EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 736-752 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Weakly Connected Closed Geodetic Domination in Graphs Under Some Binary Operations Jamil J. Hamja1,∗, Imelda S. Aniversario2, Helen M. Rara2 1 Office of the Vice Chancellor for Academic Affairs, MSU-Tawi-Tawi College of Technology and Oceanography, 7500 Tawi-Tawi, Philippines 2 Department of Mathematics and Statistics, College of Science in Mathematics, MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a simple connected graph. For S ⊆ V (G), the weakly connected closed geodetic dominating set S of G is a geodetic closure IG[S] which is between S and is the set of all vertices on geodesics (shortest path) between two vertices of S. We select vertices of G sequentially as follows: Select a vertex v1 and let S1 = {v1}. Select a vertex v2 ̸= v1 and let S2 = {v1, v2}. Then successively select vertex vi /∈ IG[Si−1] and let Si = {v1, v2, ..., vi} for i = 1, 2, ..., k until we select a vertex vk in the given manner that yields IG[Sk] = V (G). Also, the subgraph weakly induced ⟨S⟩w by S is connected where ⟨S⟩w = ⟨N [S], Ew⟩ with Ew = {u, v ∈ E(G) : u ∈ S or v ∈ S} and S is a dominating set of G. The minimum cardinality of weakly connected closed geodetic dominating set of G is denoted by γwcg(G). In this paper, the authors show and investigate the concept weakly connected closed geodetic dominating sets of some graphs and the join, corona, and Cartesian product of two graphs are characterized. The weakly connected closed geodetic domination numbers of these graphs are determined. Also, some relationships between weakly connected closed geodetic dominating set, weakly connected closed geodetic set, geodetic dominating set, and geodetic connected dominating set are established. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: weakly connected closed geodetic dominating set, and weakly con- nected closed geodetic domination number. 1. Introduction In this paper we explore a parameter that is, defined in the same manner that the well- known weakly connected closed geodetic number of a graph G is. Indeed, while a weakly connected closed geodetic set of a graph G necessitates a geodetic closure IG[S] which is between S and is the set of all vertices on geodesics (shortest path) between two vertices of S and the subgraph weakly induced ⟨S⟩w by S is connected where ⟨S⟩w = ⟨N [S], Ew⟩ with ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4356 Email addresses: jamil.hamja@g.msuiit.edu.ph (J. Hamja), imelda.aniversario@g.msuiit.edu.ph (I. Aniversario), helen.rara@g.msuiit.edu.ph (H. Rara) https://www.ejpam.com 736 © 2022 EJPAM All rights reserved. J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 737 Ew = {u, v ∈ E(G) : u ∈ S or v ∈ S}. The motivation of introducing the concept is to give a further investgation on weakly connected domination, closed geodetic domination and some of its variations. In fact, it can be shown that every weakly connected closed geodetic dominating set is a weakly connected closed geodetic set of a graph G. Thereupon, the weakly connected closed geodetic number of a graph G is at most equal to the weakly connected closed geodetic domination number of a graph G. The concept of weakly connected closed geodetic nmbers was introduce and studied by Patangan, et. al [12]. Some concept and its number are also introduced by Aniversario, et.al [1], Chellathurai, et. al [6], Dunbar, et.al [7], Jamil, et.al [11], and Sandueta, et.al [13]. Furthermore, the weakly connected closed geodetic number of a graph may be used to give bounds on some weakly connected closed geodetic domination related parameters. Moreover, this newly concept may be applied to introduce some concepts (say, a variant of weakly connected closed geodetic domination) in the future. 2. Terminology and Notation A set is a dominating set of G if NG[S] = V (G). The domination number of G, denoted by γ(G), is the minimum cardinality among the dominating sets of G. A dominating set S with |S| = γ(G) is said to be γ-set of G. A connected dominating set S of a graph G is a dominating set such that the subgraph ⟨S⟩ induced by S in G is connected. The minimum cardinality of a connected dominating set of G is called the connected domination number of G, denoted by γc(G). A connected dominating set S with |S| = γc(G) is called γc-set of G Tarr, et.al [14], and Duckworth, et.al [8]. Let S ⊆ V (G). The subgraph weakly induced by S is the graph ⟨S⟩w = (NG[S], Ew), where Ew = {uv ∈ E(G) : u ∈ S or v ∈ S}. The symbol Ew(S) means Ew, Patangan, et.al [12]. A dominating set S ⊆ V (G) is a weakly connected dominating set in G if the subgraph ⟨S⟩w weakly induced by S is connected. The weakly connected domination number γw(G) of G is the minimum cardinality among all weakly connected dominating sets of G. A weakly connected dominating set S with |S| = γw(G) is said to be γw-set of G, Sandueta, et.al [13]. Let u, v ∈ V (G). A shortest path from u to v in G is called a u-v geodesic of G. The set IG[u, v] consists of u, v, and all vertices lying in some u-v geodesic of G. For a nonempty subset S of V (G), IG[S] = ⋃ u,v∈S I[u, v], Chartrand, et.al [4]. Let G be a connected graph, then set S ⊆ V (G) is a geodetic set of G if IG[S] = V (G). The set IG[S] is called the geodetic closure of G. The minimum cardinality of a geodetic set is the geodetic number of G, and is denoted by g(G). The geodetic number of a disconnected graph is the sum of the geodetic numbers of its components. A geodetic set of cardinality g(G) is called a g-set. Henceforth, the set IG(u, v) denotes the set IG[u, v] \ {u, v}. A set S ⊆ V (G) is called a geodetic dominating set of G if S is both a geodetic set and a dominating set. The minimum cardinality of a geodetic dominating set of G is the geodetic domination number of G, and is denoted by γg(G). A geodetic dominating set S with |S| = γg(G) is said to be a γg-set G.A set of vertices in S in a graph G is said to be geodetic connected dominating set J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 738 of G if S is both a geodetic set and connected dominating set. The minimum cardinality of a geodetic connected dominating set of G is called a geodetic connected domination number of G, denoted by γgc(G). A geodetic connected dominating set S with |S| = γgc(G) is said to be a γgc-set of G. The geoodesic set, geodetic dominating set and geodetic connected dominating set are studied by Escuadro, et.al [9], Patangan et.al, [12], and Tejaswini, et.al [15]. The set S is a closed geodetic cover of a graph G if S = {v1, v2, ..., vk} and is obtained by choosing the vertices v1, v2, ..., vk such that the following hold: (i) v1 ̸= v2; (ii) vi /∈ IG[Si−1] for 3 ≤ i ≤ k; and (ii) IG[Sk] = V (G), where Si = {v1, v2, ..., vi} for all i = 1, 2, ..., k If S ⊆ V (G) satisfies (i) and (ii) of the definition above, then S is a closed geodetic subset of V (G). The collection of all closed geodetic covers of G is denoted by C∗(G). The closed geodetic number of G, is given by cgn(G) = min{|S| : S ∈ C∗(G)}. A set S ∈ C∗(G) with |S| = cgn(G) is called the closed geodetic basis of G and is denoted by cgb(G) Aniversario, et.al [1], and Patangan, et.al [12]. A vertex v in a connected G is an extreme vertex if the neighborhood N(v) of v induces a complete subgraph of G. The set of all extreme vertices in G is denoted by Ext(G). By a neighborhood N(v) of a vertex v in G is the set of all vertices x in G suh that dG(v, x) ≤ 1. A set S ⊆ V (G) is said to be a closure absorbing set in G if for every v ∈ V (G) \ S, there exist u,w ∈ N(v) ∩ S with dG(u,w) = 2, Cagaanan [3], and Aniversario, et.al [1]. Let G be the connected graph and S ⊆ V (G). The 2 - path closure P2[S]G of S is that set P2[S]G = S ∪ {w ∈ V (G) : w ∈ IG[u, v] for some u, v ∈ S with dG(u, v) = 2}. The set S is called 2 - path closure absorbing set if P2[S]G = V (G), Canoy, et.al [5], and Aniversario, et.al [1]. A set S is called a weakly connected closed geodetic set of G, if it satisfies the following properties: (i) S ∈ C∗(G); and (ii) ⟨S⟩w is connected. The minimum cardinality of a weakly connected closed geodetic set is called the weakly connected closed geodetic number of G, denoted by wcgn(G). In a weakly connected closed geodetic set S, for every v ∈ S, there exists u ∈ S such that dG(u, v) ≤ 2. Moreover, if {S1, S2, ..., Sk} is the sequence corresponding to the weakly connected geodetic set S, ⟨Si⟩w is connected for each i = 1, 2, ..., k, Patangan, et.al [12]. 3. Results Definition 1. A weakly connected closed geodetic set of G which is dominating is called a weakly connected closed geodetic dominating set of G. The minimum cardinality of a weakly J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 739 connected closed geodetic dominating set is called weakly connected closed geodetic domi- nation number of G, denoted by γwcg(G). A weakly connected closed geodetic dominating set S with |S| = γwcg(G) is said to be a γwcg-set of G. Example 1. Let G be the graph in Figure 1 and S = {u2, u4, u6}. Then u2 ̸= u4 with u6 /∈ IG[u2, u4] and IG[u2, u4] = {u2, u4}, IG[u2, u6] = {u2, u1, u6} ∪ {u2, u5, u6} = {u1, u2, u5, u6} and IG[u4, u6] = {u4, u5, u6} ∪ {u4, u3, u6} = {u3, u4, u5, u6}. Thus, IG[S] = {u1, u2, u3, u4, u5, u6} = V (G). Since u6 /∈ IG[u2, u4], IG[S] is a geodetic closure of S. Also, NG[S] = V (G), ⟨S⟩w is connected. In fact, it can be verified that there is no set of lesser cardinality than S that is a weakly connected. Note that u6 /∈ IG[u2, u4]. Thus, S = {u2, u4, u6} is a weakly connected closed geodetic set and dominating. u1 u2 u3 u4u5 u6 G : u1 u2 u3 u4u5 u6 ⟨S⟩w : Figure 1: Graph G with γwcg(G) = 3 Remark 1. Every weakly connected closed geodetic dominating set of a graph G is weakly connected closed geodetic set. So, wcgn(G) ≤ γwcg(G). Remark 2. For any nontrivial connected graph G of order n, 2 ≤ max{γ(G), wcg(G)} ≤ γwcg(G) ≤ n. Remark 3. Every superset of a weakly connected closed geodetic dominating set is weakly connected closed geodetic dominating set. Lemma 1. Aniversario, et al [1] Every geodetic cover of a connected graph G contains all its extreme vertices. Theorem 1. Let G be a connected graph of order n. Then (i) every weakly connected closed geodetic dominating set of G contains its extreme ver- tices. J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 740 (ii) if the set S of extreme vertices of G is a weakly connected closed geodetic dominating set of G. Then S is a unique minimum weakly connected closed geodetic dominating set of G and γwcg(G) = |S|. Proof. (i) Let S be a weakly connected closed geodetic dominating set and let v be an extreme vertex of G. Assume that v /∈ S. Then by Lemma 1, S is not a geodetic cover of G. Thus, S is not a closed geodetic dominating set of G. Hence, S is not weakly con- nected closed geodetic dominating set of G, which is a contradiction. Therefore, each extreme vertex of G belongs to every weakly connected closed geodetic dominating set of G. (ii) Let S be a set of extreme vertices of G. Suppose S is a weakly connected closed geodetic dominating set of G and let v be an extreme vertex of G. Claim 1: S is a minimum weakly connected closed geodetic dominating set of G. Suppose S is not γwcg-set of G. Then there exists v ∈ S such that S \{v} is a weakly connected closed geodetic dominating set of G. So, v /∈ IG[u,w] for some u, w ∈ S and v ̸= x, y for all x, y ∈ V (G) since v is an extreme vertex of G. Then v /∈ V (G), which is a contradiction. Consequently, S is a minimum weakly connected closed geodetic dominating set of G. Claim 2: S is unque minimum weakly connected closed geodetic dominating set of G. Let S be a unique weakly connected closed geodetic dominating set of G. Then since S contains its extreme vertices and is a minimum weakly connected closed geodetic dominating set of G. Therefore, S is unque minimum weakly connected closed geodetic dominating set of G. Furthermore, unique. The next result immediately follows from Theorem 1. Corollary 1. Every weakly connected closed geodetic dominating set of G contains its extreme vertices, then (i) the complete graph Kn has γwcg(Kn) = n for n ≥ 2. (ii) the path Pn of order n has γwcg(Pn) = ⌈ n+1 2 ⌉ . (iii) the cycle Cn of order n has γwcg(Cn) = ⌈ n 2 ⌉ for n ≥ 4. (iv) the complement of a cycle Cn of order n has γwcg(Cn) = 3 for n ≥ 5. (v) the fan Fn of order n has γwcg(Fn) = ⌈ n 2 ⌉ for n ≥ 4. (vi) the wheel Wn of order n has γwcg(Wn) = ⌈ n−1 2 ⌉ for n ≥ 5. (vii) the Petersen graph G has γwcg(G) = 4. J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 741 Theorem 2. Let G be a connected graph of order n. Then γwcg(G) = n if and only if G = Kn. Proof. Suppose that γwcg(G) = n. Assume that G ̸= Kn. Then there exist x, y ∈ V (G) such that dG(x, y) = 2. Now, construct a set S = {v1, v2, ..., vk} where S ∈ C∗(G) and x = v1 and y = v2. Since IG[v1, v2] ̸= {v1, v2} and vi ∈ IG[Si−1] for all i = 3, 2, 4, ..., k, we have IG[S] ̸= S. In fact, IG[S] = V (G). Thus, k < n. Moreover, Since NG[S] = V (G) and Ew(S) of G is induces a connected subgraph, it follows that ⟨S⟩w is also connected, Furthermore, since IG[S] = V (G), it follows that S is a dominating set of G. Therefore. γwcg(G) = n which is a contradiction to the assumption. Consequently, G = Kn. The converse follows from Corollary 1 (i). Lemma 2. Let m,n ≥ 2 and let U and W be the partite sets of Km,n. A subset S of V (Km,n) is a weakly connected closed geodetic dominating set of Km,n if and only if S is any of the following: (i) S = U ; (ii) S = W ; (iii) S = U ∪ {w} for some w ∈ W ; (iv) S = W ∪ {u} for some u ∈ U . Theorem 3. Let m,n ≥ 2 and let U and W be the partite sets of Km,n. Then γwcg(Km,n) = min{|S| : S ∈ W(Km,n)}. Proof. Let m,n ≥ 2 and let U and W be the partite sets of Km,n. By Lemma 2, γwcg(Km,n) = min{|U |, |W |, |U ∪ {w}| for some w ∈ W and |W ∪ {u}| for some u ∈ U}. Corollary 2. Let m,n ≥ 2 and let U and W be the partite sets of Km,n. Then γwcg(Km,n) = min{m,n}. Theorem 4. Let m,n ≥ 2 and S ⊆ V (Km,n). Then S is a γwcg-set of Km,n if and only if S is wcg-set of Km,n. Theorem 5. For the complete bipartite Km,n, (i) γwcg(Km,n) = 2, for m = n = 1. (ii) γwcg(Km,n) = n, for n ≥ 2, m = 1. (iii) γwcg(Km,n) = m, for m ≥ 2, n = 1. Corollary 3. The star K1,n−1 of order n has γwcg(K1,n−1) = n− 1. J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 742 Theorem 6. For a helm Hn, γwcg(Hn) = n+ 1 for n ≥ 3. Proof. Let ui be the vertices of a cycle Cn where n = 1, 2, 3, ..., n, v ′ be the center vertex of Hn and vi be the pendant vertices of Hn where n = 1, 2, 3, ..., n. Then every vi is connected to each vertex ui. Let S1 = {v1} and S2 = {v1, v3} where IHn [S2] = {v1, v2, v3}. Continuing this process we obtain a set Sn ∈ C∗(G) where IHn [Sn] = V (Hn). Therefore Sn is a closed geodetic cover of Hn. However, Sn is not a dominating set of Hn since Sn does not dominate the vertex v ′ . Now, we need to pick v ′ /∈ Sn for Sn+1 = {v′ , v1, ..., vn}. Let S1 = {v′}, S2 = {v′ , v2} where IHn [S2] = {v′ , v1, v2}. Continuing this process we obtain a set Sn+1 ∈ C∗(G) where IH2 [Sn+1] = V (Hn). Thus, Sn+1 is both closed geodetic cover and dominating set of Hn. Clearly, NG[Sn+1] = V (Hn) and Ew(S) induces a connected subgraph. It follows that ⟨S⟩w is connected. Therefore, Sn+1 is weakly connected closed geodetic dominating set of Hn. Furthermore, γwcg(Hn) = |Sn+1| = n+ 1. The next results present some relationships between γwcg(G), wcgn(G), γw(G), γg(G), γgc(G) and γ(G). Theorem 7. Let G be any connected graph of order n ≥ 2. Then γwcg(G) = 2 if and only if wcgn(G) = 2. Theorem 8. If G is a connected graph with γ(G) = 1, then γwcg(G) = wcgn(G). Proposition 1. For a complete bipartite graph Km,n with integers m,n ≥ 2, γwcg(Km,n) = min{m,n} = wcgn(Km,n). Theorem 9. Let G be a connected graph of order n. Then, γg(G) ≤ γwcg(G). Proof. Let G be a connected graph. Suppose that γwcg(G) < γg(G). Let S = {v1, v2, v3, ..., vi} is a γg-set of G. Then γwcg(G) < |S| = γg(G). Hence, by removing an element in S, say v1 we have γwcg(G) ≤|S|. If |S \ {u1}| = γwcg(G), then S \ {u1} is a geodetic dominating set of G, a contradiction. If γwcg(G) < |S \ {u1}|, then repeat the process above until we get |S \ {u′is}| = γwcg(G). Therefore, S \ {u′is} is a geodetic dominating set of G, which is a contradiction. Consequently, we have γg(G) ≤ γwcg(G) in any case. Theorem 10. Let G be a complete graph Kn for n ≥ 2, if G = Kn, then γg(Kn) = γwcg(Kn). Corollary 4. The γwcg(G) = γw(G) for some special graphs given as follows: (i) The complement of a cycle Cn of order n has γwcg(Cn) = 3 = γg(Cn) for n ≥ 5. J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 743 (ii) The star graph K1,n−1 of order n has γwcg(K1,n−1) = n− 1 = γg(K1,n−1). (iii) The wheel graph Wn has γwcg(Wn) = ⌈ n−1 2 ⌉ = γg(Wn) for n ≥ 5. Theorem 11. The complete bipartite Km,n has γg(Km,n) ≤ γwcg(Km,n), for m,n ≥ 2. Proposition 2. Let G be a complete graph Kn for n ≥ 2 vertices. Then γwcg(Kn) = γgc(Kn). Proposition 3. Let G be a path Pn, then γwcg(Pn) < γgc(Pn). Theorem 12. Let G be a cycle Cn, then γwcg(Cn) ≤ γgc(Cn) for n ≥ 4. Theorem 13. Let G be a complete bipartite Km,n for 2 ≤ m,n ≤ 4. Then γwcg(Km,n) ≤ γgc(Km,n). Corollary 5. If G is a complete bipartite Km,n for m, n ≥ 5. Then γwcg(Km,n) ≥ γgc(Km,n). The join 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) = E(G) • ∪E(H) • ∪ {uv : u ∈ V (G), v ∈ V (H)}, Harary [2]. Lemma 3. Aniversario, et.al [1] If G is a connected graph and diam(G) = 2, then every geodetic cover of G is a 2-path closure absorbing set in G. Theorem 14. Aniversario, et.al. [1] Let H be a connected noncomplete graph, and let G = H +Kp. Let S ⊆ V (H). If S is a 2-path closure absorbing set in H and S ∈ C∗(H), then S ∈ C∗(G). Theorem 15. Let H be a connected noncomplete graph and let G = H + Kp. If S is a 2-path closure absorbing in H and S ∈ W(H), then S ∈ W(G). Proof. Let H be a connected noncomplete graph and let G = H +Kp. Suppose that S is a 2 - path closure absorbing in H and S ∈ W(H). Then by Theorem 14, S ∈ C∗(G). To show that S is a weakly connected dominating set of G. Since G is connected, for every u, v ∈ S, dG(u, v) = 2 and for all y ∈ V (Kp), y is in u-v geodesic. Thus, V (⟨S⟩w) = V (G). It remains to show that for every u, v ∈ S there is an edge mu or nv with m,n ∈ V (G) \ S. Suppose there exists x ∈ S such that mx, nx /∈ Ew(S) for any m, n ∈ V (G) \ S. If m, n ∈ V (Kp), then mx, nx ∈ Ew(S). However, if m, n ∈ V (H) \ S, then mx, nx ∈ Ew(S). Further, if without loss of generality, m ∈ V (Kp) and n ∈ V (H) \ S, J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 744 then mx, nx ∈ Ew(S). In either case, mx, nx ∈ Ew(S). Hence, ⟨S⟩w is connected. It follows that S is a weakly connected set of G. Since every vertex in H is adjacent to every vertex in Kp, there exist u, v ∈ S such that dH(u, v) = 2 such that every vertex in Kp lies in the u − v geodesic of G and dG(u, v) = 2. It follows that V (Kp) ⊆ N [S]. Hence, V (G) = V (H) ∪ V (Kp) ⊆ N [S]. Hence, S is a dominating set of G. Thus, S is a γwcg - set, that is S ∈ W(G). Theorem 16. Let H be a connected noncomplete graph and G = H+Kp. If S is a γwcg-set of G, then S ⊆ V (H) and S is a 2-path closure absorbing set in H. Corollary 6. Let H be a connected noncomplete graph and G = H +Kp, then γwcg(H +Kp) = min{|S|: S ⊆ V (H), S ∈ W(G) and P2[S]H = V (H)}. Proof. Define ω = min{[S] : S ⊆ V (H), S ∈ W(G) and P2[S]H = V (H)}. Case 1. Suppose that H is a connected noncomplete graph and G = H + Kp, then γwcg(G) ≤ ω. Case 2. Suppose that S ∈ W(G). Let S ⊆ V (G) be a γwcg-set of G. Then by Theorem 16, S ⊆ V (H) and S is a 2-path closure absorbing set in H. Hence γwcg(G) =|S|≥ ω. Consequently, by combining these two inequalities the conclusion follows. Corollary 7. Let H be a connected noncomplete graph and let G = H+Kp. If diam(H) = 2, then γwcg(G) = γwcg(H). Proof. Suppose that G = H +Kp where H is a noncomplete graph with diam(H) = 2. Case 1. Let S ⊆ V (H) such that S ∈ W(G). Then by Corollary 6, γwcg(G) =|S|≥ γwcg(H). Case 2. Let S be a γwcg(G)-set of G. Then by Theorem 16 , S ⊆ V (H) and S is a 2-path closure absorbing set in H. Thus, by Theorem 15, S ∈ W(G). Hence, by Corollary 6, γwcg(G) =|S|≤ γwcg(H). Consequently, combining these two inequalities the conclusion follows. Theorem 17. Let G = H +K where H and K are connected noncomplete graphs. If S is a γwcg-set of G, then either (i.) S ⊆ V (H), where S is a 2-path closure absorbing set in H, or (ii.) S ⊆ V (K), where S is a 2-path closure absorbing set in K. Proof. Let G = H +K where H and K are connected noncomplete graphs. Suppose γwcg(G) = k and let S = {y1, y2, ..., yk} ∈ W(G). If ⟨S⟩ is a complete subgraph of G and IG[S] = V (G), then NG[S] = V (G) and Ew(S) is connected which implies that ⟨S⟩w is connected. Hence, V (⟨S⟩w) = V (G), a contradiction. Thus, there exist integers i, j, J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 745 1 ≤ i < j ≤ k such that dG(yi, yj) = 2. Either yi, yj ∈ V (H) or yi, yj ∈ V (K). Suppose yi, yj ∈ V (H). We claim that S ∩ V (K) = ∅. Clearly, V (K) ⊆ IG[yi, yj ]. Suppose that S ∩ V (K) = {z} and let z = yl. Then l < j. We consider the set S∗ = {xi, x2, ..., xk−1} where xn = { yn, if 1 ≤ n ≤ l − 1 yn+1, if l ≤ n ≤ k − 1 . Since dG(yl, yn) = 1 for all n = 1, 2, ..., l − 1, l + 1, ..., k, IG[S ∗] = V (G). This implies that S∗ ∈ C∗(G). Since G is connected, for every xi, xj ∈ S∗, dG(xi, xj) = 2. Then there exists z ∈ V (G) \ S such that z lies in xi-xj geodesic. Thus, for NG[S ∗] = V (G) and xi, xj ∈ Ew(S) for all z ∈ V (G)\S. This implies that S∗ ∈ W(G), contrary to the assumption that γwcg(G) = k. Suppose that |S ∩ V (K)| ≥ 2. In here, we consider two subcases, Subcase 1. When dG(x, y) = 1 for all x, y ∈ S ∩ V (K); and Subcase 2. When for some x, y ∈ S ⋂ V (K), dG(x, y) = 2. Suppose that dG(x, y) = 1 for all x, y ∈ S ∩ V (K) = {yr1 , yr2 , ..., yrl}. Then rn < j for all n = 1, 2, ..., l. We consider the set S∗ = S ∩ V (H). Write S∗ = {x1, x2, ..., xk−l} such that if xn = yp and xm = yq, then n < m if and only if p < q. Since yi, yj ∈ S∗, we have for every n = 1, 2, ..., l, IG[x, yrn ] = {x, yrn} ⊆ IG[S ∗] for all x ∈ S. Thus, IG[S ∗] = IG[S] = V (G). Hence, there exists z ∈ V (G) \ S∗ such that z lies in x-yrn geodesic. Thus, NG[S ∗] = NG[S] = V (G) and xz, yz ∈ E(⟨S⟩w) for all z ∈ V (G) \ S∗. This means that S∗ ∈ W(G). The fact that k-l < k, a contradiction. Lastly, suppose that dG(ym, yn) = 2 for some ym, yn ∈ S ∩ V (K) with m < n. Again, we must have n < j. But, if dG(ym, yn) = 2, then V (H) ⊆ IG[ym, yn], and in particular, yj ∈ IG[ym, yn]. But by definition of S, yj /∈ IG[Sn]. It follows that yj /∈ NG[Sn]. Hence, NG[Sn] ̸= V (G). Thus, Sn /∈ W(G), a contradiction. Now, we are left to show that S is a 2-path closure absorbing in H. Suppose that S ⊆ V (H). By Theorem 16 and Lemma 3, P2[S]G = V (G). Let z ∈ V (H) \ S. Then z ∈ V (G) \ S, and there exist x, y ∈ S such that z ∈ IG[x, y] and dG(x, y) = 2. This implies that [x, z, y] is a x-y geodesic in H. Thus, z ∈ IH [x, y] and dH(x, y) = 2. This means that P2[S]H = V (H), and so S is a 2-path closure absorbing in H. Similarly, if yi, yj ∈ V (K), then S ⊆ V (K). Moreover, if S ⊆ V (K), then S is a 2-path closure absorbing in K. Theorem 18. Let G = H + K, where H and K are connected noncomplete graphs. If either (i.) S ⊆ V (H), where S is a 2-path closure absorbing set in H and S ∈ W(H) or (ii.) S ⊆ V (K), where S is a 2-path closure absorbing set in K and S ∈ W(K), then S ∈ W(G). J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 746 Theorem 19. Let G = H + K, where H and K are connected noncomplete graphs. Then γwcg(G) = min{Γ(H),Γ(K)}, where Γ(H) = min{|S|: S ⊆ V (H), S ∈ W(G) and P2[S]H = V (H)} and Γ(K) = min{|S|: S ⊆ V (K), S ∈ W(G) and P2[S]K = V (K)}. Proof. Let G be a connected graph and let G = H +K where H and K are connected noncomplete graphs. Assume that S ⊆ V (G) is a γwcg-set of G. Then by Theorem 17, we have S ⊆ V (H) and S is a 2-path closure absorbing set in H or S ⊆ V (K) and S is a 2-path closure absorbing set in K. Hence, γwcg(G) ≥ min{Γ(H),Γ(K)}, where Γ(H) = min{|S|: S ⊆ V (H), S ∈ W(G) and P2[S]H = V (H)} and Γ(K) = min{|S|: S ⊆ V (K), S ∈ W(G) and P2[S]K = V (K)}. By Theorem 18, γwcg(G) ≤ min{Γ(H),Γ(K)}. Consequently, γwcg(G) = min{Γ(H),Γ(K)}. Corollary 8. The weakly connected closed geodetic domination number of the join graph of the Path graph Pn, cycle graph Cn, and complete bipartite graph Km,n are given as follows. (i.) γwcg(Pm + Pn) = min{⌈m+1 2 ⌉, ⌈n+1 2 ⌉}, for m,n > 2. (ii.) γwcg(Cm + Cn) = min{⌈m2 ⌉, ⌈ n 2 ⌉}, for m,n > 3. (iii.) γwcg(Km,n +Kp) = min{m,n}, for m,n > 2. (iv.) γwcg(Km,n +Kp,q) = min{m,n, p, q}, for m,n, p, q ≥ 2. Corollary 9. Let H and K are connected noncomplete graphs and G = H + K. If diam(H) = diam(K) = 2, then γwcg(G) = min{γwcg(H), γwcg(K)}. The corona of graphs G and H, G ◦ H, is the graph obtained by taking one copy of G and |V (G)| copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. For every v ∈ V (G), denote by Hv the copy of H whose vertices are attached one by one to the vertex v. Subsequently, denote by v + Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv, v ∈ V (G), Harary [2]. Theorem 20. Jamil, et.al [11] Let G = H ◦K, where H is a nontrivial connected graph and K a noncomplete graph, and let S ⊆ V (G). Then S ∈ C∗(G) if and only if S = ( ⋃ v∈V (H) Sv) ∪ S0, where Sv ⊆ V (Kv) and Sv ∈ C∗(v + Kv), and S0 is a closed geodetic subset of V (H). Lemma 4. Let G = H ◦K, where H is a nontrivial connected graph, and K a noncomplete graph. If S ∈ W(G), then S ∩ V (Kv) ∈ W(v +Kv) for all v ∈ V (H). Lemma 5. Let G = H ◦K, where H is a nontrivial connected graph of order m and K a noncomplete graph. Let Sv ⊆ V (Kv) for all v ∈ V (H). If Sv ∈ W(v + Kv) for each v ∈ V (H), then S = ⋃ v∈V (H) Sv ∈ W(G). J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 747 Theorem 21. Let G = H ◦K, where H is a nontrivial connected graph and K a noncom- plete graph, and let Sv ⊆ V (G). Then S ∈ W(G) if and only if S = ( ⋃ v∈V (H) Sv)∪S0, where Sv ⊆ V (Kv) and Sv ∈ W(v +Kv), and S0 is a weakly connected closed geodetic subset of V (H). Proof. Suppose that S ∈ W(G). Then S ∈ C∗(G). By Theorem 20, S = ( ⋃ v∈V (H) Sv) ∪ S0 . where Sv ⊆ V (K) and Sv ∈ C∗(v + Kv) and S0 is a closed geodetic subset. By Lemma 4, S ∩ V (Kv) ∈ W(v+Kv) for all v ∈ V (H). Thus, S0 = S \ ⋃ v∈V (H) Sv is a closed geodetic subset is also a weakly connected closed geodetic subset of V(H). It remeains to show that Sv ∈ W(v +Kv). That is, Sv is a weakly connected closed geodetic dominating set of v +Kv. Now for any x, y ∈ Sv there exists z ∈ Sv such that xz, yz ∈ E(v+Kv). Thus, Ew(SV ) will induce a connected subgraph since N [Sv] = V (v +Kv), we have Sv ∈ W(v +Kv). Conversely, suppose that S = ( ⋃ v∈V (H) Sv)∪S0 . where Sv ⊆ V (K) and Sv ∈ W(v+Kv) and S0 is a weakly connected closed geodetic subset of V (H). By Lemma 5, ⋃ v∈V (H) Sv ∈ W(G). If S0 = ∅, then we are done. Suppose that S0 ̸= ∅. By Theorem 20 and and Lemma 3, S = ( ⋃ v∈V (H) Sv) ∪ S0 gives S0 = S \ ⋃ v∈V (H) , where S ∈ C∗(G) and ⋃ v∈V (H) ∈ C∗(G) and S0 is a weakly connected closed geodetic subset of V(H). Thus, for any x, y ∈ V (G) \ S there exists s ∈ S such that xs, ys ∈ E(G). Hence, Ew(S) will induce a weakly connected subgraph of G. Therefore, S = ( ⋃ v∈V (H) Sv) ∪ S0 ∈ W(G). Corollary 10. Let G = H ◦K, where H is a connected graph and K a noncomplete graph. Then S is γwcg-set of G if and only if S = ⋃ v∈V (H) Sv, where each Sv ⊆ V (v + Kv) is γwcg-set of v +Kv. Corollary 11. Let G = H ◦ K, where H is a connected graph of order m and K a noncomplete graph. Then γwcg(G) = m · γwcg(K1 ◦K). Theorem 22. Let G = H ◦K, where H is a connected graph of order m. If n ≥ 3, then γwcg(H ◦ Cn) = m · ⌈n2 ⌉. Proof. Let G = H ◦ K, where H is a connected graph of order m and K = Cn be a noncomplete graph. Then, we have γwcg(H ◦ Cn) = m · γwcg(K1 ◦ Cn), by Corollary 11 J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 748 = m · γwcg(Wn+1) = m · ⌈n+ 1− 1 2 ⌉ , by Corolary 1 (vi) = m · ⌈n 2 ⌉ Corollary 12. If G = Pm ◦ Cn. Then γwcg(G) = m · ⌈ n 2 ⌉ for n ≥ 3. Theorem 23. Let G = H ◦K, where H is a connected graph of order m. If n ≥ 3, then γwcg(H ◦ Pn) = m · ⌊n+2 2 ⌋. Proof. Let G = H ◦ K, where H is a connected graph of order m and K = Pn be a noncomplete graph. Then, we have γwcg(H ◦ Pn) = m · γwcg(K1 ◦ Pn), by Corollary 11 = m · γwcg(Fn+1) = m · ⌈n+ 1 2 ⌉ , by Corolary 1 (v) Corollary 13. If G = Cm ◦ Pn. Then γwcg(G) = m · ⌈ n+1 2 ⌉ , n ≥ 3. Theorem 24. Let H be a nontrivial connected graph of order n and K = Kn. Then S = ⋃ v∈V (H) V (v +Kv) is a γwcg-set of H ◦Kn. Proof. Let H be a nontrivial connected graph of order n and K = Kn. Suppose that S = ⋃ v∈V (H) Sv, Sv = V (v + Kv). By Lemma 5, S = ⋃ v∈V (H) Sv ∈ W(H ◦ Kn). Then, by Corollary 10, S = ⋃ v∈V (H) V (v +Kv) is a γwcg-set of H ◦Kn. Corollary 14. Let G = H ◦K, where H is a nontrivial connected graph of order m and K = Kn with n ≥ 4. Then γwcg(G) = m · (n+ 1) . The Cartesian product of two graphs G and H, denoted by G□H is the graph with 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), Harary [2]. J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 749 Lemma 6. Chellathurai, et.al [6] Let G = (V,E) be the Cartesian product H□K of connected graphs H = (V1, E1) and K = (V2, E2). If S ⊆ V , then IG[S] ⊆ IG[S1]□IG[S2]. Lemma 7. Chellathurai, et.al [6] Let G = (V,E) be the Cartesian product H□K of connected graphs H = (V1, E1) and K = (V2, E2). If S ⊆ V ,then NG[S] ⊆ NG[S1]□NG[S2] Lemma 8. Let H and J be graphs of order m and n respectively, and let G = H□J be the Cartesian product of graphs H and J . (i.) If S ⊆ V (H) (or S ⊆ V (J)), then V [S×{vi}] ⊆ V (Hi)(orV (Ji)) for v ∈ Ji (or Hi). (ii.) If S ⊆ V (H) (or S ⊆ V (J)) is a γwcg-set of a graph H (or J), then V [S × {vi}] is a γwcg-set of graph Hi (or Ji). But, V [S × {vi}] is not a γwcg-set of G. Remark 4. Let H and J be graphs of order m and n respectively, and let H□J be the Cartesian product of graphs H and J . If S ⊆ V (H) (or S ⊆ V (J)) is a γwcg-set of graphs H (or J), then S × {vi} is a γwcg-set of graph Hi (or Ji). Theorem 25. Let H and J be connected graphs. Then γwcg(H□J) ≥ max{γwcg(H), γwcg(J)}. Equality holds if H and J are complete graphs. Proof. Let S ⊆ V (H□J) be a γwcg-set of H□J . Then by Lemma 6 and 7, V (H□J) = IG[S] ⊆ IG[S1]□IG[S2] and V (H□J) = NG[S] ⊆ NG[S1]□NG[S2]. Since G is connected and NG[S] = V (H□J), there exists xv, vy ∈ E(H□J) such that x ∈ S or y ∈ S for some v ∈ V (H□J). Hence, ⟨S⟩w ⊆ ⟨S1⟩w□⟨S2⟩w is also connected. Thus, S1 and S2 are γwcg-sets of H and J respectively, with γwcg(H) ≤ |S1| and γwcg(J) ≤ |S2|. Therefore, γwcg(H□J) = |S| ≥ max{|S1| |S2|} ≥ max{γwcg(H), γwcg(K)}. So, equality holds. Corollary 15. For every nontrivial connected graph H, γwcg(H) ≤ γwcg(H□Kn). Theorem 26. Let H be a connected graph of order at least 3 and diameter at most 2. Then H has γwcg-set S with a vertex x such that every vertex of H lies on some u- v geodesic in H for some w ∈ S and ⟨S⟩w = ⟨NH [S], Ew⟩ is connected if and only if γwcg(H) = γwcg(H□K2). Proof. Let H□K2 be formed from two copies H1 and H2 of H and S be a minimum weakly connected closed geodetic dominating set of H1 such that S contains a vertex v with the property that every vertex of H1 lies on some u−v geodesic in H1 for some v ∈ S. Let D consists of vertex x together with those vertices of H2 corresponding to those vertices in S −{u}. Hence, |D| = |S|. We show that D is weakly connceted closed geodetic dominat- ing set of H□K2. Let x /∈ D be a vertex of H□K2. First, suppose that x ∈ V (H1). Since, IH [S] = V (H1) and diam(H1) ≤ 2, it follows that, x ∈ IH [u, v] = IH [S] and v ̸= x. Since J. Hamja, I. Aniversario, H. Rara / Eur. J. Pure Appl. Math, 15 (2) (2022), 736-752 750 v ′ is the corresponding vertex of v ∈ S, v′ ∈ D and x ∈ N [D] where v ̸= x. Also, since NH [S] = V (H1) and Ew = {uv′ ∈ E(H1) : u ∈ S or v′ ∈ S} which implies that ⟨S⟩w is con- nected, and diam(H1) ≤ 2, x ∈ NH [D] where v ̸= x. Therefore, D is a weakly connected closed geodetic dominating set of H□K2. Next, suppose that x ∈ IH [u ′ , v ′ ], where u′ is the vertex in V (H2) corresponding to v and v ′ ∈ D. Since diam(H2) ≤ 2, x ∈ IH [u, v ′ ] ⊆ IH [D] and x ∈ NH [v ′ ] ⊆ NH [D], and NH [S] = V (H1) and Ew = {uv′ ∈ E(H1) : u ∈ S or v′ ∈ S} which implies that ⟨S⟩w is connected. Therefore D is a weakly connected closed geodetic dominating set of H□K2. Now, γwcg(H□K2) ≤ |D| = |S| = γwcg(H). Consequently, by Corollary 15, γwcg(H) = γwcg(H□K2). Conversely, suppose that γwcg(H) = γwcg(H□K2) where H□K2 is formed from two copies of H1 and H2 of H. Let D be a minimum weakly connected closed geodetic domi- naing set of H□K2. Clearly, D∩V (Hi) ̸= ∅, i = 1, 2. Let x ∈ D∩V (H1) and let S consist of vertices of D∩V (H1) together with those vertices in D∩V (H2). Clearly, S is a weakly connected closed geodetic dominating set of H1 and |S| = |D|. Since, D is a minimum weakly connected closed geodetic dominating set of H1. We show that every vertex of H1 lies on some u− v geodesic for some v ∈ S and ⟨S⟩w = ⟨NH [S], Ew⟩ is connected. Suppose that there exists a vertex x ∈ V (H1) such that x ∈ IH [u, v] for all v ∈ S. Then x /∈ NH [u] and d(u, x) = d(u, v) + d(u, x) > 2, a contradiction, Consequently, diam(H1) ≤ 2. Conlusion: The paper has introduced the concept of weakly connected closed geode- tic dominating sets of some graphs and the join, corona, and Cartesian product of two graphs are characterized. The weakly connected closed geodetic domination numbers of these graphs are determined. Also, some relationships between weakly connected closed geodetic dominating set, weakly connected closed geodetic set, geodetic dominating set, and geodetic connected dominating set are established. 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