EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 2, 2021, 578-589 ISSN 1307-5543 – ejpam.com Published by New York Business Global On k-Fair Total Domination in Graphs Wardah M. Bent-Usman1,∗, Rowena T. Isla2 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, 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 non-empty graph. For an integer k ≥ 1, a k-fair total dominating set (kftd-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkftd(G), is the minimum cardinality of a kftd-set. A k-fair total dominating set of cardinality γkftd(G) is called a minimum k-fair total dominating set or a γkftd-set. We investigate the notion of k-fair total domination in this paper. We also characterize the k-fair total dominating sets in the join, corona, lexicographic product and Cartesian product of graphs and determine the exact values or sharp bounds of their corresponding k-fair total domination number. 2020 Mathematics Subject Classifications: 05C69, 05C76 Key Words and Phrases: k-fair domination, k-fair total 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) = ⋃ v∈X NG(v) and its closed neighborhood is the set NG[X] = NG(X) ∪X. A set S ⊆ V (G) is a dominating set in G if for every v ∈ V (G)\S, there exists u ∈ S such that uv ∈ E(G), that is, NG[S] = V (G). The minimum cardinality of a dominating set in G, denoted by γ(G), is the domination number of G. Any dominating set in G of cardinality γ(G) is referred to as a γ-set in G. For a connected graph G, a set S ⊆ V (G) is a total ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i2.3967 Email addresses: wardah.bentusman@msumain.edu.ph (W. Bent-Usman), rowena.isla@g.msuiit.edu.ph (R. Isla) http://www.ejpam.com 578 c© 2021 EJPAM All rights reserved. W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 579 dominating set in G if NG(S) = V (G). A domination variant called fair domination was introduced by Caro, Hansberg and Henning [2] in 2012. For an integer k ≥ 1, a k-fair dominating set (kfd-set) is a dominat- ing set S ⊆ V (G) such that |NG(u)∩S| = k for every u ∈ V (G)\S. The k-fair domination number of G, denoted by γkfd(G), is the minimum cardinality of a kfd-set. In 2014, Maravilla et al.[5] characterized the k-fair dominating sets in the join, corona, lexicographic product, and Cartesian product of graphs and determined the bounds or exact values of the k-fair domination numbers of these graphs. Two variants of k-fair domination, namely connected k-fair domination and neighborhood connected k-fair dom- ination, were studied by Bent-Usman et al. [1, 6] in 2018 and 2019, respectively. Recently, Ortega and Isla [7] introduced and investigated the concepts of semitotal k-fair domination and independent k-fair domination in graphs. Maravilla et al. [4] introduced the notion of k-fair total domination in graphs. For a non-empty graph G and an integer k ≥ 1, a k-fair total dominating set (kftd-set) is a total dominating set S ⊆ V (G) such that |NG(u) ∩ S| = k for every u ∈ V (G)\S. The k-fair total domination number of G, denoted by γkftd(G), is the minimum cardinality of a kftd-set. A k-fair total dominating set of cardinality γkftd(G) is called a minimum k-fair total dominating set or a γkftd-set. In this paper, we investigate the concept of k-fair total domination and characterize the k-fair total dominating sets in graphs under some binary operations. We also determine the exact values or sharp bounds of their corresponding k-fair total domination number. A comprehensive treatment of the theoretical, algorithmic, and application (e.g., facil- ity location) aspects of domination in graphs was provided by Haynes et al.[3] in 1998. 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)}. 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). W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 580 2. Preliminary Results Remark 1. For any connected graph G of order n ≥ 2 and a positive integer k , γkfd(G) ≤ γkftd(G) and γkftd(G) ≥ 2. Remark 2. Any kftd-set is a kfd-set, where k is a positive integer. Theorem 1. Let n and r be positive integers where n ≥ 2 and r ≥ 1. Then γ1ftd(Pn) =  2, n = 2, 3 2r, n = 4r 2r + 1, n = 4r + 1 2r + 2, otherwise. Proof. Let G = Pn = {v1, v2, v3, ..., vn}. If n = 2 or n = 3, then clearly, γ1ftd(Pn) = 2 . Let n ≥ 4 and consider the following cases: Case 1: n = 4r Group the first 4r vertices of Pn into r disjoint subsets. S1 = {v1, v2, v3, v4} S2 = {v5, v6, v7, v8} S3 = {v9, v10, v11, v12} ... Sr−1 = {v4r−7, v4r−6, v4r−5, v4r−4} Sr = {v4r−3, v4r−2, v4r−1, v4r} For every induced subgraph 〈vi, vi+1, vi+2, vi+3〉 of Pn, where i = 1, 5, 9, ..., 4r − 3, the vertices vi+1 and vi+2 are in a 1-fair total dominating set of Pn. Thus, the set T = {v2, v3, v6, v7, ..., v4r−2, v4r−1} is a 1-fair total dominating set of Pn. Since |T | = 2r, γ1ftd(Pn) ≤ 2r. Note that every pair of adjacent vertices in Pn can dominate at most 2 vertices. Thus, every 1-fair total dominating set of Pn contains at least dn2 e vertices. Hence, γ1ftd(Pn) ≥ dn2 e = 2r since n = 4r. Thus, γ1ftd(Pn) = 2r. Case 2: n = 4r + 1 The set T in Case 1 is no longer a γ1ftd-set of Tn here since v4r+1 is not adjacent to any vertex in T , but clearly, T ∪ {v4r} is a γ1ftd-set. Thus, γ1ftd(Pn) = 2r + 1. Case 3. n = 4r + 2 The set S = T ∪ {v4r} is not a γ1ftd-set of Pn here since v4r+2 is not adjacent to any vertex in S, but T ∪ {v4r, v4r+1} is clearly a γ1ftd-set. Hence,γ1ftd(Pn) = 2r + 2. Case 4. n = 4r + 3 Consider the 1-fair total dominating set T in Case 1. Add v4r+2 and v4r+3 to the vertices in T so that T ∪ {v4r+2, v4r+3} is a γ1ftd-set of Pn. Hence, γ1ftd(Pn) = 2r + 2. � W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 581 Theorem 2. Let n and r be positive integers where n ≥ 3 and r ≥ 1. Then γ1ftd(Cn) =  3, n = 3 2r, n = 4r 2r + 1, n = 4r + 1 2r + 2, n = 4r + 2 2r + 3, n = 4r + 3. Proof. Suppose that Cn = [v1, v2, ..., vn, v1]. If n = 3, then clearly, γ1ftd(C3) = 3. The proof for n = 4r, n = 4r + 1, and n = 4r + 2 is similar to the proof of Cases 1 to 3 of Theorem 1. When n = 4r + 3, let T = {v2, v3, v6, v7, ...., v4r−2, v4r−1}. It can be verified that T ∪ {v4r, v4r+1, v4r+2} is a γ1ftd-set of Cn. Thus, γ1ftd(Cn) = 2r + 3. � Lemma 1. [5] Let Kn be the complete graph of order n and k a positive integer with k ≤ n. Then γkfd(Kn) = k. Theorem 3. Let n and k be positive integers, 2 ≤ k ≤ n. Then, γkftd(Kn) = k. Proof. Clearly, γ2ftd(K2) = 2, γ2ftd(K3) = 2, and γ3ftd(K3) = 3. Let n > 3. Let V (Kn) = {v1, v2, ..., vn}, and S = {v1, v2, ..., vk}. Note that each vertex in S is adjacent to the remaining k − 1 vertices in S. Moreover, for each vi ∈ V (Kn)\S, that is, for each vi, k+ 1 ≤ i ≤ n, |N(vi) ∩ S| = k. Thus, S is a kftd-set in Kn and γkftd(Kn) ≤ k. However, γkftd(Kn) ≥ γkfd(Kn) = k by Remark 1 and Lemma 1. Thus, γkftd(Kn) = k. � Theorem 4. Let a and b be positive integers such that a ≤ b. Then there exists a connected graph G such that γ1fd(G) = a and γ1ftd(G) = b. Proof. Consider the following cases: Case 1. a = b Let G = G1 be the graph shown in Figure 1. It is clear that the set A = {xi : i = 1, 2, ...a} is both a γ1fd-set and a γ1ftd-set in G1. It follows that γ1fd(G1) = γ1ftd(G1) = a. W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 582 Case 2. a < b Let G = G2 be the graph shown in Figure 2. Let A = {x1, x2, ..., xa−1}. It is clear that the set B = A ∪ {za} is a γ1fd-set and the set C = A∪{xa}∪{y1, y2, ..., yb−a} is a γ1ftd-set in G2. It follows that γ1fd(G2) = |B| = a and γ1ftd(G2) = |C| = b. � Corollary 1. γ1ftd − γ1fd can be made arbitrarily large. 3. Known Results The following characterizations of k-fair dominating sets in the join, corona, and lexi- cographic product of two nontrivial, connected graphs are found in Maravilla et al. [5]. Theorem 5. [5] Let G and H be nontrivial connected graphs of orders m and n, respec- tively, and k a positive integer with 1 ≤ k ≤ max{m,n}. Then S ⊆ V (G+H) is a kfd-set in G+H if and only if one of the following holds: (a) S = V (G+H). (b) S ⊆ V (G), |S| = k and S is a kfd-set in G. (c) S ⊆ V (H), |S| = k and S is a kfd-set in H. (d) S = SG ∪ SH , where SG is a (k − |SH |)fd-set in G and SH is a (k − |SG|)fd-set in H. (e) S = V (G) ∪ T , where |V (G)| = m < k and T is a (k −m)fd-set in H. (f) S = D ∪ V (H), where |V (H)| = n < k and D is a (k − n)fd-set in G. Theorem 6. [5] Let G and H be nontrivial connected graphs and let k be a positive integer with k ≤ |V (H)|. Then C ⊆ V (G ◦ H) is a kfd-set in G ◦ H if and only if one of the following holds: W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 583 (a) C = V (G) ∪B, where B = ∅ or B = ⋃ v∈V (G) Sv, where each Sv is a (k − 1)fd-set in Hv. (b) C = ⋃ v∈V (G) Sv, where each Sv is a kfd-set in Hv and |Sv| = k. Theorem 7. [5] Let G and H be nontrivial connected graphs. Then C = ⋃ x∈S ({x}×Tx) ⊆ V (G[H]) is a kfd-set in G[H] if and only if the following hold: (i) S is a dominating set in G. (ii) For each x ∈ S ∩NG(S), Tx = V (H) and |V (H)| = r ≤ k whenever C 6= V (G[H]) or Tx is an rfd-set and ∑ z∈NG(x)∩S |Tz| = k − r. (iii) For each x ∈ S\NG(S), Tx = V (H) and |V (H)| ≤ k or |Tx| = k and Tx is a kfd-set in H. (iv) For each y ∈ V (G)\S, ∑ v∈NG(y)∩S |Tv| = k. 4. Main Results We characterize the k-fair total dominating sets in the join, corona, and lexicographic product of graphs in this section, as well as some such sets in the Cartesian product of graphs. We also determine the k-fair total domination number of the join and corona of any two connected graphs and establish sharp bounds of the k-fair total domination number of the lexicographic and Cartesian products of graphs. Theorem 8. Let G and H be nontrivial connected graphs of orders m and n, respectively, and k a positive integer with 2 ≤ k ≤ max{m,n}. Then S ⊆ V (G + H) is a kftd-set in G+H if and only if one of the following holds: (a) S = V (G+H). (b) S ⊆ V (G), |S| = k and S is a kftd-set in G. (c) S ⊆ V (H), |S| = k and S is a kftd-set in H. (d) S = SG ∪ SH , where SG is a (k − |SH |)fd-set in G and SH is a (k − |SG|)fd-set in H. (e) S = V (G) ∪ T , where |V (G)| = m < k and T is a (k −m)fd-set in H. (f) S = D ∪ V (H), where |V (H)| = n < k and D is a (k − n)fd-set in G. W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 584 Proof. Suppose that S ⊆ V (G + H) is a kftd-set in G + H, where k ≥ 2. Then S is a kfd-set in G + H. Suppose further that S 6= V (G + H). If S ⊆ V (G), then |S| = k and S is a kfd-set in G by Theorem 5. Since S is a total dominating set in G + H, it is a total dominating set in G. Hence, S must be a kftd-set in G. Similarly, if S ⊆ V (H), then |S| = k and S is a kftd-set in H. Suppose S ∩ V (G) 6= ∅ and S ∩ V (H) 6= ∅. Then by Theorem 5, S = SG ∪ SH , where SG is a (k − |SH |)fd-set in G and SH is a (k− |SG|)fd-set in H, or S = V (G)∪ T , where |V (G)| = m < k and T is a (k−m)fd-set in H, or S = D ∪ V (H), where |V (H)| = n < k and D is a (k − n)fd-set in G. Conversely, suppose one of Statements (a) to (f) holds. Then S is a kfd-set in G+H by Theorem 5. If Statement (a) holds, then S is clearly a kftd-set in G + H. Suppose Statement (b) holds. Since S is a kftd-set in G, S is a kftd-set in G + H. Similarly, if Statement (c) holds, then the same conclusion follows. If Statement (d) is satisfied, then every vertex in SG is adjacent to each vertex in SH and vice versa, hence S = SG ∪ SH is a kftd-set in G + H. If Statement (e) holds, then every vertex in T is adjacent to each of the vertices in G and each vertex in G is adjacent to some vertex in G and to each of the vertices in T , hence S = V (G)∪ T is a kftd-set in G+H. Similarly, if Statement (f) holds, then S = D ∪ V (H) is a kftd-set in G+H. This proves the assertion. � Corollary 2. Let G and H be connected nontrivial graphs of orders m and n, respectively, and k a positive integer with 2 ≤ k ≤ max{m,n}. If G or H has a kftd-set S with |S| = k, then γkftd(G+H) = k. Theorem 9. Let G be a nontrivial connected graph and H a nontrivial graph, and let k be a positive integer with k ≤ |V (H)|. Then C ⊆ V (G ◦H) is a kftd-set in G ◦H if and only if one of the following holds: (a) C = V (G) ∪ B, where B = ∅ when k = 1 and B = ⋃ v∈V (G) Sv, where each Sv is a (k − 1)fd-set in Hv when k ≥ 2. (b) C = ⋃ v∈V (G) Sv, where each Sv is a kftd-set in Hv and |Sv| = k. Proof. Suppose that Statement (a) holds. Then by Theorem 6, C is a kfd-set in G◦H when k = 1. If B = ∅, then C = V (G) is clearly a kftd-set in G◦H when k = 1. Suppose B = ⋃ v∈V (G) Sv, where each Sv is a (k− 1)fd-set in Hv. Each vertex v in V (G) is adjacent to some vertex u in V (G), and each x ∈ Sv is adjacent to v. Thus, C = V (G) ∪ B is a kftd-set in Hv. Suppose Statement (b) holds. Since each Sv is a kfd-set in Hv and |Sv| = k, C = ⋃ v∈V (G) Sv is a kfd-set in G ◦H by Theorem 6. Moreover, since each Sv is a kftd-set in Hv, it follows that C is a kftd-set in G ◦H. Conversely, suppose C ⊆ V (G ◦ H) is a kftd-set in G ◦ H. Then C is a kfd-set in G ◦H and by Theorem 6, either Statement (a) holds, or C = ⋃ v∈V (G) Sv, where each Sv is W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 585 a kfd-set in Hv and |Sv| = k. Suppose that there is a vertex x in Sv that is not adjacent to another vertex in Sv. Then C is not a kftd-set in V (G ◦H), contrary to assumption. Thus, each Sv must be a kftd-set in Hv and Statement (b) holds. � The next result is an immediate consequence of Theorem 9. Corollary 3. Let G be a nontrivial connected graph of order m and let H be a nontrivial graph of order n, and let k be a positive integer with 1 ≤ k ≤ n. Then γkftd(G ◦H) =  m, if k = 1 mk, if k ≥ 2 and H has a kftd-set S with |S| = k m(1 + γ(k−1)fd(H)), if k ≥ 2 and H has no kftd-set S with |S| = k . Theorem 10. Let G and H be nontrivial connected graphs and let k ≥ 2. Then C = ⋃ x∈S ({x} × Tx) ⊆ V (G[H]) is a kftd-set in G[H] if and only if the following hold: (i) S is a dominating set in G, (ii) for each x ∈ S ∩ NG(S) such that Tx 6= V (H), Tx is an rfd-set and∑ z∈NG(x)∩S |Tz| = k − r, (iii) for each x ∈ S\NG(S) with Tx 6= V (H), |Tx| = k and Tx is a kftd-set in H, and (iv) for each y ∈ V (G)\S, ∑ v∈NG(y)∩S |Tv| = k. Proof. Suppose C = ⋃ x∈S ({x} × Tx) ⊆ V (G[H]) is a kftd-set in G[H]. Then C is a kfd-set in G[H] and by Theorem 7, Statements (i), (ii), and (iv) hold. Moreover, for each x ∈ S\NG(S), Tx = V (H) and |V (H)| ≤ k or |Tx| = k and Tx is a kfd-set in H. Suppose there is a vertex a ∈ Tx which is not adjacent to any other vertex in Tx. Then (x, a) is not adjacent to any vertex in C, contrary to assumption. Hence, Tx is a kftd-set in H and Statement (iii) holds. Conversely, suppose Statements (i) to (iv) hold. Then Tx is a kfd-set in H. Thus, C is a kfd-set in G[H] by Theorem 7. Suppose C 6= V (G[H]). Let (x, a) ∈ C. Consider the following cases. Case 1: x ∈ S ∩NG(S) If Tx = V (H) where |V (H)| = r ≤ k, then there exists a b ∈ Tx such that ab ∈ E(H) since H is a nontrivial connected graph. It follows that (x, b) ∈ C and (x, a)(x, b) ∈ E(G[H]). If Tx is an rfd-set and ∑ z∈NG(x)∩S |Tz| = k − r, then there is a z ∈ NG(x) ∩ S and there is a d ∈ Tz such that (z, d) ∈ C. Clearly, (x, a)(z, d) ∈ E(G[H]). Case 2: x ∈ S\NG(S) If Tx = V (H) where |V (H)| ≤ k, then similar to Case 1, there exists a b ∈ Tx such that ab ∈ E(H), (x, b) ∈ C, and (x, a)(x, b) ∈ E(G[H]). If |Tx| = k and Tx is a kftd-set in H, W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 586 then there exists a d ∈ Tx such that ad ∈ E(H), (x, d) ∈ C, and (x, a)(x, d) ∈ E(G[H]). Therefore, in both cases, C is a kftd-set in G[H]. � Corollary 4. Let G and H be nontrivial connected graphs with γ1fd(H) = 1. If G has a γ2ftd-set S with |NG(x) ∩ S| = 1 for all x ∈ S, then γ2ftd(G[H]) ≤ γ2ftd(G). Proof. For each x ∈ S, let Tx = {a}, where {a} is a γ1fd-set of H, and let C = ⋃ x∈S [{x} × Tx]. Since S is a total dominating set, |NG(x) ∩ S| = 1 and |Tx| = 1 for all x ∈ S, Conditions (i), (ii), and (iii) of Theorem 10 are satisfied. Moreover, since S is a γ2fd-set, |NG(y) ∩ S| = 2 for each y ∈ V (G)\S. Hence, Condition (iv) of Theorem 10 is also satisfied. Therefore, by Theorem 10, C is a 2ftd-set of G[H]. Accordingly, γ2ftd(G[H]) ≤ |C| = ∑ x∈S |Tx| = |S| = γ2ftd(G). � Remark 3. The bound given in Corollary 4 is sharp. To see this, consider the graph P5[P3] shown in Figure 3. The shaded vertices in P5[P3] form a γ2ftd-set. Thus, γ2ftd(P5[P3]) = 4 = γ2ftd(P5). Corollary 5. Let G and H be nontrivial connected graphs such that |V (H)| ≥ 3 and γ2ftd(H) = 2. If G has a γ-set S such that NG(S) ∩ S = ∅ and |NG(y) ∩ S| = 1 for all y ∈ V (G)\S, then γ2ftd(G[H]) ≤ 2γ(G). Proof. Let {a, b} be a γ2ftd-set of H and let Tx = {a, b} for each x ∈ S. Let C = ⋃ x∈S [{x} × Tx]. Since NG(S) ∩ S = ∅, S\NG(S) = S, |Tx| = 2 and Tx is a 2ftd-set of H for each x ∈ S, Conditions (i), (ii), and (iii) of Theorem 10 are sat- isfied. Also, since |NG(y) ∩ S| = 1 for each y ∈ V (G)\S, Condition (iv) of Theo- rem 10 is also satisfied. Thus, by Theorem 10, C is a 2ftd-set of G[H]. Therefore, γ2ftd(G[H]) ≤ |C| = ∑ x∈S |Tx| = 2|S| = 2γ(G). � Remark 4. The bound given in Corollary 5 is sharp. To see this, consider the graph P6[C3] shown in Figure 3. The shaded vertices in P6[C3] form a γ2ftd-set. Thus, γ2ftd(P6[C3]) = 4 = 2γ(P6). W. Bent-Usman, R. Isla / Eur. J. Pure Appl. Math, 14 (2) (2021), 578-589 587 Theorem 11. Let G and H be nontrivial connected graphs of orders m and n, respectively. Then C1 = S1 × V (H) and C2 = V (G)× S2 are kftd-sets in G�H if and only if S1 and S2 are kfd-sets in G and H, respectively. Proof. Suppose S1 is a kfd-set in G and C1 = S1 × V (H). Let (x, a) ∈ (G�H)\C1. Then x /∈ S1. Since S1 is a kfd-set in G, |NG(x) ∩ S1| = k. Since NG�H((x, a)) ∩ C = ⋃ y∈NG(x)∩S [{y} × {a}], it follows that |NG�H(x, a) ∩ C1| = |NG(x) ∩ S| = k, showing that C1 is a k-fair dominating set in G�H. Let (z, c) ∈ C1. Since H is a nontrivial connected graph, there exists d ∈ V (H) such that cd ∈ E(H). Thus, (z, d) ∈ C1 and (z, c)(z, d) ∈ E〈C1〉. Hence, C1 is a kftd-set in G�H. Similarly, C2 = V (G)× S2, where S2 is a kfd-set in H, is a kftd-set in G�H. For the converse, suppose that C1 = S1 × V (H) is a kftd-set in G�H. Suppose further that S1 is not a kfd-set in G. If S1 is not a dominating set in G, then there exists an x ∈ V (G)\S1 such that xy /∈ E(G) for every y ∈ S1. Let a ∈ V (H). Then (x, a) ∈ V (G�H)\C1 and (x, a)(y, a) /∈ E(G�H) for any (y, a) ∈ C1, contrary to the assumption that C1 is a kftd-set, hence a dominating set. Thus, S1 is a dominating set. If S1 is not a kfd-set, then there exists a u ∈ V (G)\S1 such that |NG(u) ∩ S1| = r 6= k. Let a ∈ V (H). Then (u, a) ∈ V (G�H)\C1 and |NG�H(u, a) ∩ C1| = r 6= k, contrary to the assumption that C1 is a kftd-set. Therefore, S1 is a kfd-set in G. Similarly, if C2 = V (G)× S2 is a kftd-set in G�H, then S2 is a kfd-set in H. � Corollary 6. Let G and H be nontrivial connected graphs of orders m and n, respectively, and k a positive integer with k ≤ min{m,n}. Then γkftd(G�H) ≤ min{m · γkfd(H), n · γkfd(G)}. Remark 5. The bound given in Corollary 6 is sharp. To see this, consider the graphs shown in Figure 4. The shaded vertices in each graph form a γkftd-set. Thus, γ1ftd(P4�C3) = 4 = min{4, 6} = {4 · 1, 3 · 2} = min{m · γ1fd(C3), n · γ1fd(P4)} = m · γ1fd(C3), and γ2ftd(P5�P3) = 9 = min{10, 9} = {5 · 2, 3 · 3} = min{m · γ2fd(P3), n · γ2fd(P5)} = n · γ2fd(P5). 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