EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2118-2131 ISSN 1307-5543 – ejpam.com Published by New York Business Global J2- Hop Domination in Graphs: Properties and Connections with other Parameters Javier A. Hassan1,∗, Alcyn R. Bakkang2, Amil-Shab S. Sappari1 1Mathematics and Sciences Department, College of Arts and Sciences, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines 2 Secondary Education Department, College of Education, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines Abstract. A subset T = {v1, v2, · · · , vm} of vertices of a graph G is called a J2-set if N2 G[vi] \ N2 G[vj ] ̸= ∅ for every i ̸= j, where i, j ∈ {1, 2, . . . ,m}. A J2-set T is called a J2-hop dominating in G if for every a ∈ V (G) \ T , there exists b ∈ T such that dG(a, b) = 2. The J2-hop domination number of G, denoted by γJ2h(G), is the maximum cardinality among all J2-hop dominating sets in G. In this paper, we initiate the study on J2-hop domination and we establish its properties and connections with other known parameters in graph theory. We show that every maximum hop independent set is a J2-hop dominating, hence, this parameter is greater than compare to the hop independence parameter on any graph. Moreover, we derive some lower and upper bounds of the parameter for a generalized graph, join and corona of two graphs, respectively. Finally, we obtain exact values of the parameter for some special graphs and shadow graph using the characterization results that are formulated in this study. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: J2-set, J2-hop dominating set, J2-hop domination number 1. Introduction Hop domination was introduced by Natarajan et al. in [9]. A subset S of a vertices of a graph G is called a hop dominating if for every a ∈ V (G)\S, there exists b ∈ S such that dG(a, b) = 2. The minimum cardinality among all hop dominating sets of G, denoted by γh(G), is called the hop domination number of G. This parameter had studied on some families of graphs and graphs obtained from some operations in [1, 2, 9]. Researchers in the field had further investigated this concept, and introduced new variants and obtained some significant results that contributed a lot to the hop domination theory (see [3–8, 10–12]). In this paper, new parameter called J2-hop domination in a graph will be introduced and investigated. We will establish its relationships with other known parameters in graph DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4905 Email addresses: javierhassan@msutawi-tawi.edu.ph (J. Hassan) alcynbakkang@msutawi-tawi.edu.ph (A. Bakkang) amilshabsappari@msutawi-tawi.edu.ph (A. Sappari) https://www.ejpam.com 2118 © 2023 EJPAM All rights reserved. J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2119 theory. Moreover, we will determine its bounds or exact values on some special graphs, shadow graph and join of two graphs. We believe that this parameter and its results would give additional insights to researchers in the field and would help them for more research directions in the future. 2. Terminology and Notation A path graph is a non-empty graph with vertex-set {x1, x2, . . . , xn} and edge-set {x1x2, x2x3, . . . , xn−1xn}, where the x ′ is are all distinct. The path of order n is denoted by Pn. If G is a graph and u and v are vertices of G, then a path from vertex u to vertex v is sometimes called a u-v path. The cycle graph Cn is the graph of order n ≥ 3 with vertex-set {x1, x2, . . . , xn} and edge-set {x1x2, x2x3, . . . , xn−1xn, xnx1}. Let G = (V (G), E(G)) be a simple and undirected graph. The distance dG(u, v) in G of two vertices u, v is the length of a shortest u-v path in G. The greatest distance between any two vertices in G, denoted by diam(G), is called the diameter of G. Two vertices x, y of G are adjacent, or neighbors, if xy is an edge of G. The open neighborhood of x in G is the set NG(x) = {y ∈ V (G) : xy ∈ E(G)}. The closed neighborhood of x inG is the setNG[x] = NG(x)∪{x}. IfX ⊆ V (G), the open neighborhood of X in G is the set NG(X) = ⋃ x∈X NG(x). The closed neighborhood of X in G is the set NG[X] = NG(X) ∪X. A vertex a in G is a hop neighbor of a vertex b in G if dG(a, b) = 2. The set N2 G(a) = {b ∈ V (G) : dG(a, b) = 2} is called the open hop neighborhood of a. The closed hop neighborhood of a in G is given by N2 G[a] = N2 G(a)∪{a}. The open hop neighborhood of S ⊆ V (G) is the set N2 G(S) = ⋃ a∈S N2 G(a). The closed hop neighborhood of S in G is the set N2 G[S] = N2 G(S) ∪ S. A subset S of V (G) is a hop dominating of G if for every a ∈ V (G)\S, there exists b ∈ S such that dG(a, b) = 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 of G. A subset S of V (G) is called a hop independent if for every pair of distinct vertices x, y ∈ S, dG(x, y) ̸= 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 an αh-set of G. Let G and H be any 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 J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2120 and write v +Hv for ⟨{v}+Hv⟩. The shadow graph S(G) of graph G is constructed by taking two copies of G, say G1 and G2, and then joining each vertex u ∈ V (G1) to the neighbors of its corresponding vertex u′ ∈ V (G2). 3. Results We begin this section by introducing the concept of J2-hop domination in a graph. Definition 1. Let G be an undirected graph andm ∈ N . A subset T = {v1, v2, · · · , vm} of vertices of G is called a J2-set if N2 G[vi] \ N2 G[vj ] ̸= ∅ for every i ̸= j, where i, j ∈ {1, 2, . . . ,m}. A J2-set T is called a J2-hop dominating in G, if T is a hop dominat- ing set in G. The J2-hop domination number of G, denoted by γJ2h(G), is the maximum cardinality among all J2-hop dominating sets in G. Any J2-hop dominating set T with |T | = γJ2h(G) (resp. |T | = γh(G)), is called a γJ2h-set or the maximum (resp. minimum) J2-hop dominating set of G. Example 1. Consider the graph G in Figure 1 and let T = {u1, u2, . . . , u6}. Notice that ui ∈ N2 G[ui] \ N2 G[uj ] ∀ i ̸= j where i, j ∈ {1, 2, . . . , 6}. Thus, T is a J2-set of G. Since N2 G[T ] = V (G), it follows that T is a J2-hop dominating set of G. Observe that N2 G[u7] ⊆ N2 G[u3], N 2 G[u8] ⊆ N2 G[u3], N 2 G[u9] ⊆ N2 G[u1], and N2 G[u10] ⊆ N2 G[u3]. Thus, T is a maximum J2-hop dominating set of G. Hence, γJ2h(G) = 6. G : u3 u5 u1 u2 u4 u6 u7 u8 u9 u10 Figure 1: Graph G with γJ2h(G) = 6 J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2121 Theorem 1. Let G be any graph of order m ≥ 1. Then each of the following holds: (i) N ⊆ V (G) is a γh-set in G if and only if N is a minimum J2- hop dominating set in G. (ii) γh(G) ≤ γJ2h(G). (iii) 1 ≤ γJ2h(G) ≤ m. Proof. (i) Suppose that N ⊆ V (G) is a γh-set in G. Then N is a minimum hop dominating set in G. It suffices to show that N is a J2-set in G. Suppose on the contrary that N is not a J2-set in G. Then there exist x, y ∈ N such that either N2 G[x]\N2 G[y] = ∅ or N2 G[y] \ N2 G[x] = ∅. This means that either N2 G[x] ⊆ N2 G[y] or N2 G[y] ⊆ N2 G[x]. If N2 G[x] ⊆ N2 G[y], then D′ = N \ {x} is a hop dominating set in G, contradicting the minimality of N . Similarly, when N2 G[y] ⊆ N2 G[x]. Consequently, N is a minimum J2-hop dominating set of G. Conversely, suppose that N is a minimum J2-hop dominating set of G. Then |N | = γh(G) (by definition). It follows that N is a γh-set in G. (ii) Let S be a γh-set of G. Then by (i), S is a minimum J2-hop dominating set in G. Since γJ2h(G) is the maximum cardinality among all J2-hop dominating sets in G, it follows that γh(G) = |S| ≤ γJ2h(G). (iii) Since γh(G) ≥ 1 for any graph G of order m ≥ 1, we have γJ2h(G) ≥ 1 by (ii). Since any J2-hop dominating set N of G is always a subset of V (G), it follows that γJ2h(G) ≤ |V (G)| = m. Therefore, 1 ≤ γJ2h(G) ≤ m. Theorem 2. Let G be any graph. Then N ⊆ V (G) is a maximum J2-set if and only if N is a γJ2h-set of G. Proof. Let N be a maximum J2-set of G. Assume that N is not a hop dominating set in G. Then there exists u ∈ V (G) \N such that u /∈ N2 G[N ]. This implies that u /∈ N2 G[v] for every v ∈ N . Let N ′ = {u}∪N . Since N is a J2-set in G and u ∈ N2 G[u], it follows that N2 G[a] \ N2 G[b] ̸= ∅ and N2 G[b] \ N2 G[a] ̸= ∅ for every a ̸= b, where a, b ∈ N ′. This means that N ′ is a J2-set in G, contradicting the maximality of N . Hence, N is a hop dominating set of G. Since N is a maximum J2-set of G, N is a maximum J2-hop dominating set of G, that is, N is a γJ2h-set of G. Conversely, suppose that N is a γJ2h-set of G. Then N is a maximum J2-hop domi- nating set of G. Hence, the assertion follows. The following result follows from Theorem 2. Corollary 1. Let G be a graph and let N = {x1, x2, . . . , xk} be a J2-set of G. Then |N | = k ≤ γJ2h(G). J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2122 Proposition 1. Given any positive integer k ≥ 1, we have γJ2h(Pk) =  1 if k = 1 2 if k = 2, 3, 4 3 if k = 5 4 if k = 6, 7 k − 4 if k ≥ 8 Proof. Clearly, γJ2h(P1) = 1, γJ2h(Pk) = 2 for k = 2, 3, 4, γJ2h(P5) = 3 and γJ2h(Pk) = 4 for k = 6, 7. Suppose that k ≥ 8. Let Pk = [a1, a2, . . . , ak] and let S = {a3, a4, . . . , ak−3, ak−2}. Then N2 Pk [S] = V (Pk), showing that S is a hop domi- nating set in Pk. Observe that ai−2 ∈ N2 Pk [ai] \ N2 Pk [aj ] and aj+2 ∈ N2 Pk [aj ] \ N2 Pk [ai] for all j > i, where i, j ∈ {3, 4, . . . , k − 3, k − 2}. Thus, N2 Pk [ai] \ N2 Pk [aj ] ̸= ∅ for all i ̸= j, where i, j ∈ {3, 4, . . . , k − 3, k − 2}, that is, S is a J2-set in Pk. Therefore, S is a J2-hop dominating set in Pk. Since N2 Pk [a1] ⊆ N2 Pk [a3], N 2 Pk [a2] ⊆ N2 Pk [a4], N 2 Pk [ak] ⊆ N2 Pk [ak−2], and N2 Pk [ak−1] ⊆ N2 Pk [ak−3], it follows that S is a maximum J2-hop dominating set of Pk. Hence, γJ2h(Pk) = k − 4 for all k ≥ 8. Theorem 3. Let G be any graph of order n and N be any J2-hop dominating set of G. Then each of the following holds: (i) a ∈ N if and only if N2 G[a] ⊈ N2 G[b] and N2 G[b] ⊈ N2 G[a] ∀ b ∈ N \ {a}. (ii) γJ2h(G) = |V (G)| = n if and only if N2 G[vi] ⊈ N2 G[vj ] ∀ i ̸= j where i, j ∈ {1, 2, . . . , n}. (iii) If G is Kn or Kn, then γJ2h(G) = n for all n ≥ 1. Proof. (i) Let G be a graph and N be a J2-hop dominating set of G. Suppose that a ∈ N . Then N2 G[a] \ N2 G[b] ̸= ∅ and N2 G[b] \ N2 G[a] ̸= ∅ ∀ b ∈ N \ {a}. It follows that N2 G[a] ⊈ N2 G[b] and N2 G[b] ⊈ N2 G[a] ∀ b ∈ N \ {a}. Conversely, suppose that N2 G[a] ⊈ N2 G[b] and N2 G[b] ⊈ N2 G[a] ∀ b ∈ N \{a}. This means that N2 G[a] \N2 G[b] ̸= ∅ and N2 G[b] \N2 G[a] ̸= ∅ ∀ b ∈ N \ {a}. Hence, a ∈ N . (ii) Suppose that γJ2h(G) = |V (G)| = n. Then N = V (G) = {v1, v2, . . . , vn} is the γJ2h-set of G. Thus, N2 G[vi]\N2 G[vj ] ̸= ∅ ∀ i ̸= j, where i, j ∈ {1, 2, . . . , n}. It follows that N2 G[vi] ⊈ N2 G[vj ] ∀ i ̸= j, where i, j ∈ {1, 2, . . . , n}. Conversely, suppose that N2 G[vi] ⊈ N2 G[vj ] ∀ i ̸= j, where i, j ∈ {1, 2, . . . , n}. Then N2 G[vi] \N2 G[vj ] ̸= ∅ ∀ i ̸= j, i, j ∈ {1, 2, . . . , n}. It follows that vi, vj are in J2-set S of G ∀ i ̸= j, where i, j ∈ {1, 2, . . . , n}. Thus, S = V (G). Consequently, γJ2h(G) = |S| = |V (G)| = n. J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2123 (iii) Let G = Kn and V (G) = {a1, a2, . . . , an}. Then {ai} = N2 G[ai] ⊈ N2 G[aj ] = {aj} ∀ i ̸= j, where i, j ∈ {1, 2, . . . , n}. Thus, by (ii), γJ2h(G) = |V (G)| = n. Similarly, if G = Kn, then γJ2h(G) = |V (G)| = n. Theorem 4. Let a, b be positive integers with 2 ≤ a ≤ b. Then there exists a connected graph G such that γh(G) = a and γJ2h(G) = b. In other words, γJ2h(G) − γh(G) can be made arbitrarily large. Proof. For a = b, consider Ka. Then by Theorem 3, γJ2h(Ka) = a = γh(Ka). Suppose that a < b. Consider the following two cases: Case 1: a is odd. Consider the graph G in Figure 2. Let m = b − a and let S = {x1, x2, . . . , xa} and S′ = {x1, x2, . . . , xa−3, u, v, xa, c1, c2, . . . , cm}. Then S and S′ are γh-set and γJ2h-set in G, respectively. Hence, γh(G) = a and γJ2h(G) = a+m = b. Consequently, γh(G) < γJ2h(G). . . . G : x2x1 v c1 xa−1x3 x4 . . . xa xa−2 c2 u cm Figure 2: Graph G with γh(G) < γJ2h(G) Case 2: a is even. Consider the graph H in Figure 3. Let t = b − a and let C = {x1, x2, . . . , xa} and C ′ = {x1, x2, . . . , xa−2, v, w, c1, c2, . . . , ct}. Then C and C ′ are γh-set and γJ2h-set in H, re- spectively. Therefore, γh(H) = a and γJ2h(H) = a + t = b, showing that γh(H) < γJ2h(H). J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2124 . . . H : x2x1 v c1 xax3 x4 . . . w xa−1 c2 ct Figure 3: Graph H with γh(H) < γJ2h(H) Theorem 5. Let G be any graph and let S ⊆ V (G). Then every hop independent set S is a J2-set in G. In particular, every αh-set is a J2-hop dominating set. Moreover, αh(G) ≤ γJ2h(G). Proof. Let S be a hop independent set in G. Then dG(a, b) ̸= 2 for every a, b ∈ S. Suppose on the contrary that S is not a J2-set in G. Then there exist x, y ∈ S such that N2 G[x]\N2 G[y] = ∅ or N2 G[y]\N2 G[x] = ∅. It follows that N2 G[x] ⊆ N2 G[y] or N 2 G[y] ⊆ N2 G[x]. In either case, we have dG(x, y) = 2, a contradiction to the fact that S is a hop independent set in G. Therefore, S is a J2-set in G. Next, let S′ be an αh-set of G. Then S′ is a maximum hop independent set of G (by definition). Thus, S′ is a J2-set in G by the first part. Now, suppose on the contrary that S′ is not a hop dominating set of G. Then there exists x ∈ V (G) \ S′ such that x /∈ N2 G[y] ∀ y ∈ S′. This means that dG(x, y) ̸= 2 for all y ∈ S′. Thus, S∗ = {x} ∪ S′ is a hop independent set in G, contradicting the maximality of S′. Hence, S′ is a hop dominating set of G, showing that S′ is a J2-hop dominating in G. Consequently, αh(G) ≤ γJ2h(G). Theorem 6. Let G and H be two connected graphs. If N = NG∪NH ⊆ V (G+H), where NG and NH are J2-sets in G and H, respectively, then N is a J2-set in G+H. Proof. Let a, b ∈ N . Suppose that a, b ∈ NG. If dG(a, b) = 1, then a ∈ N2 G+H [a] \ N2 G+H [b] and b ∈ N2 G+H [b] \ N2 G+H [a]. Since a, b are arbitrary, the as- sertion follows. Assume that dG(a, b) = 2. Since NG is a J2- set in G, there exist w, z ∈ V (G) such that w ∈ N2 G[a]\N2 G[b] and z ∈ N2 G[b]\N2 G[a]. Let s ∈ NG(w)∩NG(a) and t ∈ NG(z)∩NG(b). Then s ∈ N2 G+H [b]\N2 G+H [a] and t ∈ N2 G+H [a]\N2 G+H [b]. Since a, b are arbitrary, N is a J2-set of G + H. Next, suppose that dG(a, b) ≥ 3. Let u ∈ NG(a) and v ∈ NG(b), then u ∈ N2 G+H [b]\N2 G+H [a] and v ∈ N2 G+H [a]\N2 G+H [b]. Since a, b are arbi- trary, N is a J2-set of G+H. Similarly, if a, b ∈ NH , thenN is a J2-set of G+H. Next, sup- pose that a ∈ NG and b ∈ NH . Then a ∈ N2 G+H [a] \N2 G+H [b] and b ∈ N2 G+H [b] \N2 G+H [a]. Since a, b are arbitrary, it follows that N is a J2-set of G+H. J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2125 Theorem 7. Let G and H be two connected graphs. If N = NG∪NH ⊆ V (G+H), where NG and NH are J2-hop dominating sets in G and H, respectively, then N is a J2-hop dominating set in G+H. Moreover, γJ2h(G+H) ≥ γJ2h(G) + γJ2h(H). Proof. Let N = NG ∪ NH , where NG and NH are J2-hop dominating sets in G and H, respectively. Since NG and NH are J2-sets in G and H, respectively, it follows that N is a J2-set in G + H by Theorem 6. Since NG and NH are hop dominating sets in G and H, respectively, we have N2 G[NG] = V (G) and N2 G[NH ] = V (H). Observe that N2 G[NG] ⊆ N2 G+H [NG] and N2 H [NH ] ⊆ N2 G+H [NH ]. Thus, N2 G+H [N ] = N2 G+H [NG ∪NH ] = V (G+H), showing that N is a hop dominating set in G+H. Therefore, N is a J2-hop dominating set in G+H. Next, let N ′ = N ′ G ∪ N ′ H , where N ′ G and N ′ H are γJ2h-sets in G and H, respectively. Then by the first part, N ′ is a J2-hop dominating set in G+H. Consequently, γJ2h(G+H) ≥ |N ′| = |N ′ G|+ |N ′ H | = γJ2h(G) + γJ2h(H). Remark 1. The bound given in Theorem 7 is sharp. Moreover, strict inequality is attain- able. For the sharpness, consider the join graph P3 + P4 in Figure 4. Let S = {a, b, e, f}. Then N2 P3+P4 [S] = V (P3 + P4), showing that S is a hop dominating set in P3 + P4. Observe that x ∈ N2 P3+P4 [x] \ N2 P3+P4 [y] and y ∈ N2 P3+P4 [y] \ N2 P3+P4 [x] for every x ̸= y where x, y ∈ S. This means that N2 P3+P4 [x]\N2 P3+P4 [y] ̸= ∅ and N2 P3+P4 [y]\N2 P3+P4 [x] ̸= ∅ for every x ̸= y where x, y ∈ S. Thus, S is a J2-hop dominating set in P3 + P4. Since N2 P3+P4 [c] ⊆ N2 P3+P4 [a], N2 P3+P4 [f ] ⊆ N2 P3+P4 [d], and N2 P3+P4 [e] ⊆ N2 P3+P4 [g], it follows that S is a maximum J2-hop dominating set of P3 + P4. Hence, γJ2h(P3 + P5) = 4 . By Proposition 1, γJ2h(P3) = 2 and γJ2h(P4) = 2. Consequently, γJ2h(P3 + P4) = 4 = γJ2h(P3) + γJ2h(P4). J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2126 P3 + P4 : ba e fd g c Figure 4: Graph P3 + P4 with γJ2h(P3 + P4) = 4 = γJ2h(P3) + γJ2h(P4) For strict inequality, consider the graph P2+P8 in Figure 5. Let S′ = {a, b, d, e, f, g, h, i}. Then S′ is a γJ2h-set in P2+P8. Thus, γJ2h(P2+P8) = 8. By Proposition 1, γJ2h(P2) = 2 and γJ2h(P8) = 4. Hence, γJ2h(P2 + P8) = 8 > 6 = γJ2h(P2) + γJ2h(P8). P2 + P8 : b e a c gd hf i j Figure 5: Graph P2 + P8 with γJ2h(P2 + P8) > γJ2h(P2) + γJ2h(P8) Theorem 8. Let G be any non-trivial connected graph and H be any connected graph. If T = ⋃ v∈V (G) Tv, where Tv is a maximum J2-set in Hv for each v ∈ V (G), then T is a J2-hop dominating set in G ◦H. Moreover, γJ2h(G ◦H) ≥ |V (G)| · γJ2h(H). Proof. Suppose that T = ⋃ v∈V (G) Tv, where Tv is a maximum J2-set in Hv for each v ∈ V (G). Let a, b ∈ T . Suppose that a, b ∈ Tu for some u ∈ V (G). If dH(a, b) = 1, then a ∈ N2 G◦H [a] \ N2 G◦H [b] and b ∈ N2 G◦H [b] \ N2 G◦H [a]. It follows that T is a J2-set in G ◦H. Assume that dH(a, b) = 2. Since Tu is a J2- set in Hu, there exist w, z ∈ V (Hu) such that w ∈ N2 Hu [a]\N2 Hu [b] and z ∈ N2 Hu [b]\N2 Hu [a]. Let s ∈ NHu(w) ∩ NHu(a) and t ∈ NHu(z) ∩ NHu(b). Then s ∈ N2 G◦H [b]\N2 G◦H [a] and t ∈ N2 G◦H [a]\N2 G◦H [b]. Since a, b are arbitrary, T is a J2-set of G ◦H. Next, suppose that dH(a, b) ≥ 3. Let u ∈ NH(a) and J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2127 v ∈ NH(b), then u ∈ N2 G◦H [b]\N2 G◦H [a] and v ∈ N2 G◦H [a]\N2 G◦H [b]. Since a, b are arbitrary, T is a J2-set of G ◦H. Next, assume that a ∈ Tx and b ∈ Ty for some x, y ∈ V (G), x ̸= y. Then a ∈ N2 G◦H [a] \ N2 G◦H [b] and b ∈ N2 G◦H [b] \ N2 G◦H [a]. Thus, N2 G◦H [a] \ N2 G◦H [b] ̸= ∅ and N2 G◦H [b] \ N2 G◦H [a] ̸= ∅. Since a and b are arbitrary, T is a J2-set in G ◦ H. Now, since Tv is a maximum J2-set in Hv for each v ∈ V (G), it follows that Tv is a maximum J2-hop dominating set in Hv for every v ∈ V (G) by Theorem 2. Thus,⋃ v∈V (G) V (Hv) ⊆ N2 G◦H [T ]. Now, let r ∈ V (G ◦ H) \ ⋃ v∈V (G) V (Hv). Then r ∈ V (G). Since G is a non-trivial connected graph, there exists q ∈ Ts such that dG◦H(r, q) = 2 for some s ∈ V (G). Hence, N2 G◦H [T ] = V (G ◦ H), and so T is a J2-hop dominating set in G ◦ H. Consequently, γJ2h(G ◦H) ≥ |V (G)| · γJ2h(H). Lemma 1. [7] Let G be a non-trivial connected graph and let G1 and G2 be two copies of G in the graph S(G). If w ∈ V (G1) and w′ ∈ V (G2) is the corresponding vertex of w, then N2 S(G)[w] = N2 G1 [w] ∪N2 G2 [w′] = N2 S(G)[w ′]. Lemma 2. Let G be a non-trivial connected graph and let G1 and G2 be two copies of G in the graph S(G). If N2 G1 [a] ⊆ N2 G1 [b] or N2 G2 [a] ⊆ N2 G2 [b], then N2 S(G)[a] ⊆ N2 S(G)[b]. Proof. Let a, b ∈ V (G1) and suppose that N2 G1 [a] ⊆ N2 G1 [b]. Let x ∈ N2 S(G)[a]. Then dS(G)(a, x) = 2. If x ∈ V (G1), then dG1(a, x) = 2. So, x ∈ N2 G1 [a]. Thus, by assumption, x ∈ N2 G1 [b]. By Lemma 1, x ∈ N2 S(G)[b], and we are done. Suppose that x ∈ V (G2). Then x ∈ N2 G2 [a′] for some a′ ∈ V (G2). Since N2 G2 [a′] ⊆ N2 G2 [b′] ⊆ N2 S(G)[b], it follows that x ∈ N2 S(G)[b], and so N2 S(G)[a] ⊆ N2 S(G)[b]. Similarly, if N2 G2 [a] ⊆ N2 G2 [b], then N2 S(G)[a] ⊆ N2 S(G)[b]. Theorem 9. Let G be a connected non-trivial graph. Then T ⊆ V (S(G)) is a J2-set in S(G) if and only if T satisfies one of the following conditions: (i) T is a J2-set in G1. (ii) T is a J2-set in G2. (iii) T = TG1∪TG2, where TG1∪T ′ G2 and T ′ G1 ∪TG2 are J2-sets in G1 and G2, respectively, where T ′ G2 = {x ∈ V (G1) : x ′ ∈ TG2} and T ′ G1 = {y ∈ V (G2) : y ′ ∈ TG1}. Proof. Suppose that T is a J2-set in S(G). Let TG1 = T ∩V (G1) and TG2 = T ∩V (G2). If TG2 = ∅, then T = TG1 is a J2-set in G1. If TG1 = ∅, then T = TG2 is a J2-set in G2, showing that (i) or (ii) holds. Assume that TG1 ̸= ∅ and TG2 ̸= ∅. Suppose on the contrary that S = TG1 ∪ T ′ G2 is not a J2-set in G1. Then there exist a, b ∈ S such J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2128 that N2 G1 [a] \ N2 G1 [b] = ∅ or N2 G1 [b] \ N2 G1 [a] = ∅. It follows that N2 G1 [a] ⊆ N2 G1 [b] or N2 G1 [b] ⊆ N2 G1 [a]. If a, b ∈ TG1 , then a, b ∈ T . Since N2 G1 [a] ⊆ N2 G1 [b] or N2 G1 [b] ⊆ N2 G1 [a], we have N2 S(G)[a] ⊆ N2 S(G)[b] or N2 S(G)[b] ⊆ N2 S(G)[a] by Lemma 2. Thus, N2 S(G)[a] \N 2 S(G)[b] = ∅ or N2 S(G)[b] \N 2 S(G)[a] = ∅, a contradiction to the fact that T is a J2-set in S(G). Suppose that a, b ∈ T ′ G2 . Then a′, b′ ∈ TG2 ⊆ T . Since N2 G1 [a]\N2 G1 [b] = ∅ or N2 G1 [b] \ N2 G1 [a] = ∅, it follows that N2 G2 [a′] \ N2 G2 [b′] = ∅ or N2 G2 [b′] \ N2 G2 [a′] = ∅. Thus, N2 G2 [a′] ⊆ N2 G2 [b′] or N2 G2 [b′] ⊆ N2 G2 [a′], and so N2 S(G)[a ′] ⊆ N2 S(G)[b ′] or N2 S(G)[b ′] ⊆ N2 S(G)[a ′] by Lemma 2, which is a contradiction. Now, suppose that a ∈ TG1 and b ∈ T ′ G2 . Then b′ ∈ TG2 . Since N2 G1 [a] ⊆ N2 G1 [b] or N2 G1 [b] ⊆ N2 G1 [a], it follows that N2 S(G)[a] ⊆ N2 S(G)[b ′] or N2 S(G)[b ′] ⊆ N2 S(G)[a] by Lemma 1 and Lemma 2, a contradiction. Thus, S = TG1 ∪ T ′ G2 is a J2-set in G1. Similarly, T ′ G1 ∪ TG2 is a J2-set in G2. Thus, (iii) holds. Conversely, if (i) or (ii) holds, then the assertion follows. Assume that (iii) holds. Let x, y ∈ T = TG1 ∪ TG2 . If x, y ∈ TG1 ⊆ TG1 ∪ T ′ G2 , then N2 G1 [x] \ N2 G1 [y] ̸= ∅ and N2 G1 [y] \ N2 G1 [x] ̸= ∅ by assumption. This means that N2 G1 [x] ⊈ N2 G1 [y] and N2 G1 [y] ⊈ N2 G1 [x]. Thus, N2 S(G)[x] ⊈ N2 S(G)[y] and N2 S(G)[y] ⊈ N2 S(G)[x], and we are done. If x, y ∈ TG2 , then x′, y′ ∈ T ′ G2 ⊆ TG1 ∪ T ′ G2 . Since TG1 ∪ T ′ G2 is a J2-set in G1, we have N2 G1 [x′] ⊈ N2 G1 [y′] and N2 G1 [y′] ⊈ N2 G1 [x′]. Thus, by Lemma 1, N2 S(G)[x] ⊈ N2 S(G)[y] and N2 S(G)[y] ⊈ N2 S(G)[x]. Now, assume that x ∈ TG1 and y ∈ TG2 . Then y′ ∈ T ′ G2 , and so x, y′ ∈ TG1 ∪ T ′ G2 . Since TG1 ∪ T ′ G2 is a J2-set in G1, we have N2 G1 [x] ⊈ N2 G1 [y′] and N2 G1 [y′] ⊈ N2 G1 [x]. Thus, by Lemma 1, N2 S(G)[x] ⊈ N2 S(G)[y] and N2 S(G)[y] ⊈ N2 S(G)[x]. Since x, y are arbitrary, it follows that T is a J2-set in S(G). Theorem 10. [5] Let G be a non-trivial connected graph. Then S is a hop dominating set in S(G) if and only if one of the following conditions holds: (i) S is a hop dominating set in G1. (ii) S is a hop dominating set in G2. (iii) S = SG1 ∪ SG2 such that SG1 ∪ S′ G2 and S′ G1 ∪ SG2 are hop dominating sets in G1 and G2, respectively, where S′ G2 = {a ∈ V (G1) : a ′ ∈ SG2} and S′ G1 = {b ∈ V (G2) : b ′ ∈ SG1}. Theorem 11. Let G be a connected non-trivial graph. Then T ⊆ V (S(G)) is a J2-hop dominating set in S(G) if and only if T satisfies one of the following conditions: (i) T is a J2-hop dominating set in G1. (ii) T is a J2-hop dominating set in G2. J. Hassan, A. Bakkang, A. Sappari / Eur. J. Pure Appl. Math, 16 (4) (2023), 2118-2131 2129 (iii) T = TG ∪TH , where TG1 ∪T ′ G2 and T ′ G1 ∪TG2 are J2-hop dominating sets in G1 and G2, respectively, where T ′ G2 = {x ∈ V (G1) : x ′ ∈ TG2} and T ′ G1 = {y ∈ V (G2) : y ′ ∈ TG1}. Proof. Suppose that T is a J2-hop dominating set in S(G). Let TG1 = T ∩ V (G1) and TG2 = T ∩V (G2). If TG2 = ∅, then T = SG1 is a J2-hop dominating set of G1. If TG1 = ∅, then T = TG2 is a J2-hop dominating set of G2, showing that (i) or (ii) holds. Now, since T is a J2-set in S(G), TG1 ∪ T ′ G2 and T ′ G1 ∪ TG2 are J2-sets in G1 and G2, respectively, by Theorem 9. Also, since T is a hop dominating set in S(G), TG1 ∪ T ′ G2 and T ′ G1 ∪ TG2 are hop dominating sets in G1 and G2, respectively, by Theorem 10. Consequently, TG1 ∪T ′ G2 and T ′ G1 ∪ TG2 are J2-hop dominating sets in G1 and G2, respectively. For the converse, suppose (i) holds. Then T is both a J2-set and a hop dominating in G1. Thus, by Theorem 9 and by Theorem 10, T is a J2-hop dominating set in S(G). Similarly, if (ii) holds, then T is a J2-hop dominating set of S(G). Suppose that (iii) holds. Then by Theorem 9 and Theorem 10, T is a J2-hop dominating set of S(G). Corollary 2. Let G be a connected non-trivial graph. Then γJ2h(S(G)) = γJ2h(G). Proof. Let T be a γJ2h-set of G. Then by Theorem 11, T is a J2-hop dominating set of S(G). Thus, γJ2h(S(G)) ≥ |T | = γJ2h(G). On the other hand, suppose T ∗ is a γJ2h-set of S(G). If T ∗ is of type (i) or (ii), then T ∗ is a J2-hop dominating set of G by Theorem 11(i) and (ii). Hence, γJ2h(S(G)) = |T ∗| ≤ γJ2h(G). Next, suppose T ∗ is of type (iii), say T ∗ = TG1 ∪TG2 . Then T ∗ G = TG1 ∪T ′ G2 is a J2-hop dominating set of G1 by Theorem 11(iii). This implies that γJ2h(S(G)) = |T ∗| = |T ∗ G| ≤ γJ2h(G). Consequently, γJ2h(S(G)) = γJ2h(G). Example 2. Consider the shadow graph S(C4) of C4 in Figure 6. Let V (C4) = {a, b, c, d} and let N = {a, b}. Then N2 C4 [N ] = V (C4), a ∈ N2 C4 [a] \N2 C4 [b] and b ∈ N2 C4 [b] \N2 C4 [a]. Thus, N is a J2-hop dominating set of C4. Since N2 C4 [d] = N2 C4 [a] and N2 C4 [c] = N2 C4 [b], it follows that N is a maximum J2-hop dominating set of C4. Hence, γJ2h(C4) = 2. Observe that a ∈ N2 S(C4) [a] \N2 S(C4) [b] and b ∈ N2 S(C4) [b] \N2 S(C4) [a], showing that N is a J2-set in S(C4). SinceN 2 S(C4) [N ] = V (S(C4)), it follows thatN is a J2-hop dominating set in S(C4). By Lemma 1, N2 S(C4) [u] = N2 S(C4) [u′] for every u ∈ V (C4). Since N 2 S(C4) [a] = N2 S(C4) [d] and N2 S(C4) [b] = N2 S(C4) [c], it follows that N is a maximum J2-hop dominating set of S(C4). Thus, γJ2h(C4) = 2 = γJ2h(S(C4)). REFERENCES 2130 a′ b′ ba c d d′c′ S(C4) : Figure 6: Graph C4 with γJ2h(C4) = 2 = γJ2h(S(C4)) 4. Conclusion The concept of J2-hop domination has been introduced and initially investigated in this study. Its bounds with respect to other known parameters in graph theory have been determined. In addition, characterizations of J2-hop dominating sets in some graphs and shadow graph have been formulated and were used to solve exact value of the parameter of each of these graphs. Interested researchers may study further this parameter on graphs that were not considered in this study. Further, researchers may consider the investigation on the complexity of solving this parameter and provide application especially in real-life situation, network and other fields. Acknowledgements The authors would like to thank Mindanao State University - Tawi-Tawi College of Technology and Oceanography for funding this research. Also, the authors would like to thank the referees for their invaluable comments and suggestions that led to the improve- ment of the paper. References [1] S. Ayyaswamy, B. Krishnakumari, B. Natarjan, and Y. Venkatakrishnan. Bounds on the hop domination number of a tree. Proceedings-Mathematical Sciences., 125(4):449–455, 2015. [2] S. Ayyaswamy, C. Natarajan, and G. 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