EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 169-179 ISSN 1307-5543 – ejpam.com Published by New York Business Global Path-Induced Closed Geodetic Domination of Some Common Graphs and Edge Corona of Graphs Jesica M. Anoche1,∗, Imelda S. Aniversario1, Catherine I. Merca1 1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines Abstract. Let G be a connected graph of order n and S ⊆ V (G). A closed geodetic cover S of G is a path-induced closed geodetic dominating set of a graph G if a subgraph ⟨S⟩ has a Hamiltonian path and S is a dominating set of G. The minimum cardinality of a path-induced closed geodetic dominating set is called path-induced closed geodetic domination number of G. This study presents the characterization of the path-induced closed geodetic dominating sets of some common graphs and edge corona of two graphs. The path-induced closed geodetic domination numbers of these graphs are also determined. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Geodetic set, geodetic dominating set, path-induced closed geodetic set, path-induced closed geodetic dominating set, path-induced closed geodetic domination number 1. Introduction Domination in graph is one of the most studied concepts in Graph Theory. It was first developed in the late 1950’s and 1960’s, beginning with C. Berge in 1958. On his study, he referred the domination number as the “coefficient of external stability”. In 1962, O. Ore introduced the terms “dominating set” and “domination number”. Years later, a new domination parameter called geodetic domination in graph was introduced by Escuadro et al. (2011) and defined that a vertex in a graph G dominates itself and its neighbors. On the other hand, O. Cauntongan and I. Aniversario [3] introduced and studied the concept on path-induced closed geodetic number of some graphs. This concept follows from the definition of geodetic numbers of graphs introduced by Buckley and Harary in [2], closed geodetic numbers in [1], and path-induced geodetic numbers in [7]. In their stud- ies, they were able to present some properties and characterized the path-induced closed geodetic set of some common graphs and determined the path-induced closed geodetic numbers of those graphs. The researchers believe that the concept of path-induced closed ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4506 Email addresses: jesica.anoche@g.msuiit.edu.ph (J.Anoche), imelda.aniversario@g.msuiit.edu.ph (I. Aniversario), catherine.merca@g.msuiit.edu.ph (C. Merca) https://www.ejpam.com 169 © 2023 EJPAM All rights reserved. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 170 geodetic numbers of graphs can be applied in travel time saving, facility location, goods distribution, project crashing and other things in which this concept will be of great help. These previous studies motivated the researchers to combine the concepts of path- induced closed geodesic and dominating sets in graphs. That is, a set S ⊆ V (G) is both a path-induced closed geodetic set and a dominating set of G. In this study, we only consider a connected simple nontrivial graph G. The distance between the vertices u and v, denoted by dG(u, v), is the shortest length of the u-v path in G. A u-v path of length dG(u, v) is called a u-v geodesic. For every two vertices u and v of G, the interval IG [u, v] denotes the set interval containing u, v and all ver- tices lying in some u-v geodesic. The geodetic closure IG [S] is the union of intervals between all pairs of vertices from S, that is, IG [S] = ⋃ {IG [u, v] : u, v ∈ S}. A geode- tic set of G is a set S with IG [S] = V (G). The geodetic number, gn (G) of a graph G is the minimum cardinality among geodetic sets of G. A set S is a closed geodetic cover of G if S = {v1, v2, · · · , vk} such that v1 ̸= v2, vi /∈ IG [Si−1] for 3 ≤ i ≤ k and IG [Sk] = V (G), where Si = {v1, v2, · · · , vi} for i = 1, 2, · · · , k. The closed geodetic number cgn (G) of G is the minimum cardinality among closed geodetic covers of G [1]. 2. Preliminary Concepts and Results Definition 1. [2] The removal of a vertex v from a graph G results in the subgraph ⟨G∖ {v}⟩ with V (G ∖ {v}) = V (G) ∖ {v} and E(⟨G∖ {v}⟩) = {uw ∈ E(G) : u ̸= v and w ̸= v}. We may use the notation G∖ v for G∖ {v}. Definition 2. [2] A vertex x of a graph G is called a cut-vertex if the removal of x increases the number of components of the graph G. We will use ω (G) to describe the number of components a graph G has. Definition 3. [2] In a graph G, the neighborhood NG(u) of a vertex u ∈ V (G) is the set consisting of all vertices v which are adjacent to u, that is, NG(u) = {v ∈ V (G)|uv ∈ E(G)}. A vertex u ∈ V (G) is an extreme vertex if the neighborhood NG(u) of u induces a complete subgraph of G. Definition 4. [2] A nontrivial connected graph without cut-vertices is called non-separable graph. Otherwise, such graphs are separable. Definition 5. [2] Let G be a nontrivial connected graph. A block B of G is a subgraph of G that is itself non-separable and which is maximal with respect to this property. Definition 6. [2] A Hamiltonian path of a graph G is a path that contains all vertices of G and passes through each vertex of G exactly once. Definition 7. [6] The edge corona G ⋄ H of G and H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each of the end vertices u and v of each edge uv of G to every vertex of the copy Huv of H. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 171 Definition 8. [4] Let G be a connected graph and S ⊆ V (G). The 2-path closure P2[S]G of set S is the set P2[S]G = S ∪ {w ∈ V (G) : w ∈ IG(u, v) for some u, v ∈ S with dG(u, v) = 2 }. A set S is called 2-path closure absorbing if P2[S]G = V (G). The minimum cardinality of a 2-path closure absorbing set of G is denoted by φ(G). Definition 9. [3] Let G be a connected graph of order n and S ⊆ V (G). A closed geodetic cover S of G is called a path-induced closed geodetic set of a graph G, denoted by picg- set, if ⟨S⟩ has a Hamiltonian path. The minimum cardinality of a path-induced closed geodetic set is called path-induced closed geodetic number of G, denoted by picgn(G). A path-induced closed geodetic set S with |S| = picgn(G) is called a path-induced closed geodetic basis of G, denoted by picgb(G). Definition 10. Let G be a connected graph of order n and S ⊆ V (G). A closed geodetic cover S of G is called a path-induced closed geodetic dominating set of a graph G, denoted by picgd-set, if S is both a path-induced closed geodetic set and a dominating set of G. The minimum cardinality of a path- induced closed geodetic dominating set is called path- induced closed geodetic domination number of G, denoted by γpicg(G). A path-induced closed geodetic dominating set with |S| = γpicg(G) is said to be a γpicg-set of G. ........................................................................ ........................................................................ ........................................................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ... ........................................................................................................................................................................................................................................................................................................................... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ........... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... . .............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ... ..................................................................................................................................... .................................................................................................................................................................................................................................................................................................................. ...................................................................................................................................................................................................................................................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... c1 c2 d1 d2 e1 e2 • • • G Figure 1: A graph G Example 1. Consider the graph G in Figure 1. Let S1 = {c1}, S2 = {c1, c2} and S3 = {c1, c2, e1}. Then IG [S2] = {c1, c2} and IG [S3] = IG[c1, c2] ∪ IG[c1, e1] ∪ IG[c2, e1] = {c1, c2} ∪ {c1, c2, d1, d2, e2, e1} ∪ {c2, e1} = {c1, c2, d1, d2, e1, e2} = V (G). Let S = {c1, c2, e1} = S3. Then IG [S] = V (G). Note that ⟨S⟩ contains a Hamiltonian path [c1, c2, e1]. Thus, S is a path-induced closed geodetic set ofG. Observe that we cannot find a path-induced closed geodetic set S of cardinality less than 3. Thus, picgn(G) = 3. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 172 Now, since each vertex of V (G) ∖ S is adjacent to at least two vertices of S, S is a dominating set of G. Therefore, S is a path-induced closed geodetic dominating set and it can be verified that γpicg(G) = 3. Remark 1. [7]. Let G be a connected nontrivial graph of order n. If G admits a path- induced geodetic set, then 2 ≤ gn(G) ≤ pign(G) ≤ n. Theorem 1. [3] Path-induced closed geodetic number of a few well-known graphs: (i) For a complete graph Kn, picgn (Kn) = n. (ii) For a path Pn on n vertices, picgn (Pn) = n. (iii) For a cycle Cn of length n, picgn(Cn) = { n 2 + 1, if n is even; n+1 2 + 1, if n is odd. Theorem 2. [3] Let G be a connected graph with cut-vertices. If S ⊆ V (G) is a path-induced closed geodetic basis of G and x is a cut-vertex of G, then every component of G∖ x contains a vertex in S. Remark 2. [3] Every cut-vertex of a connected graph G belongs to every path-induced closed geodetic set of G. Theorem 3. [3] Let G be a connected graph with cut-vertices. If G admits path-induced closed geodetic set, then G∖ x has exactly two components for each cut-vertex x of G. Theorem 4. [3] Let G be a connected graph with cut-vertices. If G admits a path-induced closed geodetic set, then each block of G admits at most 2 cut-vertices. Theorem 5. [3] Let G be a connected graph of orderm such that G admits a path-induced closed geodetic set. If every vertex of G is either an extreme vertex or a cut-vertex, then picgn (G) = m. Theorem 6. [3] Let T be a tree. Then T admits a path-induced closed geodetic sets if and only if T is a path. 3. Path-induced Closed Geodetic Domination Numbers of Some Common Graphs In view of Definition 9, a picg-set S may not be a picgd-set. Consider the cycle C8 with vertex-set {v1, v2, · · · , v8} in Figure 2. A picgb(C8) are S = {v1, v2, · · · , v5} and S∗ = {v1, v8, · · · , v5} but both sets are not dominating sets of C8. However, a picgd-set S of G is always a picg-set of G and a γpicg-set of G is always a picgb(G) of G. Hence, the next remarks follow. In general, the graph Cn does not have a path-induced closed geodetic dominating set, for all n ≥ 8. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 173 .................................... .................................................................................................................................................... ................................................................................................................ .................................... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... . .... ................................ ................................................................................................................ .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .................................... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... . .... ................................ ................................................................................................................ .................................................................................................................................................... C8 v8 v1 v2 v3 v5v6 v4 v7 Figure 2: The cycle C8 Remark 3. Every picgd-set of a graph G is a picg-set of G. Remark 4. Every γpicg-set of a graph G is a picgb(G) of G. In view of Definition 10, not all connected graphs have path-induced closed geodetic dominating set. To illustrate this, let us have the following example. ...................................................................................................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .................................... .................................... ............................................................................................................................................ .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .................................... .................................... ............................................................................................................................................ v v1 v2 v3v4 : v5K1,5 : Figure 3: A graph without a picg-set Example 2. Consider the star K1,5 in Figure 3. Observe that the only geodetic covers of K1,5 are sets S = {v1, v2, v3, v4, v5} and S∗ = V (K1,5) which are also dominating sets of K1,5. But ⟨S⟩ and ⟨S∗⟩ do not contain a Hamiltonian path. Therefore, S and S∗ are not path-induced closed geodetic dominating sets of K1,5. In general, the graph K1,n does not have a path-induced closed geodetic dominating set, for all n ≥ 3. To this extent, we will examine the properties of those graphs which admit path- induced closed geodetic dominating sets and provide some conditions that will help us determine whether a graph G admits a path-induced closed geodetic dominating set or not. Let us consider the following theorem. Theorem 7. Let G be a connected graph with cut-vertices. If S ⊆ V (G) is a γpicg-set of G and x is a cut-vertex of G, then every component of G∖ x contains an element in S. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 174 Proof: Let G be a connected graph and x ∈ V (G) be a cut-vertex of G. Let S ⊆ V (G) be a γpicg-set of G. Then by Remark 4, S is a picgb(G). Hence, by Theorem 2, every component of G∖ x contains a vertex in S. ■ As a consequence of Remark 2, Theorem 3 and Theorem 4, the next results follow. Remark 5. Every cut-vertex of a connected graph G belongs to every path-induced closed geodetic dominating set of G. Theorem 8. Let G be a connected graph with cut-vertices. If G has a path-induced closed geodetic dominating set, then ω(G− x) = 2 for every cut-vertex x of G. The contrapositive of Theorem 8 says that if there exists a cut-vertex x of G with ω(G− x) ≥ 3, then G has no path-induced closed geodetic dominating set. ............................................................................................................................................ ...................................................................................................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .................................... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........ .................................... ............................................................................................................................................ ........................................................................ .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........................................................................................................................................................ v • G : Figure 4: A graph with a cut-vertex and without a γpicg-set Figure 4 shows an illustration of the situation described in Theorem 8 with ω(G−x) = 3 and therefore does not allow G to have a path-induced closed geodetic dominating set. Theorem 9. Let G be a connected graph with cut-vertices. If G has a path-induced closed geodetic dominating set, then each block of G contains at most 2 cut-vertices. The contrapositive of Theorem 9 says that if there exists a block of G with three or more cut-vertices, then G has no path-induced closed geodetic dominating set. Figure 5 shows an illustration of the situation described in Theorem 9. Block B has three cut-vertices v1, v2, v3 and hence G has no path-induced closed geodetic dominating set. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 175 ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .................................... ........................................................................................................ .................................... ...................................................................................................................... .................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .................................... ............... .............. .............. .............. .............. .............. ........ .................................... ............................................................................................. .................................... ............... .............. .............. .............. .............. .............. ........ . ................................... ............................................................................................. .................................... .................................... .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .................................... ........................................................................................................ ........................................................................ ........................................................................................................ G : • • • B1 BB2 B3 v1 v2v3 Figure 5: A graph with a cut-vertex and without a γpicg-set The next remark is a restatement of Remark 1. Remark 6. Let G be a connected nontrivial graph of order n. If G admits a path-induced closed geodetic dominating set, then 2 ≤ gn(G) ≤ γpicg(G) ≤ n. Theorem 10. Let G be a connected nontrivial graph. Then γpicg(G) = 2 if and only if G = K2. Proof: Let S = {x, y} be a γpicg-set of G. Then by Definition 10, S is both a path- induced closed geodetic set and a dominating set. Thus, IG[S] = V (G). But note that ⟨S⟩ = K2. Hence, G = K2. Conversely, suppose G = K2. Then by Theorem 1 (i), picgn (K2) = 2. Thus, V (G) is a dominating set of G. Therefore, γpicg(G) = 2. ■ Remark 7. Every vertex of a complete graph Km is an extreme vertex. In G. J. Changa et al. [5], it is shown that every geodetic basis of a graph contains its extreme vertices. Now, since every vertex of a complete graph Km is an extreme vertex, then the next result follows. Proposition 1. For any natural number m, γpicg(Km) = m. The following theorem provides some of the necessary conditions for a graph G of order m to have γpicg(G) = m. Theorem 11. Let G be a connected graph of order m such that G has a path-induced closed geodetic dominating set. If every vertex of G is either an extreme vertex or a cut-vertex, then γpicg(G) = m. Proof: Let G be a connected graph of order m and G has a path-induced closed geodetic dominating set S. By Remark 3, S is a path-induced closed geodetic set of G. Let any v ∈ V (G) be either an extreme vertex or a cut-vertex. Then by Theorem 5, v ∈ S and picgn(G) = m. This means that S = V (G). Hence, S is a dominating set of G. Therefore, γpicg(G) = m. ■ J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 176 Remark 8. Every end-vertex in a graph G is an extreme vertex. Corollary 1. Let G = Pm. Then γpicg(G) = m for all m ≥ 1. Proof: Let G = Pm and V (G) = {u1, u2, · · · , um}. Thus, by Theorem 1(ii), picgn(G) = m. Since V (G) is a dominating set of G, then γpicg(G) = m. ■ The next theorem characterizes those trees which admit path-induced closed geodetic dominating sets. Theorem 12. Let T be a tree. Then T admits a path-induced closed geodetic dominating set if and only if T is a path. Proof: Suppose T is a tree that admits a path-induced closed geodetic dominating set. Then by Definition 10, T admits a path-induced closed geodetic set. Thus, by Theorem 6, T is a path. Conversely, suppose T is a path. Then, by Corollary 1, T admits a path-induced closed geodetic dominating set. ■ 4. Path-Induced Closed Geodetic Domination Numbers of the Edge Corona of Graphs In this section, we discuss the path-induced closed geodetic domination number of a graph obtained from the edge corona of two graphs. We remark that not all edge corona of two graphs admit path-induced closed geodetic dominating sets. Consider the edge corona G = C6 ⋄ P2 shown in Figure 6. Observe that the only connected geodetic sets in G are V (G), V (G)∖ v6, and V (G)∖ v2. But these sets are not closed geodetic sets of G. Hence, G does not admit path-induced closed geodetic dominating set. ................................................................................................................................................................................................. ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ..... ............................................................................................................................................................................................... ................................................................................................................................................................................................. ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ..... ............................................................................................................................................................................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ................. ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ............ ............................................................................................................................................................................................... .... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ................................................................................................................................................................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... ............................................................................................................................................................................................................................. ............................................................................................................................................................................................... ................. ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ................ ............ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....... .............................................................................................................................................................................................................. ................ ............... ............... ............... ............... ............... ............... ............... .... ....................................................................................................................................................................................... .................................................................................................................................................................................................................................................... ............... .............. .............. .............. .............. .............. .............. .............. ............................................................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ..... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ...... ............................. ............................ ............................ ............................ ............................ ............................ ............................ ............................ ................... ................................................................................................................. ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ..... ............................................................................................................................. ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ....... ................................................................................................................. ............................. ............................ ............................ ............................ ............................ ............................ ............................ ............................ ................... ................ ............... ............... ............... ............... ............... ............... ............... .... ....................................................................................................................................................................................... .............................................................................................................................................................................................................. .................................................................................................................................................................................................................................................... ............... .............. .............. .............. .............. .............. .............. .............. .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... .................................... v6v1 v2 v3 v4 : v5 Figure 6: G = C6 ⋄ P2 In general, the graph Cn ⋄H does not admit path-induced closed geodetic dominating set, for all n ≥ 6. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 177 To this extent, we will give necessary conditions for those graphs whose edge corona admits a path-induced closed geodetic dominating set. First, let us consider the following theorem. Here, we let G = K ⋄H, ΩG ={S ⊆ V (G) : P2[S]G = V (G) and ⟨S⟩ has a Hamiltonian path} and let Huv be the copy of the graph H for each uv ∈ E(K). Remark 9. If K is a connected graph of order 2 and H is any graph, then G = K ⋄H = K +H. Theorem 13. Let H be any graph and K be a connected noncomplete graph of order n ≥ 3 and bothK andH admit SK ∈ ΩK and SH ∈ ΩH , respectively. Then G = K⋄H ad- mits a path-induced closed geodetic dominating set S if and only if S = ( ⋃ uv∈E(K) Suv ) ∪A where the following holds: (i) A ⊆ V (K) and A ∈ ΩK ; and (ii) For all Suv ⊆ V (Huv), Suv ∈ ΩHuv for each uv ∈ E(K). Proof: Let G = K ⋄ H admits a path-induced closed geodetic dominating set S and let A = S ∩ V (K) and Suv = S ∩ V (Huv) for each uv ∈ E(K). Then A ⊆ V (K) and Suv ⊆ V (Huv). Note that IG[S] = V (G) and so S = ( ⋃ uv∈E(K) Suv ) ∪ A. By Definition 7, for each uv ∈ E(K) there exists Huv copy of H. Note that H admits SH ∈ ΩH and so for any Huv copy of H, Huv admits an element in ΩHuv . Since S is a path-induced closed geodetic dominating set of G, A and Suv must be elements of ΩK and ΩHuv , respectively. Otherwise, IG[S] ̸= V (G) or ⟨S⟩ can not contain a Hamiltonian path or S is not a dominating set of G, which is a contradiction. Conversely, suppose S = ( ⋃ uv∈E(K) Suv ) ∪A and (i) and (ii) hold. Since A and Suv are 2-path closure absorbing of K and Huv for each uv ∈ E(K), respectively, it follows that IG[S] = V (G). Note that each of ⟨Suv⟩ and ⟨A⟩ contains a Hamiltonian path. Thus, by Definition 7, ⟨S⟩ contains a Hamiltonian path. Since A is a 2-path closure absorbing of K, for any wK ∈ V (K)∖A, wK is adjacent to at least two vertices of A. This means that A is a dominating set of K. Thus, by Definition 7, A is a dominating set of G and so S is a dominating set of G. Therefore, S is a path-induced closed geodetic dominating set of G. ■ Theorem 14. Let H be any graph and K be a connected noncomplete graph with |V (K)| = m ≥ 3, |E(K)| = n and both K and H admit SK ∈ ΩK and SH ∈ ΩH , respectively. Let G = K ⋄H. Then γpicg(G) = n ·min{|SH | : SH ∈ ΩH}+min{|SK | : SK ∈ ΩH}. J. Anoche, I. Aniversario, C. Merca / Eur. J. Pure Appl. Math, 16 (1) (2023), 169-179 178 Proof: Let K and H admit SK ∈ ΩK and SH ∈ ΩH , respectively. Let G = K ⋄ H where |V (K)| = m ≥ 3 and |E(K)| = n. Then by Definition 7, there are n copies of H in G. Thus, by Theorem 13, the path-induced closed geodetic dominat- ing sets of G are of the form S = ( ⋃ uv∈E(K) Suv ) ∪A where A ⊆ V (K) and A ∈ ΩK and for all Suv ⊆ V (Huv), Suv ∈ ΩHuv for each uv ∈ E(K). Note that the minimum cardinality of S is obtained when each |Suv| and |SK | are minimum in ΩHuv and ΩK , respectively. That is, γpicg(G) = ⋃ uv∈E(K) min{|Suv| : Suv ∈ ΩHuv}+min{|SK | : SK ∈ ΩK}. Since all ⟨Suv⟩ are just copies of the graph H, it follows that min{|Suv| : Suv ∈ ΩHuv} = min{|SH | : SH ∈ ΩH}. Therefore, γpicg(G) = n ·min{|SH | : SH ∈ ΩH}+min{|SK | : SK ∈ ΩK}. ■ Corollary 2. Let m,n ≥ 3 be natural numbers. Then γpicg(Pn ⋄Km) = (n− 1)m+ n. Proof: Note that ΩPn = {V (Pn)}, |E(Pn)| = n − 1 and ΩKm = {V (Km)} for all m,n ≥ 1. Then by Theorem 14, γpicg(Pn ⋄Km) = |E(Pn)| ·min{|SKm | : SKm ∈ ΩKm}+min{|SPn | : SPn ∈ ΩPn} = (n− 1) ·min{|SKm | : S ∈ ΩKm}+ n = (n− 1) ·m+ n. ■ Theorem 15. Let G = K2 ⋄ H where H is any graph that admits S ∈ ΩH with |V (H)| = n ≥ 3 and diam(H) ≥ 2. Then γpicg(G) = γpicg(H). Proof: Let G = K2⋄H where H is any graph that admits S ∈ ΩH with |V (H)| = n ≥ 3 and diam(H) ≥ 2. Then, by Remark 9, G = K2 ⋄H = K2+H. Now, we let u, v ∈ V (K2). By Definition 7, all vertices of H are joined to u and v. Since diam(H) ≥ 2, for any non adjacent x, y ∈ S, u and v lie in some x-y geodesic. Hence, IG[S] = V (G). Note that S ∈ ΩH and so ⟨S⟩ contains a Hamiltonian path and S is a 2-path closure absorbing of H. Hence, by Definition 8, for any a ∈ V (H)∖ S, a is adjacent to at least two vertices of S. It follows that S is a dominating set of H and so S is a dominating set of G. Therefore, γpicg(G) = S = γpicg(H). ■ Corollary 3. For all integers n ≥ 3, γpicg(K2 ⋄ Pn) = n. REFERENCES 179 Proof: Note that ΩPn = {V (Pn)}, for all integers n ≥ 1. Hence, by Theorem 15, γpicg(K2 ⋄ Pn) =γpicg(Pn) =|V (Pn)| =n. ■ Corollary 4. Let H be a graph that has a path-induced closed geodetic dominating set. If diam(H) = 2, then γpicg(K2 ⋄H) = γpicg(H). Proof: Let S be a path-induced closed geodetic dominating set of H where diam(H) = 2. That is, for every u, v ∈ V (H), dH(u, v) ≤ 2. Thus, P2[S]H = V (H). This means that S ∈ ΩH is with minimum cardinality. Therefore, by Theorem 15, γpicg(K2 ⋄H) = |S| = γpicg(H). ■ Example 3. Since diam(Fn) = 2 for all n ≥ 3 and diam(Wm) = 2 for all m ≥ 4, by Corollary 4, we have the following: (i) γpicg(K2 ⋄ Fn) = n, for all n ≥ 3. (ii) γpicg(K2 ⋄Wm) = m, for all m ≥ 4. Acknowledgements This research is funded by the Department of Science and Technology - Accelerated Sci- ence and Technology Human Resource Development Program (DOST-ASTHRDP), Philip- pines. References [1] I. S. Aniversario, F. P. Jamil, and S. R. Canoy Jr. The Closed Geodetic Numbers of Graphs. Utilitas Mathematica, 74:3–18, 2007. [2] F. Buckley and F. Harary. Distance in Graphs. Redwood City, CA: Addison-Wesley, 1990. [3] O. I. Cauntongan and I. S. Aniversario. Path-Induced Closed Geodetic Numbers of Some Graphs. Advances and Applications in Discrete Mathematics., 22(1):41–53, 2019. [4] M. P. Laurente F.P. Jamil and M. B. Macababat. Strongly Closed Geodetic Numbers of Graphs. International Mathematical Forum, 26:1277–1290, 2010. [5] L.D. Tong G. J. Changa and H.H. Wang. Geodetic Spectra of Graphs, volume 25. 2004. [6] Y. Pabilona and H. Rara. Total Hop Dominating Sets in the Join, Corona, and Lexi- cographic Product of Graphs. Journal of Algebra and Applied Mathematics, 2017. [7] R. N. Villarante and I. S. Aniversario. Path-Induced Geodetic Numbers of Some Graphs., volume 29(5). 2017.