EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1817-1829 ISSN 1307-5543 – ejpam.com Published by New York Business Global Connected Outer-Hop Independent Dominating Sets in Graphs Under Some Binary Operations Jahiri U. Manditong1, Javier A. Hassan1,∗, Ladznar S. Laja1, Amy A. Laja1, Nurijam Hanna M. Mohammad1, Sisteta U. Kamdon1 1 Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines Abstract. Let G be a connected graph. A set D ⊆ V (G) is called a connected outer-hop inde- pendent dominating if D is a connected dominating set and V (G) \D is a hop independent set in G. The minimum cardinality among all connected outer-hop independent dominating sets in G, denoted by γohi c (G), is called the connected outer-hop independent domination number of G. In this paper, we initiate the study and investigation of connected outer-hop independent domination in some families of graphs and graphs under some binary operations. We construct properties and determine its connections with other known concepts and parameters in graph theory. Moreover, we characterize this type of sets in the join and corona of two graphs, and we use these results to determine the exact values or bounds of the parameters of these graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Outer-hop independent, connected outer-hop independent dominating set, connected outer-hop independent domination number 1. Introduction The concept of domination in a graph has been one of the interesting topics of research in graph theory. Let G be a graph. A subset D of V (G) is called a dominating of G if for every v ∈ V (G) \D, there exists u ∈ D such that uv ∈ E(G), that is, a set D is called a dominating set of G if NG[D] = V (G). The domination number of G, denoted by γ(G), is the minimum cardinality among all dominating sets in G. Researchers have been studied this concept and introduced new variants by imposing additional conditions to the usual concept of domination. Some studies on domination and its variants can be found in these references [1–6, 8–14]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4766 Email addresses: jahirimanditong@msutawi-tawi.edu.ph (J. Manditong), javierhassan@msutawi-tawi.edu.ph (J. Hassan), ladznarlaja@msutawi-tawi.edu.ph (L. Laja), amylaja@msutawi-tawi.edu.ph (A. Laja), hannamohammad@msutawi-tawi.edu.ph (N.H. Mohammad) sistetakamdon@msutawi-tawi.edu.ph (S. Kamdon) https://www.ejpam.com 1817 © 2023 EJPAM All rights reserved. J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1818 Recently, Hassan et al. [7] introduced the concept of hop independent sets in a graph. Let G be a graph. A subset S of V (G) is called a hop independent if for every pair of distinct vertices v, w ∈ S, 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. They have shown that the maximum hop independent set in a graph is a hop dominating set, that is, the hop independence number is at least equal to the hop domination number. Moreover, they have found that that hop independence number is incomparable to the independence number of a graph. In fact, they have shown that the absolute difference between the independence number and hop independence number of a graph can be made arbitrarily large. In this study, the concept of connected outer-hop independent domination in a graph will be introduced and investigated. This will be investigated for some special graphs including those graphs obtained from some binary operations. Moreover, exact values or bounds for the parameter will be given for some families of graphs and graphs under some binary operations. 2. Terminology and Notation Let G be a simple graph. Two vertices u, v of a graph G are adjacent, or neighbors, if uv is an edge of G. The set of neighbors of a vertex u in G, denoted by NG(u), is called the open neighborhood of u in G. The closed neighborhood of u in G is the set NG[u] = NG(u) ∪ {u}. If X ⊆ V (G), the open neighborhood of X in G is the set NG(X) = ⋃ u∈X NG(u). The closed neighborhood of X in G is the set NG[X] = NG(X)∪X. A subset D of V (G) is called a dominating of 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 among all dominating sets in G. Any dominating set D with cardinality equal to γ(G) is called a γ-set of G. A dominating set D of G is called a connected dominating set if the induced subgraph ⟨D⟩ of D is connected. The connected domination number of G, denoted by γc(G), is the minimum cardinality of a connected dominating set of G. Any connected dominating set D with cardinality equal to γc(G) is called a γc-set of G. A subset B of V (G) is an independent if for every pair of distinct vertices v, w ∈ B, 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 B with cardinality equal to α(G) is called an α-set of G. Let G be a connected graph. Then D ⊆ V (G) is called a connected outer-independent dominating set if D is connected dominating set and V (G) \ D is an independent set in G. The minimum cardinality of a connected outer-independent dominating set in G, denoted by, γoic (G) is called the connected outer-independent domination number of G. Any connected outer-independent dominating set with cardinality equal to γoic (G) is called a γoic -set of G. A subset S of V (G) is called a hop independent if for every pair of distinct vertices J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1819 v, w ∈ S, 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 S with cardinality equal to αh(G) is called a αh-set of G. 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, 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⟩. 3. Results We begin this section by introducing the concept of connected outer-hop independent domination in a graph. Definition 1. Let G be a connected graph. Then D ⊆ V (G) is called a connected outer- hop independent dominating set if D is connected dominating set and V (G) \D is a hop independent set in G. The minimum cardinality of a connected outer-hop independent dominating set in G, denoted by, γohic (G) is called the connected outer-hop independent domination number of G. Any connected outer-hop independent dominating set D with cardinality equal to γohic (G) is called a γohic -set of G. It is worth mentioning that every connected graph G admits a connected outer-hop in- dependent domination. The following first remark is the result concerning the relationship between connected domination number and connected outer-hop independent domination number of a graph G. Remark 1. Let G be a connected graph. Then γc(G) ≤ γohic (G). It is clear since every connected outer-hop independent dominating set is connected dominating. Remark 2. The bound given in Remark 1 is tight. Moreover, strict inequality can be attained. For the equality, consider the graph H given in Figure 1. Let D = {d, g, h, k}. Then D is both γc-set and γohic -set of H. Thus, γc(H) = 4 = γohic (H). J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1820 a b c d e f g h i j k l H : Figure 1: A graph G with γc(H) = γohi c (H) For strict inequality, consider the graph G given in Figure 2. Let C = {b, c, f} and C ′ = {b, c, f, g, h}. Then C and C ′ are γc-set and γohic -set of G, respectively. Hence, γc(G) = 3 < 5 = γohic (G). a b c d e f g h G : Figure 2: A graph G with γc(G) < γohi c (G) Theorem 1. Let G be a connected graph on n vertices. Then 1 ≤ γohic (G) ≤ n − 1. Moreover, each of the following statements holds. (i) γohic (G) = 1 if and only if G is complete. (ii) γohic (G) = 2 if and only if for each pair of adjacent vertices a, b ∈ V (G) such that NG[a] ̸= NG[b], D = {x, y} is a dominating set of G and V (G)\D is hop independent set in G. Proof. Let G be any connected graph. Since ∅ is not a connected outer-hop indepen- dent dominating set in G, it follows that γohic (G) ≥ 1. Let a be a non-cutting vertex of G. Then V (G) \ {a} is a connected outer-hop independent dominating set in G. Thus, γohoc (G) ≤ n− 1. Consequently, 1 ≤ γohic (G) ≤ n− 1. (i) Assume that γohic (G) = 1. Suppose G is not a complete graph. Then there exists a, b ∈ V (G) such that dG(a, b) = 2. Let c ∈ NG(a) ∩ NG(b). Clearly, γohic (G) ≥ 2, a contradiction. Hence, G is complete. J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1821 Conversely, suppose G is complete. Then every v ∈ V (G) is a connected outer-hop independent dominating vertex of G. Thus, γohic (G) ≤ 1. Consequently, γohic (G) = 1. (ii) Assume that γohic (G) = 2. Let a and b be two distinct adjacent vertices of G such that NG[a] ̸= NG[b]. Suppose there exists x ∈ V (G) \ (NG[a] ∪NG[b]). Since a and b are arbitrary, it follows that γohic (G) ≥ 3, a contradiction. Therefore, {a, b} is a dominating set of G. By letting D = {a, b} to be the γohic -set of G, it would imply that V (G) \D is a hop independent set in G. Conversely, suppose that for each pair of distinct adjacent vertices a, b ∈ V (G) such that NG[a] ̸= NG[b], {a, b} is a dominating set of G and D = {a, b} is a hop independent set in G. Then G is non-complete and D is a connected outer-hop independent dominating set of G. Hence, by (i), γohic (G) = 2. The next result follows from Theorem 1. Corollary 1. Let G be a non-trivial connected graph on n vertices such that G is con- nected. Then each of the following statements holds. (i) γohic (G) ≥ 2 if and only if G is non-complete. (ii) If G is non-complete, then (a) 4 ≤ γohic (G) + γohic (G) ≤ 2n− 2, and (b) 4 ≤ γohic (G) · γohic (G) ≤ n2 − 2n+ 1. Proposition 1. For any positive integer n ≥ 1, γohi c (Pn) =  1 if n = 1, 2 2 if n = 3 n− 2 if n ≥ 4 Proof. Clearly, γohic (Pn) = 1 for n = 1, 2 and γohic (P3) = 2. Suppose n ≥ 4. Let Pn = [v1, v2, . . . , vn] and let D = {v2, · · · , vn−1}. Clearly, D is a connected dominating set of Pn. Since n ≥ 4, it follows that dPn(v1, vn) ≥ 3. Thus, V (Pn) \D = {v1, vn} is a hop independent set of Pn. Thus, D is a connected outer-hop independent dominating set in Pn, and so γohic (Pn) ≤ n− 2. Observe that every connected dominating set of Pn contains D. Therefore, γohic (Pn) = n− 2 by Remark 1. Proposition 2. For any positive integer n ≥ 3, γohi c (Cn) = { 1 if n = 3 n− 2 if n ≥ 4 Proof. Clearly, γohic (C3) = 1. Suppose that n ≥ 4. Let Cn = [v1, v2, . . . , vn, v1] and consider D′ = {v1, v2, · · · , vn−2}. Then D′ is a connected dominating set of Cn. Since dCn(vn−1, vn) = 1, it follows that V (Cn) \ D′ = {vn−1, vn} is a hop independent set in Cn. Thus, D′ is a connected outer-hop independent dominating set in Cn, and so J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1822 γohic (Cn) ≤ n− 2. Since γc(Cn) = n− 2 for all n ≥ 4, it follows that γohic (Cn) = n− 2 for all n ≥ 4 by Remark 1. The next theorem is a realization result involving connected domination number and connected outer-hop independent domination number of a graph. Theorem 2. Let a and b be positive integers such that 2 ≤ a ≤ b. Then there exists a connected graph G such that γc(G) = a and γohic (G) = b. In other words, γohic (G)− γc(G) can be made arbitrarily large. Proof. For a = b, consider a path graph Pa+2. Then γc(Pa+2) = a = γohic (Pa+2) by Proposition 1. Suppose a < b. Consider the following two cases: Case 1: a is odd. Let m = b − a and consider the graph G1 given in Figure 3. Let D1 = {d1, d2, . . . , da} and D2 = {d1, d2, . . . , da, v1, v2, . . . , vm}. Then D1 and D2 are γc-set and γohic -set of G1, respectively. Hence, γc(G1) = a and γohic (G1) = m+ a = b. . . . . . . d1 d2 d3 dada−1 v1 v2 vm G1 : Figure 3: A graph G1 with γc(G1) < γohi c (G1) Case 2: a is even. Let m = b − a and consider the graph G2 given in Figure 4. Let D = {u1, u2, . . . , ua} and D∗ = {u1, u2, . . . , ua, w1, w2, . . . , wm}. Then D and D∗ are γc-set and γohic -set of G2, respectively. Therefore, γc(G2) = a and γohic (G2) = m+ a = b. J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1823 . . . . . . u1 u2 u3 uaua−1 w1 w2 wm G2 : u4 Figure 4: A graph G2 with γc(G2) < γohi c (G2) The next theorem is a realization result involving connected outer-independent domi- nation number and connected outer-hop independent domination number of a graph. Theorem 3. Let a and b be positive integers such that 2 ≤ a ≤ b. Then (i) there exists a connected graph G such that γohic (G) = a and γoic (G) = b. (ii) there exists a connected graph G such that γoic (G) = a and γohic (G) = b. In other words, |γoic (G)− γohic (G)| can be made arbitrarily large. Proof. (i) Suppose a < b. Let m = b − a and consider the graph G in Figure 5. Let D1 = {x1, x2, . . . , xa} and D2 = {x1, x2, . . . , xa, y1, y2, . . . , ym}. Then D1 and D2 are γohic -set and γoic -set of G, respectively. Hence, γohic (G) = a and γoic (G) = m+ a = b. G : x2 y1 . . . xaxa−1x3 y2 ym+1 x1 . . . Figure 5: A graph G with γohi c (G) < γoi c (G) (ii) Suppose a < b. Let m = b − a and consider the graph G∗ in Figure 6. Let D′ = {x1, x2, . . . , xa} and D′′ = {x1, x2, . . . , xa, z1, z2, . . . , zm}. Then D′ and D′′ are γoic -set and γohic -set of G∗, respectively. Therefore, γoic (G∗) = a and γohic (G∗) = m+ a = b. J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1824 xa−1 G∗ : x2x1 xa . . . z1 z2 zm . . . Figure 6: A graph G∗ with γoi c (G∗) < γohi c (G∗) The following concept will be used in characterizing the connected outer-hop indepen- dent dominating sets in the join and corona of two graphs. Definition 2. Let G be a non-complete graph. A non-empty subset O ⊆ V (G) is called an outer-clique set if V (G) \O is clique in G. The smallest cardinality of an outer-clique set of G, denoted by ω̃(G), is called the outer-clique number of G. Any outer-clique set O of G with cardinality equal to ω̃(G), is called an ω̃-set of G. Remark 3. Let n ≥ 2 be any positive integer. Then each of the following holds. (i) ω̃(G) = n− 1 if G = Kn (ii) ω̃(Pn) = { 1 if n = 3 n− 2 if n ≥ 4; and (iii) ω̃(Cn) = n− 2 for all n ≥ 4. Theorem 4. Let G and H be two non-complete graphs. Then D ⊆ V (G + H) is a connected outer-hop independent dominating in G+H if and only if D = DG∪DH , where DG and DH are outer-clique sets in G and H, respectively. Proof. Suppose D ⊆ V (G + H) is a connected outer-hop independent dominating set in G + H. Let DG = V (G) ∩ D and DH = V (H) ∩ D. Since G and H are non- complete, it follows that DG ̸= ∅ and DH ̸= ∅. Suppose V (G) \ DG is not a clique in G. Then there exist a, b ∈ V (G) \ DG such that dG(a, b) = 2 = dG+H(a, b). Since V (G) \DG ⊆ V (G+H) \D, it follows that V (G+H) \D is not a hop independent set, a contradiction to the fact that D is a connected outer-hop independent dominating set in G+H. Therefore, V (G) \DG is clique in G. Similarly, V (H) \DH is clique in H. Conversely, suppose D = DG ∪DH , where DG and DH are outer-cliques in G and H, respectively. Clearly, D is a connected dominating set of G+H. Suppose that V (G+H)\D J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1825 is not a hop independent set in G+H. Then there exist x, y ∈ V (G+H) \D such that dG+H(x, y) = 2. This means that either x, y ∈ V (G) \ DG or x, y ∈ V (H) \ DH , and this is a contradiction to our assumption that DG and DH are outer-cliques in G and H, respectively. Therefore, V (G+H) \D is a hop independent set in G+H. Consequently, D is a connected outer-hop independent dominating in G+H. The next result follows from Theorem 4 Corollary 2. Let G and H be two non-complete graphs. Then γohic (G+H) = ω̃(G) + ω̃(H). In particular, we have (i) γohic (Pn + Pm) = n+m− 4 for all n,m ≥ 3; (ii) γohic (Cn + Cm) = n+m− 4 for all n,m ≥ 4; and (iii) γohic (Pn + Cm) = n+m− 4 for all n,m ≥ 4. The following concept will be used in characterizing connected outer-hop independent dominating sets in the join of complete and non-complete graphs. Definition 3. Let G be a connected graph. A connected dominating set C ⊆ V (G) is called a connected outer-clique dominating if V (G) \C is a clique set in G. The connected outer-clique domination number of G, denoted by γocc (G), is the minimum cardinality of a connected outer-clique dominating set of G. Any connected outer-clique dominating set C with cardinality equal to γocc (G), is called a γocc -set of G. Theorem 5. Let G be a complete graph and H be any non-complete connected graph. Then D ⊆ V (G + H) is a connected outer-hop independent dominating set in G + H if and only if D = DG ∪DH and satisfies one of the following conditions: (i) If DG = ∅, then DH is a connected outer-clique dominating set in H. (ii) If DG ̸= ∅, then DH is an outer-clique set in H. Proof. Suppose S ⊆ V (G + H) is a connected outer-hop independent dominating in G +H. Then V (G +H) \ S is a hop independent set in G +H. Let DG = ∅. Suppose on the contrary that DH is not a connected outer-clique dominating set in H. Then DH is either not a connected, not a dominating or V (H) \ DH not a clique sets in H, respectively. Assume first that DH is not a dominating set in H. Then there exists a ∈ V (H) \DH such that a /∈ NH [DH ]. Since DG = ∅, it follows that a /∈ NG+H [D], a contradiction. Therefore, DH is a dominating set in H. Similarly, a contradiction follows if DH is not a connected or V (H)\DH is not a clique in H. Hence, (i) holds. Next, suppose that DG ̸= ∅ and suppose that DH is not an outer-clique set in H. Then there exist J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1826 x, y ∈ V (H) \DH ⊆ V (G+H) \D such that dH(x, y) = dG+H(x, y) = 2, a contradiction. Hence, DH must be an outer-clique set in H showing that (ii) holds. For the converse, suppose (i) holds. Since G is complete, it follows that D is an outer- hop independent set in G+H. Clearly, D is a connected dominating set in G+H. Hence, D is a connected outer-hop independent dominating set in G+H. Similarly, if (ii) holds, then D is a connected outer-hop independent dominating set in G+H. The next result follows from Theorem 5. Corollary 3. Let G be a complete graph and H be any non-complete connected graph. Then γohic (G+H) = γocc (H). In particular, we have (i) γohic (Wn) = γohic (K1 + Cn) = n− 2 for all n ≥ 4; (ii) γohic (Fn) = γohic (K1 + Pn) = n− 1 for all n ≥ 3; (iii) γohic (Kn + Cm) = m− 2 for all n ≥ 2,m ≥ 4; and (iv) γohic (Kn + Pm) = m− 1 for all n ≥ 2,m ≥ 3. Theorem 6. Let G be a non-trivial connected graph and H be any non-complete graph. A set D ⊆ V (G ◦H) is a connected outer-hop independent dominating set in G ◦H if and only if D = V (G) ∪ ( ⋃ v∈V (G)Dv), where Dv ⊆ V (Hv) and V (Hv) \ Dv is clique in Hv for each v ∈ V (G). Proof. Assume that D is a connected outer-hop independent dominating set in G ◦H and let Dv = V (Hv) ∩ D for each v ∈ V (G). Since ⟨D⟩ is connected and H is non- complete, it follows that D = V (G) ∪ ( ⋃ v∈V (G)Dv). Suppose V (Hv) \Dv is not a clique in Hv for some v ∈ V (G). Then there exists u,w ∈ V (Hv) \ Dv ⊆ V (G ◦ H) \ D such that dHv(u,w) = dG◦H(u,w) = 2 for some v ∈ V (G), a contradiction to the fact that D is an outer-hop independent set in G ◦H. Therefore, V (Hv) \Dv is clique in Hv for every v ∈ V (G). Conversely, suppose D = V (G) ∪ ( ⋃ v∈V (G)Dv), where Dv ⊆ V (Hv) and V (Hv) \Dv is clique of Hv for each v ∈ V (G). Clearly, D is a connected dominating set of G ◦ H. Since V (Hv) \ Dv is clique of Hv for each v ∈ V (G), it follows that V (G ◦ H) \ D =⋃ v∈V (G)(V (Hv) \ Dv) is a hop independent set of G ◦ H. Therefore, D is a connected outer-hop independent dominating set in G ◦H. The next result follows from Theorem 6. Corollary 4. Let G be a non-trivial connected graph with |V (G)| = s and H be any non-complete graph with |V (H)| = t. Then γohic (G ◦H) = s+ s(ω̃(H)). In particular, we have J. Hassan et al. / Eur. J. Pure Appl. Math, 16 (3) (2023), 1817-1829 1827 (i) γohic (Ps ◦ Pt) = γohic (Cs ◦ Ct) = γohic (Ps ◦ Ct) = s+ s(t− 2) for all s ≥ 2 and t ≥ 4; (ii) γohic (Ks ◦ Pt) = γohic (Ks ◦ Ct) = s+ s(t− 2) for all s ≥ 2 and t ≥ 4; (iii) γohic (G ◦Wt) = |V (G)|+ |V (G)|(t− 2) for all t ≥ 4; and (iv) γohic (G ◦ Ft) = |V (G)|+ |V (G)|(t− 2) for all t ≥ 3. Theorem 7. Let G be a non-trivial connected graph and H be any complete graph. A set D ⊆ V (G ◦H) is a connected outer-hop independent dominating set in G ◦H if and only if D = V (G) ∪ ( ⋃ v∈V (G)Dv), where Dv ⊆ V (Hv) such that Dv = ∅ or Dv ̸= ∅ for each v ∈ V (G). Proof. Assume that D is a connected outer-hop independent dominating set in G ◦H and let Dv = V (Hv) ∩ D for each v ∈ V (G). Since ⟨D⟩ is connected, it follows that D = V (G)∪ ( ⋃ v∈V (G)Dv). Since H is complete, either Dv = ∅ or Dv ̸= ∅ holds for each v ∈ V (G). Conversely, suppose that D = V (G) ∪ ( ⋃ v∈V (G)Dv), where Dv ⊆ V (Hv). If Dv = ∅ for each v ∈ V (G), then D = V (G). Since H is complete, it follows that D = V (G) is connected outer-hop independent dominating set in G ◦H. Similarly, if Dv ̸= ∅ for each v ∈ V (G), then D connected outer-hop independent dominating set in G ◦H. The next result follows from Theorem 7. Corollary 5. Let G be a non-trivial connected graph with |V (G)| = s and H be any complete graph. Then γohic (G ◦H) = s. In particular, we have (i) γohic (Pn ◦Km) = n = γohic (Cn ◦Km) for all n ≥ 3,m ≥ 1; and (ii) γohic (Fn ◦Km) = n+ 1 = γohic (Wn ◦Km) for all n ≥ 3,m ≥ 1. 4. Conclusion The concept of connected outer-hop independent domination in a graph has been introduced and investigated in this study. It was shown that the connected outer-hop independent domination number is at least equal to the connected domination number of a graph. Connected outer-hop independent dominating sets in some special graphs, join and corona of two graphs have been characterized. 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