EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6076 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Certified Perfect Domination in Graphs Jamil J. Hamja1,∗, Amy A. Laja1, Hounam B. Copel1, Bayah J. Amiruddin-Rajik1, Nurijam Hanna M. Mohammad1 1 Department of Mathematics, College of Arts and Sciences, MSU-Tawi-Tawi College of Technology and Oceanography, 7500 Tawi-Tawi, Philippines. Abstract. Let G = (V (G), E(G)) be a simple graph. A perfect dominating set J ⊆ V (G) is called a certified perfect dominating set of G if each vertex a ∈ J has either no neighbors or at least two neighbors in V (G) \J . The certified perfect domination number of G γcerp(G) represents the smallest size of a certified perfect dominating set in G. A certified perfect dominating set of G that attains this minimum size, i.e., |J | = γcerp(G) is referred to as a γcerp-set. In this paper, we first present some upper bounds for the certified perfect domination number of G, investigate the relationship between certified domination and certified perfect domination parameters, and determine graphs with small and large values of these parameters. Secondly, we characterize the graphs with γcerp(G) = n and γcerp(G) = γcer(G). Finally, we characterize the certified perfect dominating set under the lexicographic and Cartesian products of two graphs, determine its certified perfect domination number, and identify a non-γcerp-graph under these binary operations. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Certified domination, perfect domination, certified perfect domina- tion, Corona, Lexicographic product 1. Introduction In 2020, Dettlaff et al. [1, 2] introduced the concept of certified domination in graphs. They determined the exact values of the certified domination number for certain classes of graphs and provided upper bounds for this parameter in arbitrary graphs. Addition- ally, they characterized a broad class of graphs in which the domination number and the certified domination number are equal and identified graphs with large certified domina- tion numbers. They also analyzed how the certified domination number changes when a graph is modified by adding or deleting an edge or a vertex. Moreover, they established Nordhaus–Gaddum type inequalities for the certified domination number and explored ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6076 Email addresses: jamilhamja@msutawi-tawi.edu.ph (J. J. Hamja), amylaja@msutawi-tawi.edu.ph (A. A. Laja), hounamcopel@msutawi-tawi.edu.ph (H. B. Copel), bayahamiruddin@msutawi-tawi.edu.ph (B. J. Amiruddin-Rajik), hannamohammad@msutawi-tawi.edu.ph (N. H. M. Mohammad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 2 of 13 the relationships among domination, upper domination, certified domination, and upper certified domination numbers. In 2023, Hamja [3] introduced the concept of certified perfect domination in graphs. He characterized the certified perfect dominating set, determined the exact values of the certified perfect domination number for specific graphs, and examined this parameter in graphs formed by the join and corona operations. Furthermore, he investigated the rela- tionship between the perfect dominating set and the certified perfect dominating set of a graph G Suppose we have a situation modeled by a graph G, where a subset S ⊆ V (G) represents a group of officials, and the remaining vertices H = V (G) \ S represent civilians. Each civilian x ∈ H must be assigned exactly one official u ∈ S who is responsible for serving them. Additionally, when an official u serves a civilian x, there must exist another civilian y ∈ H who only observes the service provided by the official u to the civilian x. This observer, or witness y, ensures that the service is performed ethically and without any misconduct from the official. This setup prompts the question: What is the smallest number of officials needed to ensure every civilian is served in such a monitored manner within a given social structure? This question motivated Hamja [3] to define a new concept in graph theory, known as the certified perfect dominating set of a graph G. In this paper, we extend the work of Hamja [3] by presenting new results related to certified perfect domination. We establish some upper bounds for the certified perfect domination number of a graph G and explore the relationship between certified domi- nation and certified perfect domination parameters. Additionally, we identify classes of graphs with both small and large values of these parameters. Furthermore, we provide a characterization of graphs for which γcerp(G) = n and γcerp(G) = γcer(G). We also examine the behavior of certified perfect dominating sets under graph operations such as the lexicographic and Cartesian products. In doing so, we determine the certified perfect domination number for these products and identify non-γcerp-graphs arising from these operations. This work contributes to the ongoing development of domination parameters in graphs and offers new insights into their structural properties. 2. Terminology and Notation For standard graph theory terminology, we follow the book Graph Theory by Diestel [4]. Let G = (V,E) be a simple, connected graph, where V = V (G) is the vertex set and E = E(G) is the edge set of G. The degree of a vertex v, denoted by deg(v), is the number of edges incident to v. The maximum degree of G, denoted by ∆(G), is the highest degree among all vertices of G, defined as ∆(G) = max{deg(v) : v ∈ V (G)}. The open neighborhood of a vertex u, denoted by NG(u), is the set of all vertices adjacent to u, formally given by NG(u) = {v ∈ V (G) : uv ∈ E(G)}. The closed neighborhood of a vertex u, denoted by NG[u], is the set containing u along with all vertices adjacent to it, expressed as NG[u] = NG(u) ∪ {u}. Similarly, the closed neighborhood of a subset S ⊆ V (G) is de- J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 3 of 13 fined as the union of the closed neighborhoods of all vertices in S, i.e., NG[S] = ⋃ v∈S NG[v]. A vertex of degree |V (G)| − 1 is called a universal vertex of G. A vertex with degree one is referred to as a leaf, and its unique adjacent vertex is called its support vertex (or simply, its support). A support vertex that is adjacent to at least two leaves is known as a strong support, whereas one that is adjacent to only a single leaf is called a weak support. The set of all leaves in G is denoted by L(G). For a leaf v ∈ L(G), its support vertex is represented as SL, while for a weak support v, the unique leaf adjacent to v is denoted by WL. The set of all strong support vertices in G is denoted by SS , and the set of all weak support vertices is denoted by WS , as defined by Dettlaff et al. (see [1]). A subset J of V (G) is called a dominating set if every vertex in G is either in J or adjacent to at least one vertex in J , i.e., NG[J ] = V (G). A dominating set J is said to be minimal if no proper subset J ′ ⊂ J remains a dominating set. The domination number γ(G) represents the smallest size of a dominating set in G. A dominating set J that attains this minimum size, i.e., |J | = γ(G), is referred to as a γ-set, as defined by Haynes et al. (see [5]). A subset J of V (G) is called a certified dominating set if each vertex a ∈ J has either no neighbors or at least two neighbors in V (G) \ J . The certified domination number γcer(G) represents the smallest size of a certified dominating set in G. A certified dominating set of G that attains this minimum size, i.e., |J | = γcerp(G) is referred to as a γcer-set, as defined by Dettlaff et al. (see [1]). A subset J of V (G) is called an independent set if for each pair of distinct vertices a, b ∈ J , ab /∈ E(G). The independence number α(G) represents the maximum size of an independent set of G. Additionally, the independent domination number γi(G) represents the minimum size of an independent set of G, as de- fined by Paraic et al. (see[6]). A subset J of V (G) is called a perfect dominating set if each vertex in V (G) \ J is adjacent to exactly one vertex in J . The perfect domination number γp(G) represents the smallest size of a perfect dominating set in G. A perfect dominating set of G that attains this minimum size , i.e, |J | = γcerp(G) is referred to as a γp-set, as defined by Livingston et al. (see [7]). A perfect dominating set J ⊆ V (G) is called an independent perfect dominating set if no two vertices in J are adjacent. The independent perfect domination number γip(G) represents the smallest size of an independent perfect dominating set of G. An independent perfect dominating set of G that attains this minimum size, i.e., |J | = γip(G) is referred to as an γip-set, as defined by Armada et al (see [8]). An independent perfect dominating set J ⊆ V (G) is called a certified independent perfect dominating set if every a ∈ J has either zero or at least two neighbors in V (G) \ J . The certified independence perfect number the maximum size of a certified independent perfect set of G. Additionally, the certified in- dependent perfect domination number γcerip(G) represents the minimum size of a certified independent perfect dominating set of G. A certified independent perfect dominating set of G that attains this minimum size, i.e., |J | = γcerip(G) is referred to as a γcerip-set. A perfect dominating set J ⊆ V (G) is called a certified perfect dominating set of G J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 4 of 13 if each vertex a ∈ J has either no neighbors or at least two neighbors in V (G) \ J . The certified perfect domination number of G γcerp(G) represents the smallest size of a certified perfect dominating set in G. A certified perfect dominating set of G that attains this minimum size, i.e., |J | = γcep(G) is referred to as a γcerp-set, as defined by Hamja (see [3]). It is worth noting that if a graph G has no certified perfect dominating set, then we say that G is a non-γcerp-graph. 3. Preliminary Results This section presents the bounds, properties, and exact values for various graphs. We begin by providing some upper bounds for the certified perfect domination in graphs. Proposition 1. [3] For a path Pn of order n ≥ 1, γcerp(Pn) = { n 3 , if n ≡ 0 (mod 3), n, otherwise. Proposition 2. [3] For a cycle Cn of order n ≥ 3, γcerp(Cn) = { n 3 , if n ≡ 0 (mod 3), n, otherwise. Proposition 3. [3] For a complete graph Kn of order n, γcerp(Kn) = { 1, if n = 1 or n ≥ 3, 2, if n = 2. Proposition 4. [3] For a complete bipartite graph Km,n of orders m+ n, γcerp(Km,n) =  1, if m = 1 or n = 1, 4, if m = n = 2, 2, otherwise. Proposition 5. If G is a connected graph, then each support vertex of G is contained in every certified perfect dominating set of G. Proof. Let J be a certified perfect dominating set of G. Assume, for the sake of contradiction, that there exists a support vertex a ∈ V (G) such that a /∈ J . Since a is a support vertex, there must be at least one leaf b ∈ V (G) with NG(b) = {a}. As a /∈ J , it follows that b ∈ J . However, this implies that b has exactly one neighbor in V (G) \ J , say a, contradicting the definition of a certified perfect dominating set. Thus, every support vertex of G is included in every certified perfect dominating set of G. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 5 of 13 Proposition 6. Let k be a positive integer and G be a graph of order n. If the strong support vertices of G are collectively adjacent to k leaves, then γcerp(G) ≤ n− k. Moreover, γcerp(G) ≤ n− 2|SS |. Proof. Let LG be the set of all leaves adjacent to strong support vertices in G, where |LG| = k. Consider the set V (G) \ LG. Since each leaf in LG is adjacent to exactly one strong support vertex, the set V (G) \ LG forms a certified perfect dominating set of G. Therefore, γcerp(G) ≤ |V (G) \ LG| = n− k. Moreover, each strong support vertex in SS is adjacent to at least two leaves, implying k ≥ 2|SS |. Hence, it follows that γcerp(G) ≤ n− 2|SS |. Theorem 1. Let G be a graph. Then for every maximum certified independent perfect set J of G is a certified perfect dominating set of G. Moreover, γcerp(G) ≤ αcerp(G). Proof. Let G be a graph and J a maximum certified independent perfect set of G. Since J is an independent set, for any a, b ∈ J , we have dG(a, b) ̸= 1. Furthermore, because J is a certified independent perfect set, it follows that dG(a, b) ̸= 2 for all a, b ∈ D. Now, let y ∈ V (G) \ J . Since J is a maximum-size certified independent perfect set, y must be dominated by at least one vertex in J . Hence, there exists some z ∈ J such that y ∈ NG(z). Therefore, NG[J ] = V (G), which implies that J is a dominating set of G. Additionally, since J is an independent perfect set, each vertex in V (G) \ J is dominated by exactly one vertex in J . Thus, J is an independent perfect dominating set of G. Given that J is also a certified independent perfect set, it follows that J is a certified perfect dominating set of G. Let J be a maximum certified perfect independent set of G. Therefore, |J | = αcerp(G). Since J is also a certified perfect dominating set, we have γcerp(G) ≤ αcerp(G). This completes the proof. The next result shows the relationship between certified domination and certified per- fect domination parameters. Remark 1. Every certified perfect dominating set of G is also a certified dominating set of G. Hence, γcer(G) ≤ γcerp(G). Theorem 2. Let a and b be positive integers with 3 ≤ a ≤ b. Then there exists a connected graph G such that γcer(G) = a and γcerp(G) = b. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 6 of 13 Proof. For a = b, consider G = Ka. In this case, γcer(G) = a and γcerp(G) = a, so γcer(G) = γcerp(G). For a < b, we examine the following cases: Case 1. a is odd. Let m = b − a, and consider the graph G as illustrated in Figure 1. Define J1 = {x1, x2, . . . , xa} and J2 = {x1, x2, . . . , xa, y1, y2, . . . , ym}. Then, J1 and J2 are the γcer-set and γcerp-set of G, respectively. Therefore, we have γcer(G) = |J1| = a and γcerp(G) = |J2| = a+m = b. x1 x2 x3 x4 x5 x6 xa−4 xa−3 xa−2 xa−1 xa y1 y2 y3 y4 y5 y1 ym−5 ym−4 ym−2 ym−1 ym G : . . . ya−3 Figure 1: A graph G with γcer(G) < γcerp(G). Case 2. a is even. Let m = b − a, and consider the graph G′ as shown in Figure 2. Define J ′ 1 = {x1, x2, . . . , xa} and J ′ 2 = {x1, x2, . . . , xa, y1, y2, . . . , ym}. Then, J ′ 1 and J ′ 2 are the γcer-set and γcerp-set of G, respectively. Consequently, γcer(G) = |J ′ 1| = a and γcerp(G) = |J ′ 2| = a+m = b. x1 x2 x3 x4 x5 x6 xa−4 xa−5 xa−2 xa−1 xa y1 y2 y3 y4 y5 y1 ym−5 ym−4 ym−2 ym−1 ym G′ : . . . ya−3 . . . xa−3 z1 z2 zm Figure 2: A graph G′ with γcer(G ′) < γcerp(G ′). This completes the proof. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 7 of 13 The following result is a direct consequence of Theorem 2. Corollary 1. For any non-negative integer c, there exists a connected graph G such that γcerp(G)− γcer(G) = c. In other words, the difference between γcerp(G) and γcer(G) can be made arbitrarily large. We will now determine the small and large values of graph G. Theorem 3. Let G be a graph with order n ≥ 1. Then the following hold: (i) 1 ≤ γcerp(G) ≤ n; (ii) γcerp(G) = 1 if and only if γcer(G) = 1; (iii) If γcerp(G) = 2, then γcer(G) = 2. However, the reverse is not necessarily true. (iv) If γcer(G) = n, then γcerp(G) = n. However, the reverse is not necessarily true. Proof. (i) Let G be a graph with |V (G)| = n. Since the empty set is not a certified dominating set of G, we conclude that γcer(G) ≥ 1. Additionally, note that any certified perfect dominating set J is a subset of the vertex set of G, i.e., J ⊆ V (G). Thus, we have γcerp(G) ≤ |V (G)| = n. Therefore, 1 ≤ γcerp(G) ≤ n. (ii) Assume that γcerp(G) = 1. Then J = {x} is the minimum certified perfect dom- inating set of G. This implies that J is also a minimum certified dominating set of G, so γcer(G) = |J | = 1. Conversely, suppose γcer(G) = 1. Then J ′ = {x′}. For every y′ ∈ V (G) \ J ′, the only vertex dominating y′ is x′ ∈ J ′. Hence, J ′ is a perfect dominating set of G, which implies γcerp(G) = |J ′| = 1. (iii) Assume that γcerp(G) = 2. Then J = {x, y} is the minimum certified perfect dominating set of G. Since every certified perfect dominating set is also a certified domi- nating set, we conclude that γcer(G) = 2. However, the reverse does not always hold. For example, consider G = C5. In this case, γcer(C5) = 2, but γcerp(C5) = 5. (iv) Suppose γcer(G) = n. Since γcerp(G) ≥ γcer(G) and γcerp(G) ≤ n, it follows that γcerp(G) = n. To show that the reverse is not necessarily true, consider G = P5. In this case, γcerp(P5) = 5, but γcer(P5) = 2. In what follows the following result will be used. Theorem 4. [3] Let G be a graph consisting of components G1, G2, . . . , Gk, where k ≥ 2. Then γcerp(G) = k∑ i=1 γcerp(Gi). Proposition 7. Let G = Kn be the complement of a complete graph of order n. Then γcerp(Kn) = n. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 8 of 13 Proof. Let G = Kn be the complement of a complete graph of order n. Then G consists of n isolated vertices. Since every vertex is an isolated, each component is a trivial graph K1. From [3], we know that γcerp(K1) = 1 for each component. Since G has exactly n such components, Theorem 4 implies that γcerp(G) = n∑ i=1 γcerp(K1) = n. Thus, we conclude that γcerp(G) = γcerp(Kn) = n. As previously noted, for any graph G with order n, we have γcerp(G) ≤ n and γcerp(G) ̸= n − 1. Furthermore, there exist graphs such as a path Pn, a cycle Cn for n ̸≡ 0 (mod 3) (see [3]), the complement of a complete graph on n vertices, where γcerp(G) = n. This motivates the investigation into the characterization of all graphs for which γcerp(G) = n, which is addressed in this section. Recall that F. Harary (see [9]) defined the corona of two graphs G and H, denoted by G ◦ H, is constructed by taking one copy of G and |V (G)| copies of H, then connecting each vertex i of G to every vertex in the corresponding ith copy of H. For each v ∈ V (G), let Hv represent the copy of H whose vertices are individually linked to v. The subgraph of the corona G ◦ H associated with the join ⟨{v}⟩ + Hv is denoted by v + Hv, where v ∈ V (G) [9]. In particular, when H is a single-vertex graph, H = K1, the corona G ◦K1 is referred to as the corona of G. Theorem 5. [3] If G is a connected graph with |V (G)| = m, and H is a trivial graph, then γcerp(G ◦H) = 2m. Corollary 2. If G is the corona of a graph with |V (G)| = n, then γcerp(G) = n. Proof. Let D be a γcerp-set of G. Let G be the corona of some graph. Then G = H ◦k1 for some graph H. By Theorem 5, |D| = γcerp(G) = γcerp(H ◦K1) = 2m = |V (G)| = n. Therefore, γcerp(G) = n. Remark 2. Let G be a graph with |V (G)| = n. If G is either the corona of a graph, the complement of a complete graph, or the union of these two types, then γcerp(G) = n. Remark 3. It is important to observe that the result above indicates the sharpness of the upper bound in the inequality γcerp(G) ≤ 2γ(G), because for the corona G of any graph that does not contain an isolated vertex, we have γcerp(G) = 2γ(G). 4. Graphs with γcerp(G) = γcer(G) We proceed with our investigation of the certified perfect domination number by ex- amining the class of graphs where γcerp(G) = γcer(G). J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 9 of 13 Theorem 6. Let G be a connected graph with |V (G)| ≥ 3. Then γcer(G) = γcerp(G) if and only if there exists a γcer-set J of G such that each vertex u ∈ V (G) \ J is dominated by exactly one vertex v ∈ J . Proof. Assume that γcer(G) = γcerp(G). Let J be a γcerp-set of G. Since J is a γcerp-set and γ(G) = γcer(G), it follows that J is also a γcer-set of G. Now, for the sake of contradiction, assume that there exists a vertex u ∈ V (G) \ J that is dominated by at least two distinct vertices v1, v2 ∈ J . In this case, J would not be a certified perfect dominating set, which contradicts our assumption that J is a γcerp-set. Therefore, every vertex u ∈ V (G) \ J is dominated by exactly one vertex in J . Conversely, suppose that J is a γcer-set of G such that each vertex u ∈ V (G) \ J is dominated by exactly one vertex in J . For contradiction, assume that there exists a vertex v ∈ J such that |NG(v) ∩ (V (G) \ J)| = 1. This implies that the unique neighbor of v in V (G)\J contradicts our assumption that every vertex in V (G)\J is dominated by exactly one vertex in J . Hence, we conclude that |NG(v) ∩ (V (G) \ J)| ≥ 2 for all v ∈ J , meaning that J is a certified perfect dominating set. Therefore, γcerp(G) ≤ |J | = γcer(G). By Remark 1, we also have γcer(G) ≤ γcerp(G). Thus, γcer(G) = γcerp(G). Corollary 3. Let G be a connected graph with |V (G)| ≥ 3. If G has a γip-set that does not include any leaf of G, then γcer(G) = γcerp(G). Proof. Let J be a γip-set of G that does not contain any leaf of G. Since J is in- dependent, every vertex v ∈ J has all of its neighbors in V (G) \ J . This implies that NG(v) ⊆ V (G) \ J . Furthermore, since v is neither a leaf nor an isolated vertex, we have |NG(v)| ≥ 2. Thus, |NG(v)∩ (V (G) \ J)| = |NG(v)| ≥ 2. Therefore, J is a certified perfect dominating set. By Theorem 6, we conclude that γcer(G) = γcerp(G). The following result immediately follows from Theorem 6 and Corollary 3. Corollary 4. Let G be a graph where δ(G) ≥ 2. If G contains a γp-set J such that each vertex in J has at least two neighbors in V (G) \ J , then γcerp(G) = γcer(G). 5. Lexicographic Product of Two Graphs In this section, we characterize the lexicographic product of two graphs and determine its certified perfect domination number. Recall that F. Harary (see [9]) defined the Lexicographic product or composition of two graphs G and H is the graph G[H] with vertex set V (G[H]) = V (G)×V (H) and edge set E(G[H]) satisfying the following condition: (x, u)(y, v) ∈ E(G[H]) if and only if xy ∈ E(G) or x = y and uv ∈ E(H). Any subset C of V (G[H]) can be expressed as Q = ⋃ x∈S{x} × Tx, where S ⊆ V (G) and Tx ⊆ V (H) for each x ∈ S. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 10 of 13 Theorem 7. Let G and H be connected nontrivial graphs. A subset Q = ⋃ u∈J [ {u}×Tu ] of V (G[H]), where J ⊆ V (G) and Tu ⊆ V (H) for each u ∈ J , is a certified perfect dominating set of G[H] if and only if J is a certified independent perfect dominating set of G and Tu is a dominating set of H satisfying |Tx| = 1 for all u ∈ J . Proof. Suppose that Q is a certified perfect dominating set of G[H]. Let a ∈ V (G) \ J and select any b ∈ V (H). Since Q is a perfect dominating set, there exists a vertex (y, z) ∈ Q such that NG[H](a, b) ∩ Q = {(y, z)}. This implies that NG(a) ∩ J = {y}, meaning that each vertex a ∈ V (G)\J is dominated by exactly one vertex in J . Therefore, J is a perfect dominating set in G. Next, assume that there exist vertices r, s ∈ J such that rs ∈ E(G). Since H is connected, there exist vertices k, l ∈ V (H) such that kl ∈ E(H), and consequently, (r, k)(s, k) ∈ E(G[H]) with both (r, k) and (s, k) belonging to Q. This leads to a contradiction, as both (r, l) and (s, l) are dominated by (r, k) and (s, k) in Q. Thus, rs /∈ E(G) for any r, s ∈ J , meaning that J is an independent set. Therefore, J is an independent perfect dominating set in G, and hence a certified independent perfect dominating set of G since NG(J) ≥ 2. Now, let u ∈ J , and suppose that |Tu| ≥ 2. Choose distinct c, d ∈ Tu, so that both (u, f) and (u, g) belong to Q. Consider an adjacent vertex h ∈ V (G) such that uh ∈ E(G). Since J is independent, we have h /∈ J . Clearly, (h, f) /∈ Q, and it must be dominated by both (u, f) and (u, g) in Q, which contradicts the assumption that Q is a perfect dominating set. Therefore, |Tu| = 1 for each u ∈ J . Furthermore, since J is an independent perfect dominating set of G and Q is a dominating set in G[H], it follows that each Tu is a dominating set in H, with |Tu| = 1 for each u ∈ J . Conversely, suppose that J is a certified independent perfect dominating set in G and that Tu is a dominating set of H with |Tu| = 1 for all u ∈ J . Let Tu = {w} for each u ∈ J , where w is the vertex in H that dominates all other vertices of H. Consider an arbitrary vertex (r, s) ∈ V (G[H]) \Q, and consider the following cases: Case 1: r /∈ J . Since J is a perfect dominating set, there exists u ∈ J such that NG(r) ∩ J = {u}. Hence, NG[H](r, s) ∩ Q = {(u,w)}. Therefore, (r, s) is dominated by exactly one vertex in Q, and thus Q is a perfect dominating set. Case 2: r ∈ J . Since Tu = {w} for each u ∈ J and J is a perfect dominating set, we have NG[H](r, s) ∩ Q = {(r, w)}, meaning that (r, s) is dominated by exactly one vertex (r, w) in Q. Therefore, Q is a perfect dominating set. Finally, since J is a certified independent perfect dominating set in G and Tu is a dominating set of H with |Tu| = 1 for all u ∈ J , it follows that Q is a certified perfect dominating set of G[H]. This completes the proof. The following results immediately follow from Theorem 7. Corollary 5. If G and H are connected graphs. Then γcerp(G[H]) =  γcerp(G), if H = K1, γcerp(H), if G = K1, γcerp(G), if G contains γcerip − set and γ(H) = 1. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 11 of 13 Corollary 6. Let G and H be connected graphs. Then G[H] is a non-γcerp-graph if and only if one of the following conditions holds: (i) γ(H) ≥ 2 (ii) γ(H) = 1 and G does not possess a certified independent perfect dominating set. The following result is a direct consequence of Corollary 6 (ii). Corollary 7. Let G be a connected graph that does not have a γcerip-set, and let H be any graph. Then G[H] is a non-γcerp-graph. 6. Cartesian Product of Two Graphs Recall that F. Harary (see [9]) defined the Cartesian product G□H of two graphs G and H is a graph with the vertex set V (G□H) = V (G)× V (H). Two vertices (ui, vj) and (uk, vl) are adjacent in G□H if and only if one of the following conditions holds: (i) ui = uk and vjvl ∈ E(H), (ii) vj = vl and uiuk ∈ E(G). Proposition 8. Let G be a graph with γcerp(G) = 1. Then γcerp(G□K1) = 1. Proof. Let G be a graph with γcerp(G) = 1. Since G□K1 = G, it follows that γcerp(G□K1) = 1. Theorem 8. Let p ≥ 2 be an integer. Then Kp□Kp is a non-γcerp-graph. Proof. Consider the complete graph Kp with p vertices. The graph Kp□Kp has p2 vertices. For the sake of contradiction, assume that Kp□Kp is a γcerp-graph. Let (u, v) be any vertex in V (Kp□Kp), where u is a vertex in the first Kp and v is a vertex in the second Kp. The vertex (u, v) is adjacent to p − 1 vertices of the form (u, vi), where vi is one of the p− 1 vertices of the second Kp. Therefore, (u, v) dominates 2(p− 1) vertices in Kp□Kp, and so (u, v) must be included in the dominating set J . Now, we need to select additional vertices for J from the remaining p2 − 2(p − 1) vertices. Without loss of generality, assume that (u, v) is among these remaining vertices. Since (u, v) is adjacent to at least one vertex that is already dominated by another vertex in J , this contradicts the assumption that J is a perfect dominating set. Thus, no certified perfect dominating set exists in Kp□Kp, which implies that Kp□Kp is a non-γcerp-graph for all p ≥ 2. Theorem 9. Let m ≥ 3, n ≥ 2 be integers. Then Km□Pn is a non-γcerp-graph. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 12 of 13 Proof. Let V (Km) = {x1, x2, . . . , xm} and V (Pn) = {y1, y2, . . . , yn}, so that Km□Pn consists of mn vertices. Assume, for the sake of contradiction, that Km□Pn is a γcerp- graph. Without loss of generality, suppose (k1, p1) ∈ J . Then (k1, p1) dominates both (k1, p2) and (kj , p1) for some j = 2, 3, . . . ,m. Note that (ki, p2) /∈ J , as (ki, p1) is adjacent to (ki, p2) for some i = 1, 2, . . . ,m. In this scenario, (ki, p3) could dominate (ki, p2), but (k1, p2) is already dominated by (k1, p1), violating the condition for perfect domination. This contradiction implies that no certified perfect dominating set exists in Km□Pn, concluding that Km□Pn is not a γcerp-graph. The next result is a direct consequence of Theorems 8 and 9. Corollary 8. For every even integer n, K2□Pn is a non-γcerp-graph. Proposition 9. Let n be a positive odd integer. Then γcerp(K2□Pn) = ⌈n 2 ⌉ . Open Questions and Problems: Problem 1. Characterize the trees T , certain classes of graphs, and other binary oper- ations that have not yet been studied, and determine their certified perfect domination numbers. Problem 2. Establish Nordhaus-Gaddum type results for γcerp(G). Problem 3. Investigate how the certified perfect domination number is affected when a graph is modified by adding or removing an edge or vertex. Conclusion: In this paper, we extended the study of certified perfect domination in graphs by presenting new upper bounds for the certified domination number γcerp(G). We explored the relationship between certified domination and certified perfect domination parameters, identifying graphs with both small and large values of these parameters. Our results provide a deeper understanding of the structural properties of graphs in the context of certified perfect domination. We also characterize graphs for which γcerp(G) = n and γcerp(G) = γcer(G), offering insight into special classes of graphs where the certified perfect domination number attains these specific values. Furthermore, our investigation into the lexicographic and Cartesian products of graphs revealed the behavior of certified perfect dominating sets under these operations. We determined the certified perfect domination number for such products and identified non-γcerp-graphs arising from these binary operations. Acknowledgements The authors sincerely thank the reviewers for their valuable comments and sugges- tions that significantly improved the quality of this paper. Gratitude is also extended to Mindanao State University – Tawi-Tawi College of Technology and Oceanography (MSU- TCTO) for the financial support that made the publication of this work possible. J. J. Hamja et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6076 13 of 13 References [1] M. Dettlaff, M. Lemańska, J. Topp, R. 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