respectively. Consequently, γch(G ′) = a and γohich (G′) = m+ a = b, that is γch(G ′) = a < b = γohich (G′). J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1853 xa−1 G′ : x2x1 xa . . . y1 y2 ym . . . Figure 2: A graph G′ with γch(G ′) < γohi ch (G′) Corollary 2. Let n be a positive integer. Then there exists a connected graph G such that γohich (G)− γch(G) = n. In other words, γohich (G)− γch(G) can be made arbitrarily large. The next result is a realization problem involving connected outer-independent hop domination and connected outer-hop independent hop domination. Theorem 4. Let a and b be positive integers such that 2 ≤ a ≤ b. Then (i) there exists a connected graph G such that γohich (G) = a and γoich(G) = b. (ii) there exists a connected graph G such that γoich(G) = a and γohich (G) = b. Proof. (i) For a = b, consider G = Ka. Then γoich(G) = a = γohich . Suppose a < b. Let m = b − a and consider the graph J in Figure 3. Let S1 = {v1, v2, . . . , va} and S2 = {v1, v2, . . . , va, y1, y2, . . . , ym}. Then S1 and S2 are γohich -set and γoich-set of J , respectively. Hence, γ ohi ch (J) = a and γoich(J) = m+ a = b. J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1854 . . . v1 v2 va−1 va y1 w ym . . . J : va−2 Figure 3: A graph J with γohi ch (J) < γoi ch(J) (ii) For a = b, consider G = Ka. Then γohich (G) = a = γoich(G). Suppose a < b. Let m = b − a and consider the graph H in Figure 4. Let D1 = {x1, x2, . . . , xa} and D2 = {x1, x2, . . . , xa, y1, y2, . . . , ym}. Then D1 and D2 are γoich-set and γohich -set of H, respectively. Hence, γoich(H) = a and γohich (H) = m+ a = b. xa−2 H : x2x1 xa . . . y1 y2 ym . . . w aa−1 Figure 4: A graph H with γoi ch(H) < γohi ch (H) Corollary 3. Let n be a positive integer. Then each of the following statements holds. (i) There exists a connected graph G such that γoich(G)− γohich (G) = n. (ii) There exists a connected graph G such that γohich (G)− γoich(G) = n. In other words, the absolute difference |γoich(G)−γohich (G)| can be made arbitrarily large. J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1855 Proposition 2. For any positive integer n ≥ 1, γohi ch (Pn) =  1 if n = 1 2 if n = 2, 3, 4, 5 n− 4 if n ≥ 6 Proof. Clearly, γohich (P1) = 1 and γohich (Pn) = 2 for n = 3, 4, 5. Suppose that n ≥ 6. Let Pn = [v1, v2, . . . , vn] and D = {v3, v4 · · · , vn−3, vn−2}. Then N2 Pn [D] = V (Pn) and ⟨D⟩ is connected. Thus, D is a connected hop dominating set of Pn. Since n ≥ 6, it follows that dPn(a, b) ̸= 2 for every a, b ∈ V (Pn) \D. Hence, V (Pn) \D is a hop independent set of Pn, showing that D is a connected outer-hop independent hop dominating set of Pn, and so γohich (Pn) ≤ n− 4 for all n ≥ 6. On the other hand, observe that any connected outer-hop independent hop dominating set S in Pn contains D. Therefore, γohich (Pn) = n − 4 for all n ≥ 6. Proposition 3. For any positive integer n ≥ 3, γohi ch (Cn) = { 3 if n = 3 n− 2 if n ≥ 4 Proof. Clearly, γohich (C3) = 3. Suppose n ≥ 4. Let Cn = [v1, v2, . . . , vn, v1] and consider D∗ = {v1, v2, . . . , vn−2}. Then N2 Cn [D∗] = V (Cn) and ⟨D∗⟩ is connected. Thus, D∗ is a connected hop dominating set of Cn. Since dCn(vn−1, vn) = 1, it follows that D∗ is a connected outer-hop independent hop dominating set of Cn. Since the maximum connected hop independent set in Cn is of cardinality 2, it follows that D∗ is a γohich -set of Cn. Therefore, γ ohi ch (Cn) = n− 2 for all n ≥ 4. Theorem 5. Let G be any connected graph of order n ≥ 1. Then γohich (G) ≥ n− αh(G). Proof. LetD be a γohich -set of G. Then γohich (G) = |D| and V (G)\D is a hop independent set in G (by definition). It follows that αh(G) ≥ |V (G) \D|. Hence, n− αh(G) ≤ n− |V (G) \D| = n− n+ |D| = |D| = γohich (G). Remark 1. The bound in Theorem 5 is sharp. Moreover, strict inequality can be attained. For sharpness, consider the graph G in Figure 5. Let S = {c, d, e}. Then S is a minimum connected hop dominating set of G. Notice that V (G) \S is a hop independent set. It follows that S is a minimum connected outer-hop independent hop dominating set of G. Thus, γohich (G) = 3. Next, let S′ = {b, c, f, g, h}. Then S is the maximum hop independent set of G. Consequently, |V (G)| − αh(G) = 8− 5 = 3 = γohich (G). J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1856 G : a b c d e f g h Figure 5: A graph G with γohi ch (G) = |V (G)| − αh(G) For strict inequality, consider K7. Then γohich (K7) = 7 = αh(K7). Thus, γohich (K7) = 7 > (7− αh(K7)) = 7− 7 = 0. The following concept will be used in characterizing the connected outer-hop indepen- dent hop dominating sets in the join of two graphs. Definition 2. Let G be a non-complete graph. Then D ⊆ V (G) is called an outer-clique pointwise non-dominating set in G if D is pointwise non-dominating set and V (G) \D is clique set in G. The smallest cardinality of an outer-clique pointwise non-dominating set of G, denoted by ocpnd(G), is called the outer-clique pointwise non-domination number of G. Any outer-clique pointwise non-dominating set D of G with |D| = ocpnd(G), is called an ocpnd-set of G. Example 1. Consider the graph G in Figure 6. Let O = {a1, a2, a5, a6}. Then O is a pointwise non-dominating set of G. Since ⟨V (G) \ O⟩ ∼= K4, it follows that V (G) \ O is clique in G. Thus, O is an outer-clique pointwise non-dominating set of G. Next, let O′ = {a1, a2, a6}. Then O′ is a pointwise non-dominating of G. However, O′ is not an outer-clique pointwise non-dominating set of G since V (G)\O′ = {a3, a4, a5, a7, a8} is not clique in G. Moreover, since B = {a3, a4, a7, a8} is the maximum clique set in G, it follows that ocpnd(G) = 4. J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1857 a1 a2 a3 a4 G : a5 a6 a7 a8 Figure 6: A graph G with ocpnd(G) = 4 Theorem 6. Let G and H be two non-complete graphs. Then C ⊆ V (G+H) is a connected outer-hop independent hop dominating of G +H if and only if C = CG ∪ CH , where CG and CH are outer-clique pointwise non-dominating sets of G and H, respectively. Proof. Suppose C ⊆ V (G+H) be a connected outer-hop independent hop dominating set of G + H. Let CG = V (G) ∩ C and CH = V (H) ∩ C. Assume that CG = ∅. Then C = CH . Observe that V (G) ⊆ NG+H(C). It follows that V (G) /∈ N2 G+H [C], a contradiction. Hence, CG ̸= ∅. Similarly, CH ̸= ∅. Now, let a ∈ V (G) \ CG. Since C is a hop dominating set, there exists b ∈ C such that dG+H(a, b) = 2. Thus, b ∈ CG and a /∈ NG(b). This means that CG is a pointwise non-dominating set of G. Since V (G+H) \C is a hop independent set of G+H, it follows that V (G) \CG is a clique set of G. Hence, CG is an outer-clique pointwise non-dominating set of G. Similarly, CH is an outer-clique pointwise non-dominating set of H. Conversely, suppose C = CG ∪CH , where CG and CH are outer-clique pointwise non- dominating sets in G and H, respectively. Clearly, ⟨C⟩ is connected and V (G + H) \ C is a hop independent set. Now, let a ∈ V (G +H) \ C. Suppose a ∈ V (G). Since CG is a pointwise non-dominating set of G, there exists b ∈ CG \NG(a). Thus, dG+H(a, b) = 2. Similarly, when a ∈ V (H). Therefore, C is a hop dominating set of G+H. Consequently, S is a connected outer-hop independent hop dominating set of G+H. Corollary 4. Let G and H be two non-complete graphs. Then γohich (G+H) = ocpnd(G) + ocpnd(H). Proof. Suppose C ⊆ V (G+H) is a γohich -set ofG+H. Then by Theorem 6, C = CG∪CH , where CG and CH are outer-clique pointwise non-dominating sets of G andH, respectively. Thus, γohich (G+H) = |C| = |CG|+ |CH | ≥ ocpnd(G) + ocpnd(H). On the other hand, let CG and CH be ocpnd-sets of G and H, respectively. Then by J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1858 Theorem 6, C = CG ∪ CH is a connected outer-hop independent hop dominating set of G+H. Hence, ocpnd(G) + ocpnd(H) = |CG|+ |CH | = |C| ≥ γohich (G+H). Therefore, γohich (G+H) = ocpnd(G) + ocpnd(H). Theorem 7. Let G be a complete graph and H be non-complete graph. Then T ⊆ V (G + H) is a connected outer-hop independent hop dominating set in G + H if and only if T = V (G) ∪ TH , where TH is an outer-clique pointwise non-dominating set of H. Proof. Suppose that T = V (G) ∪ TH is a connected outer-hop independent hop dom- inating set in G + H. Since G is complete, it follows that T = V (G) ∪ TH , where TH ̸= ∅. Since T is a hop dominating, TH must be a pointwise non-dominating set in H. If V (H) \ TH is not clique in H, then there exist a, b ∈ V (H) \ TH ⊆ V (G+H) \ T such that dH(a, b) ≥ 2. It follows that dG+H(a, b) = 2, a contradiction to the fact that V (G + H) \ T is a hop independent in G + H. Thus, TH is an outer-clique pointwise non-dominating set of H. Conversely, assume that T = V (G) ∪ TH , where TH is an outer-clique pointwise non- dominating set of H. Then T is connected outer-hop independent hop dominating set in G+H by Theorem 6. Corollary 5. Let G be a complete graph and H be any non-complete graph. Then γohich (G+H) = |V (G)|+ ocpnd(H). Proof. Suppose T ⊆ V (G + H) is a γohich -set of G + H. Then by Theorem 7, T = V (G) ∪ TH , where TH is an outer-clique pointwise non-dominating set of H. Thus, γohich (G+H) = |T | = |V (G)|+ |TH | ≥ |V (G)|+ ocpnd(H). On the other hand, let T = V (G) ∪ TH , where TH is an ocpnd-set of H. Then by Theorem 7, T = V (G) ∪ TH is a connected outer-hop independent hop dominating set of G+H. Hence, |V (G)|+ ocpnd(H) = |V (G)|+ |TH | = |T | ≥ γohich (G+H). Consequently, γohich (G+H) = |V (G)|+ ocpnd(H). J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1859 Theorem 8. Let G be a non-trivial connected graph and H be any non-complete graph. A set C ⊆ V (G ◦H) is a connected outer-hop independent hop dominating set of G ◦H if and only if C = V (G) ∪ ( ⋃ a∈V (G)Ca), where Ca ⊆ V (Ha) and V (Ha) \ Ca is clique in Ha for each a ∈ V (G). Proof. Suppose C ⊆ V (G ◦H) is a connected outer-hop independent hop dominating set of G ◦ H and let Ca = V (Ha) ∩ C for each a ∈ V (G). Since ⟨C⟩ is connected, it follows that C = V (G) ∪ ( ⋃ a∈V (G)Ca). Since V (G ◦ H) \ C = ⋃ a∈V (G)(V (Ha) \ Ca) is a hop independent set of G ◦ H, it follows that V (Ha) \ Ca is a hop independent set of Ha for each a ∈ V (G). Suppose V (Ha) \ Ca is not a clique in Ha for some a ∈ V (G). Then there exists u, v ∈ V (Ha) \Ca ⊆ V (G ◦H) \C such that dHa(u, v) = dG◦H(u, v) = 2 for some a ∈ V (G), a contradiction to the fact that C is a connected outer-hop indepen- dent hop dominating set ofG◦H. Therefore, V (Ha)\Ca is clique inHa for every a ∈ V (G). Conversely, suppose C = V (G)∪ ( ⋃ a∈V (G)Ca), where Ca ⊆ V (Ha) and V (Ha) \Ca is clique in Ha for each a ∈ V (G). Clearly, C is a connected hop dominating set of G ◦H. Since V (Ha) \ Ca is clique in Ha for each a ∈ V (G), it follows that V (G ◦H) \ C = ⋃ a∈V (G) (V (Ha) \ Ca) is a hop independent set of G ◦ H. Therefore, C is a connected outer-hop independent hop dominating set of G ◦H. Corollary 6. Let G be a non-trivial connected graph with |V (G)| = n and H be any non- complete graph with |V (H)| = m. Then γohich (G ◦H) = n + n(m − ω(H)). In particular, we have (i) γohich (G ◦H) = n+ n(m− 2) if H = Pm,K1,m for all m ≥ 3, (ii) γohich (G ◦H) = n+ n(m− 2) if H = Cm for all m ≥ 4, (iii) γohich (G ◦Wm) = n+ n(m− 3) for all m ≥ 4, (iv) γohich (G ◦ Fm) = n+ n(m− 3) for all m ≥ 3, Proof. Let C be a γohich -set ofG◦H. Then C = V (G)∪( ⋃ v∈V (G)Cv), where Cv ⊆ V (Hv) and V (Hv) \ Cv is clique in Hv for each v ∈ V (G) by Theorem 8. Hence, γohich (G ◦H) = |C| = |V (G)|+ | ⋃ v∈V (G) Cv| = V (G) + ∑ v∈V (G) |Cv| = |V (G)|+ ∑ v∈V (G) (|V (Hv)| − |V (Hv) \ Cv|) ≥ |V (G)|+ |V (G)|(m− ω(H)) = n+ n(m− ω(H)). J. Hassan, A. Lintasan, N. H. Mohammad / Eur. J. Pure Appl. Math, 16 (3) (2023), 1848-1861 1860 Therefore, γohich (G ◦H) ≥ n+ n(m− ω(H)). On the other hand, for each v ∈ V (G), let Cv ⊆ V (Hv) such that V (Hv) \ Cv is a maximum clique of Hv. Then by Theorem 8, C = V (G) ∪ ( ⋃ v∈V (G)Cv) is a connected outer-hop independent hop dominating set of G ◦H. Thus, γohich (G ◦H) ≤ |C| = |V (G)|+ | ⋃ v∈V (G) Cv| = V (G) + ∑ v∈V (G) |Cv| = |V (G)|+ ∑ v∈V (G) (|V (Hv)| − |V (Hv) \ Cv|) = |V (G)|+ |V (G)|(|V (Hv)| − ω(H)) = n+ n(m− ω(H)). Consequently, γohich (G ◦H) = n+ n(m− ω(H)). Since ω(Pm) = ω(K1,m) = 2 for all m ≥ 3 and ω(Cn) = 2 for all m ≥ 4, statements (i) and (ii) hold. Also, since ω(Wm) = 3 for all m ≥ 4 and ω(Fm) = 3 for all m ≥ 3, statements (iii) and (iv) hold. 4. Conclusion This study has initiated the study of the concept called connected outer-hop indepen- dent hop domination in a graph. It was shown that the connected outer-hop independent hop domination number is at least equal to the connected hop domination number of a graph. This study gave some lower or upper bounds on the parameter of some graphs. In addition, exact values of the parameter have been obtained for some special graphs and graphs under some binary operations. Realization results involving connected outer- hop independent hop domination were presented. Interested researchers may consider and investigate this newly defined parameter for some products of graphs which were not considered in this study. They may also consider and study the complexity of this parameter. Acknowledgements The authors would like to thank the referees for their invaluable comments and sug- gestions that led to the improvement of the paper. 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