EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1265-1279 ISSN 1307-5543 – ejpam.com Published by New York Business Global Outer-Connected Semitotal Domination in Graphs Alkajim Ahadi Aradais1,∗, Ferdinand P. Jamil2,3 1 Intergrated Laboratory School, College of Education, MSU Tawi-Tawi College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, 3 Center for Graph Theory, Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. In this paper, we introduce and initiate the study of outer-connected semitotal domi- nation in graphs. Given a graph G without isolated vertices, a set S of vertices of G is a semitotal dominating set if every vertex outside of S is adjacent to a vertex in S and every vertex in S is of distance at most 2 units from another vertex in S. A semitotal dominating set S of G is an outer-connected semitotal dominating set if either S = V (G) or S ̸= V (G) satisfying the prop- erty that the subgraph induced by V (G) \ S is connected. The smallest cardinality γ̃t2(G) of an outer-connected semitotal dominating set is the outer-connected semitotal domination number of G. First, we determine the specific values of γ̃t2(G) for some special graphs and characterize graphs G for specific (small) values of γ̃t2(G). Finally, we investigate the outer-connected semitotal dom- inating sets in the join, corona, and composition of graphs and, as a consequence, we determine their corresponding outer-connected semitotal domination numbers. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Semitotal dominating set, Semitotal domination number, Outer- connected dominating set, Outer-connected domination number 1. Introduction In 2014, the concept of semitotal domintion was introduced and investigated by W. Goddard, M. Henning, and C. McPillan (see [6]). Accordingly, it strengthens the concept of domination but relaxes the concept of total domination. Semitotal domination was further studied by M. Henning and A. Marcon (see [8]) in 2014 and 2016, by G. Hao and W. Zhuang (see [7]) in 2018, and by I. Aniversario et al. [1] in 2019. In this present paper, inspired by the work of J. Cyman [4], on outer-connected dom- ination, we introduce and initiate the study of the outer-connected semitotal domination in graphs. We investigate the concept in some special graphs and in graphs under some binary operations, such as the join, corona, and lexicographic products of graphs. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4461 Email addresses: alkajimaradais@msutawi-tawi.edu.ph (A. Aradais), ferdinand.jamil@g.msuiit.edu.ph (F. Jamil) https://www.ejpam.com 1265 © 2022 EJPAM All rights reserved. A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1266 2. Terminology and Notation The symbols V (G) and E(G) denote the vertex set and edge set, respectively, of a graph G. For S ⊆ V (G), |S| is the cardinality of S. In particular, |V (G)| and |E(G)| are the order and size, respectively, of G. The induced subgraph ⟨S⟩ is the graph with vertex set S and such that uv ∈ E(⟨S⟩) if and only if u, v ∈ S and uv ∈ E(G). All graph terminologies that are not introduced but are being used here are adopted from [2]. Given two graphs G and H with disjoint vertex sets, the join G+H of 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) and v ∈ V (H)}. The corona of G and H is the graph G ◦ H 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. The lexicographic product or composition 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]) 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). For vertex u of G, all vertices adjacent to u constitute the set NG(u) called the open neighborhood of u. The closed neighborhood of u in G is the set NG[u] = NG(u) ∪ {u}. If S ⊆ V (G), the open neighborhood of S in G is the set NG(S) = ∪u∈SNG(u). The closed neighborhood of S in G is the set NG[S] = NG(S)∪S. We define NG(S) = ∪v∈SNG(v) and NG[S] = S ∪NG(S). A set S ⊆ V (G) is a dominating set of G if NG[S] = V (G). Thus, S is a dominating set of G if and only if for each v ∈ V (G) \ S, there exists u ∈ S such that uv ∈ E(G). A set S ⊆ V (G) is a total dominating set of G if for every v ∈ V (G), there exists u ∈ S such that uv ∈ E(G). The minimum cardinality of a dominating set (resp. total dominating set) of G, denoted by γ(G) (resp. γt(G)), is the domination number (resp. total domination number) of G. A dominating set (resp. total dominating set) S of G with |S| = γ(G) (resp. |S| = γt(G)) is called a γ-set (resp. γt-set) of G. The authors always refer to [3] for the introduction and more comprehensive discussion of the development of the concept of domination in graphs. A set S ⊆ V (G) of a graph G = (V,E) is called an outer-connected dominating set of G if the following hold: (i) S is a dominating set of G, and (ii) either S = V (G) or the induced subgraph ⟨V (G) \ S⟩ of V (G) \ S is connected. The cardinality of a minimum outer-connected dominating set of G is called the outer-connected domination number of G, and is denoted by γ̃(G). For a graph G without isolated vertices, a set S ⊆ V (G) is a total outer-connected dominating set if S is a total dominating set of G and the subgraph induced by V (G) \ S is connected. The minimum cardinality of a total outer-connected dominating set in G is the total outer-connected domination number denoted by γ̃t(G). We refer to [4] and [5] for the introduction and results concerning outer-connected domination and total outer-connected domination, respectively, that are of interest in this study. Suppose that G has no isolated vertices. A set S ⊆ V (G) is a semitotal dominating set of G if S is a dominating set in G such that for every x ∈ S there exists y ∈ S \ {x} for which dG(x, y) ≤ 2. The smallest cardinality of a semitotal dominating set in G, denoted by γt2(G), is called a semitotal domination number of G. A semitotal dominating set of G with cardinality γt2(G) is called a γt2-set. Some results on semitotal domination in graphs A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1267 are found in [1, 6–8]. A semitotal dominating set S is an outer-connected semitotal dominating set of G if either S = V (G) or S ̸= V (G) and the induced subgraph ⟨V (G) \ S⟩ is connected. The smallest cardinality of an outer-connected semitotal dominating set in G, denoted by γ̃t2(G), is called the outer-connected semitotal domination number of G. An outer- connected semitotal dominating set in G with cardinality γ̃t2(G), is called a γ̃t2-set. For the purposes of this study, we write for v ∈ V (G), N2 G(v) = {u ∈ V (G) \ {v} : dG(u, v) ≤ 2}, and write for S ⊆ V (G), N2 G(S) = ∪v∈SN 2 G(v). 3. Results Observe that an outer-connected semitotal dominating set is both a semitotal dominat- ing set and an outer-connected dominating set. On the other hand, a total outer-connected dominating set is an outer-connected semitotal dominating set. Thus, max{2, γt2(G), γ̃(G)} ≤ γ̃t2(G) ≤ γ̃t(G). (1) Strict inequalities in Equation 1 can be attained for a graph. To see this, consider the graph G in Figure 1. It can be verified that {a, b, c, d}, {x, y, z, w}, {x, y, z, w, a, b} and {a, b, c, d, x, y, z} are γt2-set, γ̃-set, γ̃t2-set and γ̃t-set, respectively. Thus, γ̃(G) = 4 = γt2(G), γ̃t2(G) = 6 and γ̃t(G) = 7. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... ................................................................... ............ ........... ........... ........... ........... ........... ................................................................... ............ ........... ........... ........... ........... ........... ......... ........ ........ ........ ........ ... ............................................ ......... ........ ........ ........ ........ ... ............................................ a b c d x y z w Figure 1: Graph G with max{2, γt2(G), γ̃(G)} < γ̃t2(G) < γ̃t(G) Proposition 1. For path Pn on n ≥ 2 vertices γ̃t2(Pn) =  2, if n = 2 n− 1, if 3 ≤ n ≤ 5 n− 2, if n ≥ 6. Proof. The case where n = 2 is obvious. Assume n ≥ 3. Let S ⊆ V (Pn), be γ̃t2 - set of Pn with S ̸= V (Pn). Then P = ⟨V (Pn) \ S⟩ is a path. Suppose that [x, y, z] is a A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1268 geodesic in P . Then y /∈ NPn [S], a contradiction. Thus, |V (P )| = 1 or 2. Consequently, γ̃t2(Pn) = |S| ≥ n − 2. It is can readily be verified that if 3 ≤ n ≤ 5, |S| = n − 1. That is, γ̃t2(Pn) = n− 1. Suppose that n ≥ 6. Put Pn = [x1, x2, . . . , xn]. Since S = {x1, x2, x5, x6, . . . , xn} is an outer-connected semitotal dominating set of Pn, γ̃t2(Pn) ≤ |S| = n − 2. Therefore, γ̃t2(Pn) = n− 2. Proposition 2. For cycle Cn on n ≥ 3 vertices γ̃t2(Cn) = { 2, if n = 3 n− 2, if n ≥ 4. Proof. The case for n = 3 is trivial. Assume that n ≥ 4, and say C = [x1, x2, . . . , xn, x1]. Since {x3, x4, . . . , xn} is an outer-connected semitotal dominating set of Cn, γ̃t2(Cn) ≤ n − 2. Following similar arguments used above, if S ⊆ V (Cn) is a γ̃t2- set of Cn and P = ⟨V (Cn) \ S⟩, then P is a path with 1 ≤ |V (P )| ≤ 2. Consequently, γ̃t2(Cn) = |S| ≥ n− 2. Proposition 3. For complete multipartite graph Kn1,n2,...,nt of order n = n1+n2+ ...+nt, where 1 ≤ n1 ≤ n2 ≤ · · · ≤ nt and t ≥ 2, γ̃t2(Kn1,n2,...,nt) = { n2, if t = 2 and n1 = 1, n2 ≥ 2 2, else. Proof. Put G = Kn1,n2,...,nt , and let U1, U2, . . ., Ut be the partite sets of G. First, observe that if nt = 1, then G = Kt and γ̃t2(G) = 2. Assume that nt ≥ 2. We consider the following cases: Case 1: Suppose that n1 = 1. We consider the following subcases: Subcase 1.1: If t = 2 and n2 ≥ 2, then G = K1,n2 , a star of order n ≥ 3. If S ⊆ V (G) is an outer-connected semitotal dominating set of G, then either |S| = n − 1 = n2 or |S| = n. Thus, γ̃t2(G) = n2. Subcase 1.2: Suppose that t ≥ 3 and n1 = n2 = 1 such that G is not a path. Pick u ∈ U2 and v ∈ U3. Then S = {u, v} is an outer-connected semitotal dominating set of G. Thus, γ̃t2(G) = 2. Subcase 1.3: Suppose that t ≥ 3 and n2 ≥ 2. Pick u ∈ U1 and v ∈ U2. Then S = {u, v} is an outer-connected semitotal dominating set of G. Thus, γ̃t2(G) = 2. Case 2: Suppose that t ≥ 2 and nk ≥ 2 for all k ∈ {1, 2, . . . , t}. Pick u ∈ U1 and v ∈ U2. Then S = {u, v} is an outer-connected semitotal dominating set of G. Proposition 4. Let G be a connected graph of order n ≥ 2. Then (i) γ̃t2(G) = 2 if and only if G can be obtained from a connected graph H of order n− 2 by adding to H vertices u and v such that dG(u, v) = 1 or 2 and {u, v} dominates V (H). A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1269 (ii) γ̃t2(G) = n if and only if G = K2. Proof. Statement (i) is clear. By Proposition 3, if n = 2, then γ̃t2(Kn) = 2 = n. Assume that γ̃t2(G) = n. Suppose that n ≥ 3. By Proposition 3, G ̸= Kn. Let [u, v, z] be a geodesic in G. Put S = V (G) \ {v}. Then S is a semitotal dominating set with V (G) \ S = {v}. Thus S is an outer-connected semitotal dominating set of G. Consequently, γ̃t2(G) ≤ n− 1, a contradiction. Therefore, n = 2. This proves (ii). Proposition 5. Let G be any graph with nontrivial components C1, C2, . . ., Ck of orders n1, n2, . . ., nk, respectively. Then γ̃t2(G) = min{γ̃t2(Cj) + k∑ i=1,i ̸=j ni : j = 1, 2, . . . , k}. Proof. Put α = min{γ̃t2(Cj) + ∑k i=1,i ̸=j ni : j = 1, 2, . . . , k}. Let j ∈ {1, 2, . . . , k}, and choose a γ̃t2-set Sj of Cj . Since S = ( ∪k i=1;i ̸=jV (Ci) ) ∪ Sj is an outer-connected semitotal dominating set of G, γ̃t2(G) ≤ |S| = γ̃t2(Cj) + ∑k i=1,i ̸=j ni. Since j is arbitrary, γ̃t2(G) ≤ α. Let S ⊆ V (G) be an outer-connected semitotal dominating set of G. Since S is a semitotal dominating set of G, Sj = S ∩ V (Cj) is a semitotal dominating set of Cj for all j = 1, 2, . . . , k. First, we claim that there exists j ∈ {1, 2, . . . , k} for which Si = V (Ci) for all i except possibly when i = j. Suppose that, to the contrary, there exist distinct i, j ∈ {1, 2, . . . , k} such that Si ̸= V (Ci) and Sj ̸= V (Cj). Pick u ∈ V (Ci) \ Si and v ∈ V (Cj) \ Sj . Observe that ⟨V (G) \ S⟩ does not have a path joining u and v, a contradiction and thus, the claim is established. This means that, S = Sj ∪( ∪k i=1,i ̸=jV (Ci) ) for some j. Next, we claim that Sj is an outer-connected semitotal dominating set of Cj . If S = V (G), then Sj = V (Cj) and we are done. Suppose that S ̸= V (G). Since Sj is a semitotal dominating set, it is left to verify that ⟨V (Cj) \ Sj⟩ is connected. But since V (Cj) \ Sj = V (G) \ S, the conclusion follows. Thus, |S| = |Sj |+ k∑ i=1,i ̸=j ni ≥ γ̃t2(Cj) + k∑ i=1,i ̸=j ni ≥ α. Since S is arbitrary, γ̃t2(G) ≥ α. Proposition 6. Let G be a nontrivial graph. (i) If G is connected, then γ̃t2(G+K1) = 2. (ii) If G is disconnected with components C1, C2, . . ., Ck of orders n1, n2, . . ., nk, A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1270 respectively, satisfying that n1 ≤ n2 ≤ · · · ≤ nk, then γ̃t2(G+K1) = min{ k∑ j=1 γ(Cj), 2 + k−1∑ j=1 nj}. Proof. Put V (K1) = {u}. To prove (i), suppose that G is connected. Since G is nontrivial, G contains at least two vertices which are not cutvertices. Pick a non-cutvertex v of G. Then S = {u, v} is an outer-connected semitotal dominating set of G +K1. By Equation 1, γ̃t2(G+K1) = 2. To prove (ii), suppose that G is disconnected with components C1, C2, . . ., Ck of orders n1, n2, . . ., nk, respectively, and satisfying that n1 ≤ n2 ≤ · · · ≤ nk. If nk = 1, then G+K1 is a star, and the result follows from (i). It is worth noting that in this case, k∑ j=1 γ(Cj) = k = |V (G)|. Assume nk ≥ 2. First, let Sj ⊆ V (Cj) be a γ-set of Cj for all j = 1, 2, . . . , k. Then S = ∪k j=1Sj is a semitotal dominating set of G +K1. Since u ∈ V (G +K1) \ S, S is an outer-connected semitotal dominating set of G+K1. Thus, γ̃t2(G+K1) ≤ |S| = k∑ j=1 γ(Cj). Next, let S = ( ∪k−1 j=1V (Cj) ) ∪ {u, v}, where v ∈ V (Ck) which is a non-cutvertex of Ck. Then S is an outer-connected dominating set of G+K1. This means that γ̃t2(G+K1) ≤ 2 + k−1∑ j=1 nj . Thus, γ̃t2(G+K1) ≤ min{ k∑ j=1 γ(Cj), 2 + k−1∑ j=1 nj}. Now, to get the other inequality, let S ⊆ V (G + K1) be a γ̃t2-set of G + K1. Then Sj = S ∩ V (Cj) is a dominating set of Cj for all j ∈ {1, 2, . . . , k}. If u /∈ S, then γ̃t2(G+K1) = |S| = k∑ j=1 |Sj | ≥ k∑ j=1 γ(Cj). Suppose that u ∈ S. Since ⟨V (G+K1) \ S⟩ is connected, V (G+K1) \ S = V (Cj) \ Sj for A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1271 some j. Moreover, since S is a γ̃t2-set of G, j = k. Thus, S = ( ∪k−1 i=1 V (Ci) ) ∪ Sk ∪ {u} so that γ̃t2(G+K1) = |S| = 1 + |Sk|+ k−1∑ j=1 nj ≥ 2 + k−1∑ j=1 nj . Therefore, γ̃t2(G+K1) ≥ min{ k∑ j=1 γ(Cj), 2 + k−1∑ j=1 nj}. Suppose that Cj = K1 for all j ∈ {1, 2, . . . , k− 1} in Proposition 3. If γ(Ck) = 1, then k∑ j=1 γ(Cj) < 2 + k−1∑ j=1 nj . On the other hand, if γ(Ck) ≥ 3, then k∑ j=1 γ(Cj) > 2 + k−1∑ j=1 nj , and attain equality if γ(Ck) = 2. Theorem 1. [1] Let G and H be nontrivial graphs, and S ⊆ V (G + H). Then S is a semitotal dominating set in G+H if and only if one of the following holds: (i) S ⊆ V (G) is a nonsingleton dominating set in G; (ii) S ⊆ V (H) is a nonsingleton dominating set in H; (iii) S ∩ V (G) ̸= ∅ and S ∩V(G) ̸= ∅. Theorem 2. Let G and H be any nontrivial graphs, and S ⊆ V (G + H), then S is an outer-connected semitotal dominating set in G+H if and only if one of the following holds: (i) S ⊆ V (G) and one of the following holds: (a) S = V (G) and H is connected; (b) S ̸= V (G) and S is a nonsingleton dominating set in G. (ii) S ⊆ V (H) and one of the following holds: (a) S = V (H) and G is connected; A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1272 (b) S ̸= V (H) and S is a nonsingleton dominating set in H. (iii) S ∩ V (G) ̸= ∅ and S ∩ V (H) ̸= ∅ such that if S ̸= V (G + H), then one of the following holds: (a) V (G) ⊆ S and ⟨V (H) \ S⟩ is connected; (b) V (H) ⊆ S and ⟨V (G) \ S is a connected; (c) V (G) \ S ̸= ∅ and V (H) \ S ̸= ∅. Proof. Assume that S is an outer-connected semitotal dominating set of G + H. Suppose that S ⊆ V (G). If S = V (G), then H = ⟨V (G+H) \ S⟩ is connected, and (i)(a) holds. Suppose that S ̸= V (G). Since S is a semitotal dominating set of G + H, S is a nonsingleton dominating set of G, and (i)(b) holds. Similarly, if S ⊆ V (H), then (ii) holds. Now, assume that SG = S ∩ V (G) ̸= ∅ and SH = S ∩ V (H) ̸= ∅. Suppose further that S ̸= V (G + H). Statement (iii)(a) follows from the fact that if V (G) ⊆ S, then ⟨V (G+H) \ S⟩ = ⟨V (H) \ SH⟩ is connected. Similarly, if V (H) ⊆ S, then (iii)(b) holds. If both (iii)(a) and (iii)(b) do not hold, then necessarily, (iii)(c) holds. Conversely, suppose that S ̸= V (G+H) satisfying condition (i). Then S ⊆ V (G) and is a nonsingleton dominating set of G. By Theorem 1, S is semitotal dominating set of G +H. If S = V (G), then ⟨V (G +H) \ S⟩ = H, which by (i)(a) is connected. Suppose that S ̸= V (G). Then ⟨V (G+H) \ S⟩ = ⟨(V (G) \ S) ∪ V (H)⟩ is clearly connected. This makes S an outer-connected semitotal dominating set of G +H. Similarly, if (ii) holds, then S is an outer-connected semitotal dominating set of G + H. Finally, suppose that (iii) holds. By Theorem 1, S is a semitotal dominating set of G + H. If V (G) ⊆ S, then ⟨V (G + H) \ S⟩ = ⟨V (H) \ S⟩, which is connected by (iii)(a). Similarly, if (iii)(b) holds, then S is an outer-connected semitotal dominating set of G+H. Following similar arguments as above, if (iii)(c) holds then S is an outer-connected semitotal dominating set of G+H. Corollary 1. For all nontrivial graphs G and H, γ̃t2(G+H) = 2. Proof. Pick u ∈ V (G) and v ∈ V (H). By Theorem 2, S = {u, v} is an outer- connected semitotal dominating set of G + H. Thus, γ̃t2(G + H) ≤ 2. Finally, by (1), γ̃t2(G+H) = 2. Theorem 3. Let G be a nontrivial connected graph and S ⊆ V (G ◦ K1). Then S is an outer-connected semitotal dominating set of G◦K1 if and only if one of the following holds for S: (i) S = V (G ◦K1) \ V (Kv 1 ) for some v ∈ V (G); (ii) S = A∪ ( ∪v∈V (G)V (Kv 1 ) ) , where A ⊆ V (G) is an outer-connected dominating set of G. A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1273 Proof. Put V (K1) = {x}. Then V (Kv 1 ) = {xv}. Assume that S is an outer-connected semitotal dominating set of G ◦K1. We consider two cases: Case 1: Suppose that xv /∈ S for some v ∈ V (G). Since S is a dominating set of G ◦K1, v ∈ S. Since xv ∈ V (G ◦K1) \S and ⟨V (G ◦K1) \S⟩ is connected, V (G ◦K1) \S = {xv}. That is, S = V (G ◦K1) \ {xv}. In this case, (i) holds. Case 2: Suppose that xv ∈ S for all v ∈ V (G). Define A = S ∩ V (G). Then S = A ∪ ( ∪v∈V (G){xv} ) . We claim that A is an outer-connected dominating set of G. First, let v ∈ V (G)\A. Then xv ∈ S. Since S is a semitotal dominating set of G ◦ K1, there exists u ∈ S for which dG◦K1(x v, u) ≤ 2. Because v /∈ S, u ∈ A ∩NG(v). Since v is arbitrary, A is a dominating set of G. Note further that , V (G) \ A = V (G ◦K1) \ S. Thus, A is an outer-connected dominating set of G. In this case, (ii) holds. Conversely, obviously, if condition (i) holds for S, then S is an outer-connected semi- total dominating set of G ◦ K1. Now, suppose that condition (ii) holds for S. Since ∪v∈V (G){xv} is a dominating set of G ◦K1, S is a dominating set of G ◦K1. Let u ∈ S. We consider the following cases: Case 1: Suppose that u = xv for some v ∈ V (G). If v ∈ S, then we pick v for dG(u, v) ≤ 2. Suppose that v /∈ S. Since A is a dominating set of G and v ∈ V (G) \ A, there exists w ∈ A ⊆ S such that wv ∈ E(G) ⊆ E(G ◦K1). Since dG◦K1(u,w) = 2, w is the desired vertex. Case 2: Suppose that u ∈ V (G). In this case, we pick xu ∈ S. Note that uxu ∈ E(G◦K1). The above cases show that S is a semitotal dominating set of G ◦ K1. Finally, since A is an outer-connected dominating set of G, ⟨V (G) \ A⟩ = ⟨V (G ◦ K1) \ S⟩ is connected. Therefore, S is an outer-connected semitotal dominating set of G ◦K1. Corollary 2. For nontrivial connected graph G of order n, γ̃t2(G ◦K1) = n+ γ̃(G). Proof. In view of Theorem 3, γ̃t2(G ◦K1) = min{2n− 1, n+ γ̃(G)} = n+ γ̃(G). Theorem 4. Let G and H be nontrivial connected graphs, and let S ⊆ V (G◦H). Then S is an outer-connected semitotal dominating set of G ◦H if and only if one of the following holds for S: A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1274 (i) There exists v ∈ V (G) and B ⊆ V (Hv) such that S = (V (G ◦H) \ V (Hv)) ∪B, where either B = V (Hv) or ⟨V (Hv) \B⟩ is connected. (ii) S = A ∪ (∪x∈AV (Hx)) ∪ ( ∪x∈V (G)\ASx ) , (2) where A ⊆ V (G) and Sx ⊆ V (Hx) for all x ∈ V (G) \A satisfying the following: (a) ⟨V (G) \A⟩ is connected; (b) For each x ∈ V (G) \ A, Sx is a dominating set of Hx. Moreover, if |Sx| = 1, then A ∩NG(x) ̸= ∅. Proof. Assume that S is an outer-connected semitotal dominating set of G ◦ H. If S = V (G◦H), then (i) holds. In what follows, we assume that S ̸= V (G◦H). We consider two cases: Case 1: Suppose that V (G) ⊆ S. Since ⟨V (G◦H)\S⟩ is connected, there exists v ∈ V (G) and B ⊆ V (Hv) such that V (G ◦H) \ S = V (Hv) \B. That is, S = (V (G ◦H) \ V (Hv)) ∪B and ⟨V (Hv) \B⟩ is connected. Case 2: Suppose that V (G) ⊈ S. Put A = S ∩ V (G). If A = ∅, then (2) trivially holds with Sx = S ∩ V (Hx) for all x ∈ V (G). Suppose that A ̸= ∅. Since ⟨V (G ◦ H) \ S⟩ is connected and V (G) \ A ̸= ∅, V (Hx) ⊆ S for all x ∈ A. Put Sx = S ∩ V (Hx) for all x ∈ V (G)\A. Then Equation (2) holds for S. Statement (ii)(a) follows immediately from the connectedness of ⟨V (G ◦H) \S⟩. Now, let x ∈ V (G) \A and u ∈ V (Hx) \Sx. Since S is a dominating set of G◦H and u /∈ S, there exists w ∈ S for which uw ∈ E(G◦H). Since x /∈ S, w ̸= x so that w ∈ Sx. This means that Sx is a dominating set of Hx. Suppose further that |Sx| = 1, say Sx = {u}. Because S is a semitotal dominating set of G ◦ H, there exists w ∈ S \ {u} such that dG◦H(u,w) ≤ 2. Since w /∈ Sx, w ∈ A ∩ NG(x), and Statement (ii)(b) holds. Conversely, suppose that condition (i) holds. Since V (G) ⊆ S, S is a semitotal domi- nating set of G◦H. Moreover, V (G◦H) \ S = V (Hv) \ B so that S is an outer-connected semitotal dominating set of G ◦H. Now, suppose that condition (ii) holds for S. By con- dition (ii)(b), S is a dominating set of G ◦H. Suppose that A = ∅. Then S = ⋃ x∈V (G) Sx, and by condition (ii)(b), Sx is a nonsingleton dominating set of Hx for all x ∈ V (G). Note that for each x ∈ V (G), dG◦H(u, v) ≤ 2 for all u, v ∈ Sx. It follows that S is a semitotal dominating set of G◦H. Further, since V (G) ⊆ V (G◦H)\S, ⟨V (G◦H)\S⟩ is connected. Finally, suppose that A ̸= ∅. Let x ∈ S. If x ∈ A, then V (Hx) ⊆ S. Pick u ∈ V (Hx). A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1275 Then we have u ∈ S and dG◦H(x, u) = 1. If x ∈ V (Hv) for some v ∈ A, then v is the desired vertex in S for which dG◦H(x, v) ≤ 2. Next, suppose that x ∈ V (Hv) for some v ∈ V (G) \A. If |Sv| ≥ 2, then pick u ∈ Sv \ {x}. Then u ∈ S and dG◦H(x, u) ≤ 2. Lastly, suppose that |Sv| = 1, i.e., Sv = {x}. By condition (ii)(b), the exists z ∈ A∩NG(v). Then z ∈ S and dG◦H(x, z) = 2. We have shown that S is a semitotal dominating set of G ◦H. Condition (ii)(a) implies further that S is an outer-connected semtiotal dominating set of G ◦H. Corollary 3. Let G and H be nontrivial connected graphs of orders n and m, respectively. (i) If γ(H) = 1, then γ̃t2(G ◦H) ≤ min{2n, n+mγ̃(G)}. (ii) If γ(H) ≥ 2, then γ̃t2(G ◦H) ≤ min{nγ(H), γ̃(G) (1 +m− γ(H)) + nγ(H)} Proof. Let A ⊆ V (G) be a γ̃-set of G. For each v /∈ A, let Sv ⊆ V (H) be a γ-set of Hv. Define S = A ∪ (⋃ v∈A V (Hv) ) ∪  ⋃ v∈V (G)\A Sv  . Since S satisfies Theorem 4(ii), S is an outer-connected semitotal dominating set of G◦H. Thus, γ̃t2(G ◦H) ≤ |S| = |A|+m|A|+ (n− |A|) γ(H) = γ̃(G) +mγ̃(G) + (n− γ̃(G))γ(H) = γ̃(G) (1 +m− γ(H)) + nγ(H). In particular, if γ(H) = 1, then γ̃t2(G ◦H) ≤ n+mγ̃(G). To complete the desired results, for the case where γ(H) = 1, let y ∈ V (H) for which NH [y] = V (H). Pick a z ∈ V (H) \ {y} and define Sv = {z, y} for all v ∈ V (G). On the other hand, if γ(H) ≥ 2, then choose Sv ⊆ V (Hv) to be a γ-set of Hv for all v ∈ V (G). In any case, S = ⋃ v∈V (G) Sv satisfies Theorem 4(ii). Thus, S is an outer-connected semitotal dominating set of G ◦H. This means that if γ(H) = 1, then γ̃t2(G ◦H) ≤ |S| = 2n. For γ(H) ≥ 2, γ̃t2(G ◦H) ≤ |S| = nγ(H). Corollary 4. Let G and H be nontrivial connected graphs of orders n and m, respectively. A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1276 (i) If γ(H) = 1, then γ̃t2 ((G+K1) ◦H) = min{2n+ 2, n+m+ 1}. (ii) If γ(H) ≥ 2, then γ̃t2 ((G+K1) ◦H) = min{(n+ 1)γ(H), γ̃(G) (1 +m− γ(H)) + nγ(H)} Proof. Put K = (G+K1) ◦H, and α = min{2n+2, n+m+1}. By Corollary 3, with γ̃(G+K1) = 1, we have γ̃t2(K) ≤ α. Now, let S ⊆ V (K) be an outer-connected semitotal dominating set of K. Since S is a dominating set of K, S ∩ V (Hx + x) ̸= ∅ for all x ∈ V (G + K1). First, suppose that S ∩ V (G +K1) = ∅. By Theorem 4(ii), S ∩ V (Hx) is a nonsingleton dominating set of Hx + x for all x ∈ V (G + K1). This means that |S| ≥ 2(n + 1) = 2n + 2 ≥ α. Next, suppose that S ∩ V (G+K1) ̸= ∅. Clearly, if V (G+K1) ⊆ S, then |S| ≥ 2n+ 2. Assume that V (G+K1)\S ̸= ∅. Let w ∈ S∩V (G+K1). Since S is an outer-connected semitotal dominating set of K and V (G+K1) \ S ⊆ V (K) \ S, V (Hw) ⊆ S. This means that |S| ≥ |V (Hw + w)|+ ∑ x∈V (G+K1)\{w} |S ∩ V (Hx + x)| ≥ m+ 1 + n ≥ α. Since S is arbitrary, γ̃t2(K) ≥ α. Similar arguments will prove (ii). It is worth noting that the wheel graphs and the fan graphs are among the graphs represented by G+K1 in Corollary 4. Theorem 5. [1] Let G and H be nontrivial connected graphs, and let C = ∪x∈S({x} × Tx) ⊆ V (G[H]). Then C is a semitotal dominating set in G[H] if and only if one of the following holds: (i) S is a total dominating set in G; (ii) S is a semitotal dominating set in G and for each x ∈ S \NG(S), Tx is a dominating set in H; (iii) S is a dominating set in G, such that, Tx is a dominating set in H for each x ∈ S \NG(S), and |Tx| ≥ 2 for each x ∈ S \N2 G(S). For C ⊆ V (G[H]), define CG = {x ∈ V (G) : (x, y) /∈ C for some y ∈ V (H)}. Theorem 6. Let G be a nontrivial connected graph and n ≥ 2, and C = ∪x∈S ({x} × Tx) ̸= V (G[Kn]). Then C is an outer-connected semitotal dominating set of G[Kn] if and only if each of the following holds: (i) One of the following holds: (a) S is a semitotal dominating set in G. A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1277 (b) S is a dominating set in G such that |Tx| ≥ 2 for each x ∈ S \ N2 G(S). (ii) Exactly one of the following holds: (a) CG = {x} for some x ∈ V (G). (b) |CG| ≥ 2 and for each distinct u, v ∈ CG, G has a u-v geodesic P for which either S ∩ V (P ) = ∅ or |Tx| < n for each x ∈ S ∩ V (P ). Proof. Assume that C is an outer-connected semitotal dominating set of G[Kn]. Since C is a semitotal dominating set of G[Kn] and every nonempty subset of V (Kn) is a dominating set of Kn, (i) holds by Theorem 5. To show (ii), if |CG| = 1, then (ii)(a) holds. Suppose that |CG| ≥ 2, and let u, v ∈ CG with u ̸= v. Pick z, w ∈ V (Kn) such that (u, z), (v, w) /∈ C. Since ⟨V (G[Kn]) \ C⟩ is connected, there exists a (u, z)-(v, w) geodesic [(u, z) = (x1, y1), (x2, y2), . . . , (xn, yn) = (v, w)] in G[Kn] such that (xk, yk) /∈ C for all k = 1, 2, . . . , n. It implies that for some k1 < k2 < · · · < kr in {1, 2, . . . , n}, P = [u = xk−1, xk2 , . . . , xkr = v] is a u-v geodesic in G. If V (P ) ∩ S = ∅, then we are done. Suppose that V (P )∩S ̸= ∅, and let xkj ∈ S ∩V (P ). Necessarily, ykj /∈ Txkj . Thus, Txkj ̸= V (Kn). This shows that (ii)(b) holds. Conversely, condition (i) implies that C is a semitotal dominating set of G[Kn] by Theorem 5. If condition (ii)(a) holds, then V (G[Kn]) \ C = { {x} × V (Kn), if x /∈ S {x} × (V (Kn) \ Tx) , if x ∈ S, and C is an outer-connected semitotal dominating set of G[H]. Now, suppose that condi- tion (ii)(b) holds. Let (u, z), (v, w) ∈ V (G[Kn]) \ C be distinct. Case 1: u ̸= v Since u, v ∈ CG, G has a u-v geodesic P = [u = x1, x2, . . . , xn = v] as being described in (ii)(b). If S∩V (P ) = ∅, then for any y ∈ V (Kn), [(u, z), (x2, y), (x3, y), . . . , (xn−1, y), (v, w)] is a (u, z)-(v, w) path in ⟨V (G[Kn]) \ C⟩. Suppose that S ∩ V (P ) ̸= ∅. Put y1 = z and w = yn. For each k ∈ {2, . . . , n − 1}, pick any yk ∈ V (Kn) whenever xk /∈ S; otherwise, pick yk ∈ V (Kn) \ Txk . Then [(u, z) = (x1, y1), (x2, y2), . . . , (xn−1, yn−1), (xn, yn) = (v, w)] is a (u, z)-(v, w) path in ⟨V (G[Kn]) \ C⟩. Case 2: u = v Pick x ∈ CG \ {u}. Let P = [x = x1, x2, . . . , xn−1, xn = u] be a x-u geodesic in G as described in condition (ii)(b). In particular, xn−1 ∈ CG. Pick y ∈ V (Kn) such that (xn−1, y) /∈ C. Then [(u, z), (xn−1, y), (v, w)] is a (u, z)-(v, w) path in ⟨V (G[Kn]) \ C⟩. The above cases imply that ⟨V (G[Kn]) \ C⟩ is connected. Therefore, C is an outer- connected semitotal dominating set of G[Kn]. Now, we provide proof for the following lemma, which is very useful to get the desired result in this section. The lemma is given without proof in [1]. A. Aradais, F. Jamil / Eur. J. Pure Appl. Math, 15 (3) (2022), 1265-1279 1278 Lemma 1. [1] If G is a nontrival connected graph and S ⊆ V (G) is a dominating set in G, then γt2(G) ≤ 2|S \N2 G(S)|+ |S ∩N2 G(S)|. Proof. Let S ⊆ V (G) be a dominating set of G. For each x ∈ S \ N2 G(S), pick ux ∈ V (G) such that xux ∈ E(G). Then S∗ = S ∪ {ux : x ∈ S \ N2 G(G)} is a semitotal dominating set of G. Thus, γt2(G) ≤ |S∗| = |S ∩N2 G(S)|+ 2|S \N2 G(S)|. Corollary 5. Let G be a nontrivial connected graph and n ≥ 2. Then γ̃t2(G[Kn]) = γt2(G). Proof. Let S ⊆ V (G) be a γt2-set of G. Choose v ∈ V (Kn) and define C = S × {v}. Since conditions (i)(a) and (ii)(b) of Theorem 6 hold for C, C is an outer-connected semitotal dominating set of G[Kn]. Consequently, γ̃t2(G[Kn]) ≤ |S| = γt2(G). Let C = ⋃ x∈S ({x} × Tx) ⊆ V (G[Kn]) be an outer-connected semitotal dominating set of G[Kn]. By Theorem 6, S is a dominating set of G. If S is a semitotal dominating set of G, then γt2(G) ≤ |S| ≤ ∑ x∈S |Tx| = |C|. Suppose that S is not a semitotal dominating set in G. Let S1 = S \ N2 G(S) and S2 = S ∩N2 G(S). By Theorem 6, C =  ⋃ x∈S1 ({x} × Tx)  ∪  ⋃ x∈S2 ({x} × Tx)  , where |Tx| ≥ 2 for all x ∈ S1. Thus, |C| = ∑ x∈S1 |Tx|+ ∑ x∈S2 |Tx| ≥ 2|S1|+ |S2| = 2|S \N2 G(S)|+ |S ∩N2 G(S)|. By Lemma 1, γt2(G) ≤ |C|. Since C is arbitrary, γt2(G) ≤ γt2(G[Kn]). REFERENCES 1279 Acknowledgements The authors would like to thank the referees for the invaluable assistance they gave us through their comments and suggestions which led to the improvement of the pa- per. Also, the authors would like to thank the Department of Science and Technology - Accelerated Science and Technology Human Resource Development Program (DOST- ASTHRDP)-Philippines, and MSU-Iligan Institute of Technology for funding this research. Furthermore, the researchers are grateful to MSU-IIT and MSU-TCTO for the support in doing this research. References [1] I. Aniversario, Jr. S. Canoy, and F.P Jamil. Semitotal domination in graphs. European Journal of Pure and Applied Mathematics., 12(4):1410–1425, 2019. [2] F. Buckley and F. Harary. Distance in graphs. Redwood City, CA: Addison-Wesley., 1990. [3] E.J. Cockayne and S.T. Hedetniemi. Towards a theory of domination in graphs. Net- works., 7:247–261, 1997. [4] J. Cyman. The outer-connected domination number of a graph. Australas J. Combin, 38:35–46, 2007. [5] J. Cyman and J. Raczek. Total outer-connected domination numbers of trees. Discrete Applied Mathematics., 157:3198–3202, 2009. [6] W. Goddard, M. Henning, and C. McPil. Semitotal domination in graphs. Utilitas Mathematica., 94:67–81, 2004. [7] G. Hao and W. Zhuang. Semitotal domination in trees. Discrete Mathematics and Theoretical Computer Science., 20(2):1–11, 2018. [8] H. Henning and A. Marcon. Semitotal domination in claw-free cubic graphs. Annals of Combinatorics., 20(4):799–813, 2016.