EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1643-1655 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Total Partial Domination in Graphs under Some Binary Operations Roselainie D. Macapodi1,∗, Rowena T. Isla2,3 1 Mathematics Department, College of Natural Sciences and Mathematics, Mindanao State University-Main Campus, 9700 Marawi City, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines 3 Center for Graph Theory, Algebra, and Analysis, Premier Research Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G = (V (G), E(G)) be a simple graph without isolated vertices and let α ∈ (0, 1]. A set S ⊆ V (G) is an α-partial dominating set in G if |N [S]| ≥ α |V (G)|. The smallest cardinality of an α-partial dominating set in G is called the α-partial domination number of G, denoted by ∂α(G). An α-partial dominating set S ⊆ V (G) is a total α-partial dominating set in G if every vertex in S is adjacent to some vertex in S. The total α-partial domination number of G, denoted by ∂Tα(G), is the smallest cardinality of a total α-partial dominating set in G. In this paper, we characterize the total partial dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharp bounds of the corresponding total partial domination number of these graphs. 2010 Mathematics Subject Classifications: 05C69, 05C76 Key Words and Phrases: Partial domination, Total partial domination, Join, Corona, Lexico- graphic product, Cartesian product 1. Introduction Let G = (V (G), E(G)) be a simple graph and v ∈ V (G). The open neighborhood of v in G is the set NG(v) = {u ∈ V (G) : uv ∈ E(G)} and the closed neighborhood of v is the set NG[v] = NG(v) ∪ {v}. For X ⊆ V (G), the open neighborhood of X in G is the set NG(X) = N(X) = ⋃ v∈X NG(v) and its closed neighborhood is the set NG[X] = N [X] = N(X) ∪ X. A set D ⊆ V (G) is a dominating set in G if for every v ∈ V (G)\D, there exists u ∈ D such that uv ∈ E(G), that is, N [D]= V (G). The mini- mum cardinality of a dominating set in G, denoted by γ(G), is the domination number of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3554 Email addresses: macapodiroselainie@gmail.com (R. Macapodi) rowena.isla@g.msuiit.edu.ph (R. Isla) http://www.ejpam.com 1643 c© 2019 EJPAM All rights reserved. R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1644 G. Any dominating set in G of cardinality γ(G) is referred to as a γ-set in G. Let G be a graph without isolated vertices. A set T ⊆ V (G) is a total dominating set in G if N(T ) = V (G). The total domination number γt(G) of G is the minimum cardinality of a total dominating set in G. Dominating sets are important in a wide range of applications where some level of service or resource must be provided to each member of a network. However, considera- tions of scarcity of resources, practicality, or profitability may lead to a necessity for less than complete coverage of the nodes in a network. This gives rise to the notion of partial domination in graphs [1]. For any simple graph G and an α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominat- ing set in G if |N [S]| ≥ α |V (G)|. Case et al. [1] and Das [2] independently worked on α-partial domination in graphs in 2017. Case et al. focused on α = 1 2 while Das dealt with general values of α ∈ (0, 1] . The α-partial domination number ∂α(G) is the minimum cardinality of an α-partial dominating set in G. ∂α(G) is denoted by pdα(G) in Das [2] and by γ 1 2 (G) in Case et al. [1] when α = 1 2 . An α-partial dominating set S in G with |S| = ∂α(G) is referred to as an ∂α-set in G. Case et al. [1] investigated the partial domination number of some special graphs and presented some bounds of the said parameter. Das [2] also studied different bounds on the partial domination number of a graph with respect to several parameters like its order, maximum degree, and domination number. Macapodi, Isla and Canoy [3] characterized the partial dominating sets in the join, corona, lexicographic product and Cartesian prod- uct of graphs and determined the exact values or sharp bounds of the corresponding partial domination number of these graphs. They also introduced and examined the concepts of total partial domination and (α, k)-partial domination, where α ∈ (0, 1] and k ∈ (−∞, 0]. Let G be a simple graph. A nonempty set S ⊆ V (G) is an (α, k)-partial dominating set in G if |N [S]| ≥ α |V (G)|+ k. Let G be a graph without isolated vertices. An α-partial dominating set S ⊆ V (G) is a total α-partial dominating set in G if every vertex in S is adjacent to some vertex in S. In this case, we also say that G is totally α-partial dominated by the vertices in S. The total α-partial domination number of G, denoted by ∂Tα(G), is the minimum cardinality of a total α-partial dominating set in G. A total α-partial dominating set S with |S| = ∂Tα(G) is referred to as a ∂Tα-set in G. LetG be a connected graph. Let α ∈ (0, 1] and k ∈ (−∞, 0]. A nonempty set S ⊆ V (G) is a total (α, k)-partial dominating set in G if |N [S]| ≥ α |V (G)|+k and every element in S is adjacent to an element in S. The total (α, k)-partial domination number of G, denoted by ∂Tα,k(G), is given by ∂Tα,k(G) = min{|S| : S is a total (α, k)-partial dominating set in G}. Any partial dominating set in G with cardinality ∂Tα,k(G) is referred to as a ∂Tα,k-set in G. The join G+H of two graphs G and 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)}. R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1645 The corona of two graphs G and H, denoted by G ◦H, is the graph obtained by taking one copy of G of order n and n copies of H, and then joining the i-th vertex of G to every vertex in the i-th copy of H. For every v ∈ V (G), we denote by Hv the copy of H whose vertices are joined or attached to the vertex v. For each v ∈ V (G), the subgraph 〈v〉+Hv of G ◦H will be denoted by v +Hv. The lexicographic product of two graphs 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]) satisfying the following conditions: (u1, v1)(u2, v2) ∈ E(G[H]) if and only if either u1u2 ∈ E(G) or u1 = u2 and v1v2 ∈ E(H). The Cartesian product of two graphs 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) satisfying the following conditions: (u1, v1)(u2, v2) ∈ E(G�H) if and only if either u1u2 ∈ E(G) and v1 = v2 or u1 = u2 and v1v2 ∈ E(H). 2. Preliminary Results Remark 1. Let m,n and p be positive integers and let α ∈ (0, 1]. Let G be a complete graph Km, a fan graph Fm, a star graph K1,n or a wheel graph Wp. Then ∂Tα(G) = 2 for m ≥ 2, n ≥ 1 and p ≥ 3. Remark 2. For any α ∈ (0, 1] and a complete bipartite graph Km,n, with m,n ≥ 2, ∂Tα(Km,n) = 2. Remark 3. Let G be a graph without isolated vertices. If γt(G) = 2, then ∂Tα(G) = 2 for all α ∈ (0, 1]. Remark 4. Let G be a nontrivial graph. Then ∂α(G) ≤ ∂Tα(G) ≤ γt(G). Theorem 1. Let n be a positive integer and α = 1 2 . Then ∂Tα(Pn) = ∂Tα(Cn) =  2, 2 ≤ n ≤ 7 2r, n = 8r 2r + 1, n = 8r + s, s = 1, 2 2r + 2, n = 8r + s, s = 3, 4, 5, 6, 7 where r and s are integers such that n = 8r + s, 1 ≤ s ≤ 7. Proof. Let Pn = [v1, v2, ..., vn]. If 2 ≤ n ≤ 7, then clearly, ∂Tα(Pn) = 2. Let n ≥ 8 and consider the following cases: Case 1: n = 8r Group the vertices of Pn into r disjoint subsets. S1 = {v1, v2, v3, v4, v5, v6, v7, v8} S2 = {v9, v10, v11, v12, v13, v14, v15, v16} ... Sr−1 = {v8r−15, v8r−14, v8r−13, v8r−12, v8r−11, v8r−10, v8r−9, v8r−8} R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1646 Sr = {v8r−7, v8r−6, v8r−5, v8r−4, v8r−3, v8r−2, v8r−1, v8r} For every induced subgraph 〈vi, vi+1, vi+2, vi+3, vi+4, vi+5, vi+6, vi+7〉 of Pn, where i = 1, 9, ..., 8r − 7, the vertices vi+1 and vi+2 form a total α-partial dominating set of Pn. Thus, the set T = {v2, v3, v10, v11, . . . , v8r−6, v8r−5} is a total α-partial dominating set of Pn. Since |T | = 2r, ∂Tα(Pn) ≤ 2r. Note that every pair of adjacent vertices in Pn can dominate at most 4 vertices. Thus, every total α-partial dominating set in Pn contains at least dn4 e vertices. Hence, ∂Tα(Pn) ≥ dn4 e = 2r since n = 8r. Thus, ∂Tα(Pn) = 2r. Case 2: n = 8r + s, s = 1, 2 In Case 1, the first 8r vertices of Pn are totally α-partial dominated by 2r vertices in T . Since n = 8r + 1 or 8r + 2, vertices v8r+1 and v8r+2 of Pn are not totally α-partial dominated by T . Hence, ∂Tα(Pn) > 2r. Now, add v8r−4 to T so that T ∪ {v8r−4} is a ∂Tα-set in Pn. Thus, ∂Tα(Pn) = 2r + 1. Case 3: n = 8r + s, s = 3, 4, 5, 6, 7 Consider the total α-partial dominating set T in Case 1. Since {v8r+s|s = 3, 4, 5, 6, 7} are not totally α-partial dominated by vertices in T ∪ {v8r−4}, ∂Tα(Pn) > 2r + 1. Now, add vertices v8r+2 and v8r+3 to T so that T ∪ {v8r+2, v8r+3} is a ∂Tα-set in Pn. Thus, ∂Tα(Pn) = 2r + 2. Finally, it can be verified that the total α-partial domination number of Pn still holds for G = Cn for n ≥ 3. � We now present a realization problem. Theorem 2. Let a and b be positive integers such that a = b or b = 2a and let α = 1 2 . Then there exists a connected graph G such that ∂α(G) = a and ∂Tα(G) = b. Proof. Consider the following cases: Case 1: a = b Subcase (i). a is even. Let G = G1 be the graph shown in Figure 1. It is clear that the set A = {xi : i = 1, 2, ..., a} is both an ∂α-set and a ∂Tα-set in G1. It follows that ∂α(G1) = ∂Tα(G1) = |A| = a = b. Subcase (ii). a is odd. Let G = G2 be the graph shown in Figure 2. It is clear that the set A = {xi : i = 1, 2, ..., a} is both an ∂α-set and a ∂Tα-set in G2. It follows that ∂α(G2) = ∂Tα(G2) = |A| = a = b. R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1647 Case 2: b = 2a Let G = G3 be the graph shown in Figure 3. Observe that the set A = {xi : i = 1, 2, ..., a} is an ∂α-set and the set B = A ∪ {zj : j = 1, 2, ..., a} is a ∂Tα-set in G3. It follows that ∂α(G3) = |A| = a and ∂Tα(G3) = |B| = 2a = b. This proves the assertion. � Corollary 1. Given a positive integer m and α = 1 2 , there exists a connected graph G such that ∂Tα(G) − ∂α(G) = m, that is, the difference ∂Tα − ∂α can be made arbitrarily large. Proof. Let a = m and b = 2m. By Theorem 2, there exists a connected graph G with ∂Tα(G)− ∂α(G) = 2m−m = m. � The following characterizations of partial dominating sets in the join, corona, lexico- graphic product and Cartesian product of graphs are found in Macapodi et al. [3]. Theorem 3. Let G and H be connected graphs of orders m and n, respectively, and let α ∈ (0, 1]. Then C ⊆ V (G+H) is an α-partial dominating set in G+H if and only if at least one of the following is true: (i) C ⊆ V (G) and C is an (α, (α− 1)n)-partial dominating set in G. (ii) C ⊆ V (H) and C is an (α, (α− 1)m)-partial dominating set in H. (iii) C ∩ V (G) 6= ∅ and C ∩ V (H) 6= ∅. Theorem 4. Let G be a non-trivial connected graph of order m and H be any graph of order n. Let α ∈ (0, 1] and C ⊆ V (G ◦H). If at least one of the following holds: (i) C = ⋃ v∈V (G) Sv, where Sv is an α-partial dominating set in Hv for each v ∈ V (G), R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1648 (ii) C ⊆ V (G) where either C is a dominating set in G and |C| ≥ αm(n+ 1)−m n or |C| ≥ αm, then C is an α-partial dominating set in G ◦H. Theorem 5. Let G and H be connected graphs. Let α ∈ (0, 1] and C = ⋃ x∈S ({x} × Tx) ⊆ V (G[H]). If either one of the following holds: (i) S is a total α-partial dominating set in G, or (ii) S is an α-partial dominating set in G and Tx is a dominating set in H for every x ∈ S\NG(S), then C is an α-partial dominating set in G[H]. Theorem 6. Let G and H be nontrivial connected graphs and α ∈ (0, 1]. Then C1 = S1 × V (H) and C2 = V (G) × S2 are α-partial dominating sets in G�H if and only if S1 and S2 are α-partial dominating sets in G and H, respectively. 3. Main Results We characterize the total partial dominating sets in the join, corona, lexicographic product and Cartesian product of graphs in this section. Remark 5. Every total (α, (α − 1)n)-partial dominating set is an (α, (α − 1)n)-partial dominating set. Remark 6. Every total α-partial dominating set is an α-partial dominating set. Theorem 7. Let G and H be connected graphs of orders m and n, respectively, and let α ∈ (0, 1]. Then C ⊆ V (G+H) is a total α-partial dominating set in G+H if and only if at least one of the following is true: (a) C ⊆ V (G) and C is a total (α, (α− 1)n)-partial dominating set in G. (b) C ⊆ V (H) and C is a total (α, (α− 1)m)-partial dominating set in H. (c) C ∩ V (G) 6= ∅ and C ∩ V (H) 6= ∅. Proof. Suppose C ⊆ V (G + H) is a total α-partial dominating set in G + H. Then by Remark 6, C is an α-partial dominating set in G+H. By Theorem 3, at least one of the following is true: (i) C ⊆ V (G) and C is an (α, (α− 1)n)-partial dominating set in G, (ii) C ⊆ V (H) and C is an (α, (α− 1)m)-partial dominating set in H, or (iii) C ∩ V (G) 6= ∅ and C ∩ V (H) 6= ∅. Suppose (i) holds. Since C is a total α-partial dominating set in G+H and C ⊆ V (G), it follows that C is a total α-partial dominating set in G. Hence, C is a total (α, (α − 1)n)-partial dominating set in G, so Condition (a) holds. Similarly, R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1649 if (ii) holds, then Condition (b) is true. Finally, Condition (iii) is the same as Condition (c). For the converse, suppose Condition (a) holds. Then C ⊆ V (G) and C is an (α, (α−1)n)-partial dominating set in G by Remark 5. Thus, by Theorem 3, C ⊆ V (G+H) is an α-partial dominating set in G+H. Let x ∈ C. Since C is a total (α, (α−1)n)-partial dominating set in G, there exists a y ∈ C such that xy ∈ E (〈C〉). Thus, C is a total α-partial dominating set in G + H. Similarly, if Condition (b) holds, it can be shown that if g ∈ C ⊆ V (H), then there exists an h ∈ C such that gh ∈ E (〈C〉). Thus, C is a total α-partial dominating set in G + H. If Condition (c) holds, then by Theorem 3, C ⊆ V (G + H) is an α-partial dominating set in G + H. Moreover, C is clearly a total α-partial dominating set in G+H. � The next result immediately follows by Remark 5 and Theorem 7. Corollary 2. Let G and H be connected graphs, α ∈ (0, 1] and let C ⊆ V (G+H) satisfying one of the following conditions: (i) C ⊆ V (G) is a total α-partial dominating set in G. (ii) C ⊆ V (H) is a total α-partial dominating set in H. (iii) |C ∩ V (G)| ≥ 1 and |C ∩ V (H)| ≥ 1. Then C is a total α-partial dominating set in G+H. Corollary 3. Let G and H be connected graphs of orders m and n, respectively, and let α ∈ (0, 1]. Then, ∂Tα(G+H) = 2. Proof. Pick x ∈ V (G), y ∈ V (H). Clearly, S = {x, y} is a γt-set, hence a ∂Tα-set, in G+H and thus, ∂Tα(G+H) = 2. � Theorem 8. Let G be a nontrivial connected graph of order m and H be any graph of order n. Let α ∈ (0, 1] and C ⊆ V (G ◦H). If at least one of the following holds: (i) C = ⋃ v∈V (G) Sv, where Sv is a total α-partial dominating set in Hv for each v ∈ V (G), (ii) C ⊆ V (G) where either C is a total dominating set in G and |C| ≥ αm(n+ 1)−m n or |C| ≥ αm and C \NG(C) = ∅, then C is a total α-partial dominating set in G ◦H. Proof. Suppose Condition (i) holds. Since C = ⋃ v∈V (G) Sv, where Sv is a total α-partial dominating set in Hv for each v ∈ V (G), by Remark 6, Sv is an α-partial dominating set in Hv for each v ∈ V (G). It follows by Theorem 4 that C is an α-partial dominating set R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1650 in G ◦H. Moreover, since Sv is a total α-partial dominating set in Hv for each v ∈ V (G), C is a total α-partial dominating set in G ◦H. Suppose Condition (ii) holds. Suppose further that C is a total dominating set in G and |C| ≥ αm(n+ 1)−m n . Since C is a dominating set in G, it follows by Theorem 4 that C is an α-partial dominating set in G ◦ H. Moreover, since C ⊆ V (G) is a total α-partial dominating set in G, C is a total α-partial dominating set in G ◦ H. Next, if |C| ≥ αm and every vertex in C is adjacent to some vertex in C, then by Theorem 4, C is an α-partial dominating set in G ◦H. Since C \ NG(C) = ∅, C is a total α-partial dominating set in G ◦H. � Remark 7. The converse of Theorem 8 is not true. To see this, consider the graph P12 ◦ P2 in Figure 4. Let α = 1 2 . The shaded vertices form a ∂Tα-set but neither of Condition (i) nor (ii) holds since C = {2, 3, 6, 7, 8} is not a total dominating set in G and |C| < αm. The next result is an immediate consequence of Theorem 8. Corollary 4. Let G be a nontrivial connected graph of order m and H be any graph of order n and α ∈ (0, 1] . Then ∂Tα(G ◦H) ≤ min { ηtG, µ t G } , where ηtG = min { |C| : C is a total dominating set in G with |C| ≥ αm(n+ 1)−m n } and µtG = min{|C ′| : C ′ ⊆ V (G) with ∣∣C ′∣∣ ≥ αm and C ′ \NG(C ′) = ∅}. Remark 8. The bound in Corollary 4 is sharp. However, the strict inequality can be attained. To see this, consider the graphs shown in Figures 5 and 6. Let α = 2 5 . The shaded ver- tices in Figure 5 form a ∂Tα-set. Then ∂Tα(P5◦P5) = 2 = min{3, 2} = min{ηtG, µtG} = µtG. Let α = 3 5 . The shaded vertices in Figure 6 form a ∂Tα-set. Then ∂Tα(P12 ◦ P2) = 6 = min{6, 8} = min{ηtG, µtG} = ηtG while ∂Tα(P5 ◦K1) = 2 < 3 = min{3, 3} = min{ηtG, µtG}. R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1651 Theorem 9. Let G be a connected graph and α ∈ (0, 1]. If S is an α-partial dominating set in G, then ∂Tα(G) ≤ |S ∩NG(S)|+ 2|S \NG(S)|. In particular, ∂Tα(G) ≤ 2∂α(G). Proof. Let S be an α-partial dominating set in G. If S is a total α-partial dominating set in G, then S ∩NG(S) = S and S \NG(S) = ∅. Hence, ∂Tα(G) ≤ |S| = |S ∩NG(S)|+ 2|S \NG(S)|. So suppose that S \ NG(S) 6= ∅. Choose vx ∈ V (G) ∩ NG(x) for each x ∈ S \ NG(S) and let SG = {vx : x ∈ S \NG(S)}. Then |SG| ≤ |S \NG(S)| and S∗ = S ∪ SG is a total α-partial dominating set in G. Thus, ∂Tα(G) ≤ |S∗| = |S|+ |SG| ≤ |S ∩NG(S)|+ |S \NG(S)|+ |S \NG(S)| = |S ∩NG(S)|+ 2|S \NG(S)|. In particular, if S is an ∂α-set in G, then ∂Tα(G) ≤ |S ∩NG(S)|+ |S \NG(S)|+ |S \NG(S)| ≤ 2|S| = 2∂α(G). � Theorem 10. Let G and H be connected graphs. Let α ∈ (0, 1] and C = ⋃ x∈S ({x} × Tx) ⊆ V (G[H]) . If either one of the following holds: (i) S is a total α-partial dominating set in G, or (ii) S is an α-partial dominating set in G and Tx is a total dominating set in H for every x ∈ S\NG(S), R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1652 then C is a total α-partial dominating set in G[H]. Proof. Suppose Condition (i) holds. Since S is a total α-partial dominating set in G, C is an α-partial dominating set in G[H] by Theorem 5. Let (x, a) ∈ C. Then x ∈ S. Since S is a total α-partial dominating set in G, there is a y ∈ S such that xy ∈ E(G). Pick (y, b) ∈ C. Then (x, a)(y, b) ∈ E(G[H]). Hence, C is a total α-partial dominating set in G[H]. Suppose Condition (ii) holds. Since Tx, being a total dominating set, is a dominating set in H for every x ∈ S \ NG(S), it follows by Theorem 5 that C is an α-partial dominating set in G[H]. Let x ∈ S \NG(S). Suppose (x, a) ∈ C. Since Tx is a total dominating set in H, there exists a b ∈ Tx such that ab ∈ E(H), hence (x, b) ∈ C and (x, a)(x, b) ∈ E(G[H]). Suppose x ∈ S ∩ NG(S). Then there exists y ∈ S such that xy ∈ E(G). Let (x, a) ∈ C. Pick (y, b) ∈ C. Then (x, a)(y, b) ∈ E (G[H]). Therefore, C is a total α-partial dominating set in G[H]. � Remark 9. The converse of Theorem 10 is not true. To see this, consider the graph P5[P5] in Figure 7. The set C = {(x, a), (x, b)} form a ∂Tα-set, where α = 1 2 , but neither of Condition (i) nor (ii) holds since {x} is not a total α-partial dominating set in G, and S = {x} is an α-partial dominating set in G but Tx = {a, b} is not a total dominating set in H. Corollary 5. Let G and H be nontrivial connected graphs and let α ∈ (0, 1]. Then ∂Tα(G[H]) = ∂Tα(G). Proof. By Theorem 10, ∂Tα(G[H]) ≤ min {∂α(G) · γt(H), ∂Tα(G)} . Clearly, ∂Tα(G[H]) ≤ ∂Tα(G). Next, let C = ⋃ x∈S ({x} × Tx) be a ∂Tα-set in G[H]. Suppose that there exists y ∈ S ∩ NG(S) such that |Ty| ≥ 2. Let T ′x = Tx for all x ∈ S \ {y}, and let T ′y = {a} where a ∈ Ty. Let C ′ = ⋃ z∈S ( {z} × T ′z ) . Since y ∈ S ∩ NG(S), NG[H][C ′] = NG[H][C]. Hence, R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1653 |NG[H][C ′]| = |NG[H][C]| ≥ α|V (G[H])|, that is, C ′ is a total α-partial dominating set in G[H]. Further, |C ′| < |C| since |T ′y| = 1 < |Ty|, a contradiction. Thus, |Tx| = 1 for all x ∈ S ∩NG(S). Now, since every element of C is adjacent to an element of C, it follows that |Tx| ≥ 2 for all x ∈ S \NG(S). By Theorem 9, ∂Tα(G[H]) = |C| = ∑ x∈S∩NG(S) |Tx|+ ∑ x∈S\NG(S) |Tx| ≥ |S ∩NG(S)|+ 2 · |S \NG(S)| ≥ ∂Tα(G). Therefore, ∂Tα(G[H]) = ∂Tα(G). � Corollary 6. If ∂α(G) = 1 and γt(H) = 2, then ∂Tα(G[H]) = 2. Theorem 11. Let G and H be nontrivial connected graphs and let α ∈ (0, 1]. Then C1 = S1 × V (H) and C2 = V (G)× S2 are total α-partial dominating sets in G�H if and only if S1 and S2 are α-partial dominating sets in G and H, respectively. Proof. Suppose C1 = S1 × V (H) and C2 = V (G)× S2 are total α-partial dominating sets in G�H. Then by Remark 6, C1 and C2 are α-partial dominating sets in G�H. By Theorem 6, S1 and S2 are α-partial dominating sets in G�H. Suppose S1 and S2 are α-partial dominating sets in G and H, respectively. Then by Theorem 6, C1 = S1 × V (H) and C2 = V (G) × S2 are α-partial dominating sets in G�H. Let (x, a) ∈ C1 = S1 × V (H). Since H is connected, there exists a vertex b ∈ V (H) such that ab ∈ E(H). Hence, (x, b) ∈ C1 and (x, a)(x, b) ∈ E(G�H). Similarly, if (y, c) ∈ C2 = V (G) × S2, then since G is connected, there exists (z, c) ∈ C2 such that (y, c)(z, c) ∈ E(G�H). Thus, C1 and C2 are total α-partial dominating sets in G�H. � Corollary 7. Let G and H be nontrivial connected graphs. Then C1 = S1 × V (H) and C2 = V (G)× S2 are α-partial dominating sets in G�H if and only if C1 and C2 are total α-partial dominating sets in G�H. Proof. Suppose C1 = S1 × V (H) and C2 = V (G) × S2 are α-partial dominating sets in G�H. Then by Theorem 6, S1 and S2 are α-partial dominating sets in G and H, re- spectively. By Theorem 11, C1 = S1 × V (H) and C2 = V (G) × S2 are total α-partial dominating sets in G�H. Conversely, suppose C1 = S1 × V (H) and C2 = V (G) × S2 are total α-partial domi- nating sets in G�H. By Remark 6, C1 = S1 × V (H) and C2 = V (G)× S2 are α- partial dominating sets in G�H. � Corollary 8. Let G and H be nontrivial connected graphs of orders m and n, respectively, and α ∈ (0, 1]. Then, ∂Tα(G�H) ≤ min { m · ∂α(H), n · ∂α(G) } . Remark 10. The bound in Corollary 8 is sharp. However, the strict inequality can be attained. R. Macapodi, R. Isla / Eur. J. Pure Appl. Math, 12 (4) (2019), 1643-1655 1654 To see this, consider the graphs shown in Figure 8. Let α = 1 2 . The shaded vertices in each graph form a ∂Tα-set. Thus, ∂Tα(P4�P6) = 4 = min {4, 6} = min {4(1), 6(1)} = min {4 · ∂α(P6), 6 · ∂α(P4)} = 4·∂α(P6), ∂Tα(P6�P3) = 3 = min {6, 3} = min {6(1), 3(1)} = min {6 · ∂α(P3), 3 · ∂α(P6)} = 3·∂α(P6), and ∂Tα(P6�P5) = 4 < 5 = min{6(1), 5(1)} = min{6 · ∂α(P5), 5 · ∂α(P6)}. Corollary 9. Let G be a connected graph of order m and Kn be the complete graph of order n ≥ 2. Then, ∂Tα(G�Kn) ≤ min{m,n · ∂α(G)}. Remark 11. The bound in Corollary 9 is sharp. However, the strict inequality can be attained. To see this, consider the graphs shown in Figure 9. Let α = 1 2 . The shaded vertices in each graph form a ∂Tα-set. Thus, ∂Tα(P2�K4) = 2 = min {2, 4(1)} = min {m, 4 · ∂α(P2)} = m, ∂Tα(P6�K3) = 3 = min {6, 3(1)} = min {m, 3 · ∂α(K3)} = 3 · ∂α(K3), and ∂Tα(P6�K4) = 3 < 4 = min{6, 4(1)} = min{m, 4 · ∂α(P6)}. REFERENCES 1655 Acknowledgements This research is funded by the Philippine Commission on Higher Education-Faculty Development Program Phase II (CHED-FDP II) and the Mindanao State University-Iligan Institute of Technology. The authors thank the anonymous reviewers for the suggestions that led to the improvement of the paper. References [1] B. Case, S. Hedetniemi, R. Laskar, and D. Lipman. Partial Domination in Graphs. Eprint, arXiv:1705.03096, 2017. [2] A. Das. Partial Domination in Graphs. Iranian Journal of Science and Technology. Transaction A: Science, pages 1713–1718, 2018 DOI: 10.1007/s40995-018-0618-5. [3] R. Macapodi, R. Isla, and S. Canoy Jr. Partial Domination in the Join, Corona, Lex- icographic and Cartesian Products of Graphs. Advances and Applications in Discrete Mathematics, 20(2):277–293, 2019.