EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6123 ISSN 1307-5543 – ejpam.com Published by New York Business Global Connected Degree Equitable Domination in Graphs Hearty M. Nuenay- Maglanque1 1 Department of Applied Mathematics, College of Science and Mathematics, University of Science and Technology of Southern Philippines, 9000 Cagayan de Oro City, Philippines Abstract. Let G be a connected graph. A subset S ⊆ V (G) is an equitable dominating set in G if for each vertex not in S there exists u ∈ S such that uv ∈ E(G) and |degG u − degG v| ≤ 1 . An equitable dominating set S ⊆ V (G) is called a connected equitable dominating set of G if the subgraph ⟨S⟩ induced by S is connected . The minimum cardinality of such connected equitable dominating sets in G is called the connected equitable domination number of G and is denoted by γce(G). This paper investigates the connected equitable domination in the join and corona of graphs. The connected equitable dominating sets in the join and corona of graphs are characterized and, as direct consequences, the connected equitable domination numbers of these graphs are obtained. In addition, a n exact value of some families of graphs and a realization problem are established. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Equitable dominating set, connected equitable dominating set, equi- table domination number, connected equitable domination number 1. Introduction Throughout this paper, we consider simple, finite and undirected graphs G = (V (G), E(G)). For a subset S ⊆ V (G), the symbol |S| refers to the cardinality of S. In particular, |V (G)| is the order of G. Let G be a connected graph. If the pair e = [u, v] is in E(G), then e is an edge in G, and e is said to join u and v. In this case, it is customary to write e = uv and say that u and v are adjacent, while u and e are incident, as v and e are. The degree of a vertex v ∈ V (G), denoted by degG(v), is the number of incident edges to v. Adjacent vertices are also called neighbors. For v ∈ V (G), the open neighborhood of v is the set NG(v) consisting of all vertices adjacent to v. Thus, degG(v) = |NG(v)| . The maximum and minimum degree of G are denoted by ∆(G) and δ(G), that is, ∆(G) = max{degG(u) : u ∈ V (G)} and δ(G) = min{degG(u) : u ∈ V (G)}. The subgraph ⟨S⟩ of G induced by a subset S of V (G) is the graph having vertex set S DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6123 Email addresses: hearty.maglanque@ustp.edu.ph (H. Nuenay-Maglanque) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 2 of 14 and whose edge set consists of those edges of G incident with two elements of G. A graph is called connected if every two vertices are joined by a path; otherwise, it is disconnected. In graph theory, domination in graphs is one of the most extensively studied concepts in graph theory with numerous variants that model influence, control and coverage, in networked systems. We may refer to [1–5] to see some of the recent studies in domination, and to [6–8] for its various applications, such as Electrical Power Networks, Communi- cation Network, Facility Location Problem, Land Surveying, and Routings. A subset S ⊆ V (G) is a dominating set of G if for every v ∈ V (G) \ S, there exists u ∈ S such that uv ∈ E(G). The domination number of G, denoted by γ(G), is the smallest cardi- nality of a dominating set of G. Traditional domination parameters focus on identifying minimal sets of vertices that dominate or control the entire graph structure. However, in many real-world contexts—such as sensor networks, facility location, and communication systems—there is often a need to ensure both fairness in vertex degrees and connectivity among dominating vertices. To address this, the notion of equitable domination has been introduced. We may refer to [9–13] to see some of the studies in equitable domination. A dominating set S ⊆ V (G) is called a (degree) equitable dominating set in G if for every v ∈ V (G) \ S there exists u ∈ S such that |degG(u) − degG(v)| ≤ 1. The equitable dom- ination number of G, denoted by γe(G), is the smallest cardinality among all equitable dominating set in G. The notion of equitable domination imposed a balance condition, that is, the sizes of the open neighborhoods of the dominating vertices must be as equal as possible, minimizing the difference in coverage. Some of the studies Among the many variations of domination is the connected domination which requires that the subgraph ⟨S⟩ induced by dominating set S is connected, reflecting situations where communication or cohesion among controlling elements is essential. We may refer to [14, 15] to see some of the studies in connected domination. Combining these two constraints ”connectedness” and equitability or fairness gives rise to the concept of connected equitable domination. The notion of connected equitable domination has been introduced where dominating sets are not only degree-balanced but also induce connected subgraphs. An equitable domi- nating set S ⊆ V (G) is called a connected equitable dominating set of G if the subgraph ⟨S⟩ induced by S is connected . The minimum cardinality among all connected equitable dominating sets in G is called the connected equitable domination number of G , and is denoted by γce(G). A subset S ⊆ V (G) is called minimal connected equitable dominating set in G if no proper subset of S is a connected equitable dominating set in G. The concept of connected equitable domination has many real life applications where both connectivity (connectedness) and fairness (equitability) in workload distribution or influence distribution are essential. Such theoretical and practical applications includes (1) Distributed Computing and Load Balancing: In distributed systems, assigning tasks to processing units can be modeled as a connected equitable dominating set which en- sures that workload distribution is equitable, that is, nearly equal, and all coordinating vertices can communicate efficiently; (2) Social Networks and Influence Modeling: In sce- narios involving opinion formation or influence spread, a connected equitable dominating set can represent a group of influential individuals who are both interconnected and exert roughly equal influence over the population; (3) Urban Planning and Facility Location: H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 3 of 14 When placing essential facilities such as hospitals, fire stations and etc. in a city modeled as a graph, a connected equitable dominating set ensures that all regions are covered by a connected and fairly distributed set of facilities. By combining the structural benefits of connectedness with the fairness of equitable load distribution, connected equitable dom- ination provides a robust and efficient framework for designing and analyzing network systems; and (4) Telecommunication and Sensor Networks: In wireless sensor networks or ad hoc communication networks, a connected equitable dominating set can represent as a core framework where central vertices (dominating vertices) are connected and share the communication or monitoring load equitably. This ensures both robustness (connectivity) and balanced energy usage or data handling. In this paper, we investigate this parameter in graphs formed through two fundamental operations: the join and the corona. We provide structural characterizations of such dominating sets in these graphs and determine exact values of γce(G) for various graph families. A realization problem is also addressed, offering insight into how graphs can be constructed to attain prescribed values of the parameter. For simplicity, we use the terms γce-set, γc-set, and γe to refer to the connected equi- table dominating set with cardinality γce(G), connected dominating set with cardinality γc(G) and equitable dominating set with cardinality γe(G), respectively. The following results are due to V. Swaminathan et.al and Sivakumar et.al Theorem 1. [15] The connected equitable domination number of some standard graphs are (i) For a complete graph Kn on n vertices, γce(G) = 1. (ii) For the paths Pn and the cycles Cn on n vertices, γce(Cn) = γce(Pn) = n− 2. (iii) If Wn denotes the wheel on n vertices, then γce(Wn) = { ⌈ n+3 3 ⌉ , if n ≥ 6 1, otherwise. (iv) For the complete bipartite graph Km,n, we have, γce(Km,n) =  1, either m = 1 or n = 1 2, if |m− n| ≤ 1 and m,n ≥ 2 m+ n, if |m− n| ≥ 2, and m,n ≥ 2. 2. Results The following result Theorem 2 is a correction of Theorem 1 (iv) found in [15]. Theorem 2. For any complete bipartite Km,n with m,n ≥ 1, γce(Km,n) = { 2, if |m− n| ≤ 1 m+ n, if |m− n| ≥ 2. H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 4 of 14 Proof. Let Km,n be a complete bipartite graph with m vertices in one partition say A and n vertices in another partition say B. Then, deg(u) = { n, if u ∈ A m, ifu ∈ B. If |m − n| ≤ 1, then any set {x, y} with x ∈ A and y ∈ B is a connected equitable dominating set in Km,n. Thus, γ ce(Km,n) = 2. Suppose that |m − n| ≥ 2. Let S be a minimum connected equitable dominating set in Km,n and suppose that |S| < m + n. Then, there exists u ∈ V which is not in S. WLOG, let u ∈ B. Then, degG(u) = m. Since S is a connected equitable dominating set in Km,n, there exists v ∈ S such that u is adjacent with v and |degG(u) − degG(v)| ≤ 1. Clearly, v ∈ A. Therefore degG(v) = n. Thus, |degG(u) − degG(v)| = |m − n| ≥ 2, a contradiction . Therefore, |S| = m + n. Consequently, γce(Km,n) = m+ n if |m− n| ≥ 2. Theorem 3. For a fan graph Fn with n vertices on a path and a single vertex u connected to each vertex in Pn, γce(Fn) =  1, if n =: 2, 3 2, if n = 4⌈ n 3 ⌉ + 1, if n ≥ 5. Proof. Let G = Fn be a fan with n vertices on a path and a single vertex u connected to each vertex in Pn. Let V (Fn) = {u, v1, v2, . . . , vn} where u is adjacent to every vertex vi (1 ≤ i ≤ n) on path Pn. When n = 2, {u}, {v1} and {v2} are minimal connected equitable dominating sets in G. Thus, γce(Fn) = 1. When n = 3, since degG(u) = degG(v2) = 3 and degG(v1) = degG(v3) = 2, the sets {u} and {v2} are the only minimal connected equitable dominating sets in G. Thus, γce(Fn) = 1. For n = 4, since the deg(vi) of any vertices vi lying on P4 is either 2 or 3 and degG(u) = 4, any connected dominating set in P4 is equitable in F4. Thus, γc(P4) = ⌈ n 3 ⌉ = 2 = γce(F4). Suppose that n ≥ 5. Then deg(u) = n and degG(vi) = { 2, if i =: 1, n 3, if 2 ≤ i < n. Thus, any dominating set in Pn is not equitable in Fn since |degG(u) − degG(vi)| > 2 for all i = 1, 2, . . . , n. Now, consider S = D ∪ {u} where D is a γe- set in Pn. Clearly, S is a connected equitable dominating set in Fn of minimum cardinality. Therefore, γce(Fn) = γe(Pn) + 1 = ⌈ n 3 ⌉ + 1. Theorem 4. For any double star graph Sr,s, where r, s ≥ 1, γce(Sr,s) =  2, if r = s = 1 s+ 2, if r = 1 and s ≥ 2 r + 2, if s = 1 and r ≥ 2 r + s+ 2, if r, s ≥ 2 Proof. Let u and v be the two central vertices of G = Sr,s, and let U and V be the sets of all leaves adjacent to u and v, respectively, with |U | = r and |V | = s. Then H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 5 of 14 NG({u}) = {v} ∪ U and NG({v}) = {u} ∪ V .If r = s = 1, then the only connected equitable dominating set in G is the set {u, v}. Thus γce(G) = 2. Suppose that r = 1 and s ≥ 2. Since every y ∈ U ∪V is of degree 1 and degG(u) = 2, the only connected equitable dominating set in G is the {u, v}∪V . Thus, γce(G) = s+2. Similarly, if s = 1 and r ≥ 2, γce(G) = r + 2. Suppose that r, s ≥ 2. Let S be a connected equitable dominating set in G with |S| = γce(G). Suppose that |S| < r + s + 2 = |V (G)| and let x ∈ V \ S. Then, either x ∈ {u, v}, x ∈ U , or x ∈ V . Suppose that x is any of the central vertices of G, say u. This means that there exists uj ∈ U which is an element of S, a contradiction to the assumption that S is connected. Thus, x must not be any of the central vertices. Now, either x ∈ U or x ∈ V . Without loss of generality, let x ∈ U . Since S is a connected equitable dominating set in G, there exists y ∈ S such that xy ∈ E(G) and |degG(x)−degG(y)| ≤ 1. Note that the only adjacent vertex of x in G is the central vertex u. Thus, y = u and |degG(x) − degG(y)| = |degG(x) − degG(u)| = |1 − (r + 1)| = r > 1, a contradiction. Therefore, S = V (G). Hence, γce(G) = |S| = r + s+ 2. Theorem 5. γce(G) = 1 if and only if there exists v ∈ V (G) such that degG(v) = |V (G)| − 1 and degG(u) ≥ |V (G)| − 2 for all u ̸= v in G. Proof. Suppose that γce(G) = 1 . Then, there exists v in G such that {v} is a connected equitable dominating set in G with degG(v) = |V (G)|− 1. Now, let u ∈ V (G) \ {v}. Since {v} is a γce-set in G, uv ∈ E(G) and | degG(u)− degG(v)| ≤ 1. That is, |degG(u)− degG(v)| ≤ 1 =⇒ | degG(u)− (|V (G)| − 1) ≤ 1 =⇒ −1 ≤ degG(u)− |V (G)|+ 1 ≤ 1 =⇒ |V (G)| − 2 ≤ degG(u) ≤ |V (G)| Thus, degG(u) ≥ |V (G)| − 2 for all u ̸= v in G. Conversely suppose that there exists v ∈ V (G) such that degG(v) = |V (G)| − 1 and degG(u) ≥ |V (G)| − 2 for all u ̸= v in G. Then, {v} is a dominating set and | degG(v)− degG(u)| ≤ |V (G)| − 1− (|V (G)| − 2) ≤ 1 for all u ̸= v in G. Thus, {v} is a connected equitable dominating set in G. Consequently, γce(G) = 1. Remark 1. If a subset S ⊆ V (G) is a connected equitable dominating set in G and |degG(u)− degG(v)| ≥ 2 for all v ∈ NG(u), then u ∈ S. Remark 2. Every equitable dominating set is a dominating set. Thus, γ(G) ≤ γe(G). Remark 3. Every connected dominating set is a dominating set. Thus, γ(G) ≤ γc(G). Remark 4. Every equitable connected dominating set is a connected dominating set. Thus , γc(G) ≤ γce(G). Theorem 6. For any graph G, γ(G) ≤ γc(G) ≤ γce(G). H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 6 of 14 Proof. Let S be a γce-set in G. Then S is a connected dominating set in G. Thus, by Remark 4, γc(G) ≤ |S| = γce(G). Furthermore, S is a dominating set in G. By Remark 3, the result follows. The next result shows that every pair of positive integers are realizable as the connected domination number and connected equitable domination number of a connected graph. Theorem 7. For every positive integers a and b with 1 ≤ a ≤ b there exists a connected graph G such that γc(G) ≤ γce(G). Proof. Suppose that a = b. Write V (K1,a−1) = {x, u1, u2, . . . , ua−1} as in Figure 1. Obtain G from K1,a−1 by adding pendant edges vjuj , j = 1, 2, . . . , a−1 as shown in Figure 1. .................................... .................................... .................................... .................................... ........................................................................ ..................................................................................................................................................................... ..................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ..... .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........................................................................................................................ . . . ua−1 u1 u2u3 u4 x K1,a−1 .................................... .................................... .................................... .................................... ........................................................................ ..................................................................................................................................................................... ..................................... ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ..... .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ........................................................................................................................ . . . .................................... ........................................................................ .................................... .................................... ........................................................... ........................................................... ......... ........ ........ ........ ........ ........ ........ .. ............................................................. ......... ......... ......... ......... ... ua−1 x va−1 u1 v1 u2 v2 u3 v3 u4v4 Figure 1: G obtained from K1,a−1 Observe that the set S = {x, u1, u2, . . . , ua−1} is the only minimal connected equitable dominating set, and a minimal connected dominating set of G. Thus, γc(G) = |S| = a− 1 + 1 = a = b = γce(G). Suppose that b = a + 1. Obtain G = G1 from K1,a−1 by adding pendant edges vjuj , j = 2, . . . , a− 1, and joining the path P3 = [w1, v1, h1] to vertex u1 of K1,a−1 as in Figure 2. Observe that the set {x, u1, u2, . . . , ua−1} is the only minimal connected dominating set of G. Thus γc(G) = a−1+1 = a. Also, note that the sets S1 = {x, u1, u2, . . . , ua−1, v1} and S2 = {x, u1, u2, . . . , ua−1, w1, h1} are the only minimal connected equitable dominating sets of G. Since S1 = a − 1 + 1 + 1 = a + 1 and |S2| = a − 1 + 1 + 2 = a + 2, we have γce(G) = |S1| = a− 1 + 1 + 1 = a+ 1 = b = γce(G). Suppose that b = a + k for k ≥ 2. Obtained G = G2 from G1 by joining k − 1 path P3j = [wj , vj , hj ] to vertex u1 of G1 where 2 ≤ j ≤ k as in Figure 3. Observe that the set {x, u1, u2, . . . , ua−1} is the only minimal connected dominating set of G. Thus γc(G) = a−1+1 = a. Also, note that the sets S1 = {x, u1, u2, . . . , ua−1}∪( k⋃ j=1 {vj} ) and S2 = {x, u1, u2, . . . , ua−1}∪ ( k⋃ j=1 {wj , hj} ) are the only minimal connected H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 7 of 14 .................................... .................................... .................................... .................................... ........................................................................ ................................................................................................................................................................ ................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ . .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .............................................................................................................. . . . ua−1 u1 u2u3 u4 x K1,a−1 .................................... .................................... .................................... .................................... ........................................................................ ................................................................................................................................................................ ................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ . .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .............................................................................................................. . . . .................................... ........................................................................ .................................... .................................... ........................................................ ........................................................ ......... ........ ........ ........ ........ ........ ....... ........................................................... ......... ......... ......... ......... .................................... .................................... .................................................................................................................................. ........ ........ ....... ........ ........ ........ ......... ......... .......... ............ ............................................................................................................................. ua−1 x va−1 u1 v1 u2 v2 u3 v3 u4v4v4 w1 h1 G = G1 Figure 2: G1 obtained from K1,a−1 .................................... .................................... .................................... .................................... ........................................................................ ................................................................................................................................................................ ................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ . .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .............................................................................................................. . . . .................................... ........................................................................ .................................... .................................... ........................................................ ........................................................ ......... ........ ........ ........ ........ ........ ....... ........................................................... ......... ......... ......... ......... .................................... .................................... .................................................................................................................................. ........ ........ ....... ........ ........ ........ ......... ......... .......... ............ ............................................................................................................................. ua−1 x va−1 u1 v1 u2 v2 u3 v3 u4v4v4 w1 h1 G1 .................................... .................................... .................................... .................................... ........................................................................ ................................................................................................................................................................ ................................ ......... ........ ........ ........ ........ ........ ........ ........ ........ ........ . .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .............................................................................................................. . . . .................................... .................................... .................................... .................................... ........................................................ .......................................................................................................................... ......... ........ ........ ........ ........ ........ ....... ........................................................... ......... ......... ......... ......... .................................... .................................... .................................... ............ ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ........... ....... ............. ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ .......... ........................................................ ......... ........ ........ ..... ......... ......... ......... ......... ........... ............................................................................................................................................................................................................................................................................................. .................................... .................................... .................................... ...................................................................................................................................................... ........................... ........................... ........................... .................. ........................................................ ......... ........ ........ ..... ......... ......... ........... .............. ............................................................................................................................................................................................ .................................... .................................... .................................... ................................................................................................................................................................................................... ................................................................................................................................................................................................................................... ........................................................ ......... ........ ........ ..... ......... ......... .......... .......... ........... ........... ............ ............ ............. ............. .............. .............. ............... ............... ................ ................ ................. ................. ............. ua−1 x va−1 u1 u2 v2 u3 v3 u4v4v4 v1 v2 vk h1 h2 hk w1 w2 wk ... G = G2 Figure 3: G2 obtained from G1 equitable dominating sets of G. Now, |S1| = a− 1+1+ k = a+ k and |S2| = a− 1+2k = a+ 2k − 1. Since k > 1, |S2| = a+ 2k − 1 > a+ k = |S1|. Thus, γce(G) = |S1| = a+ k = b > γc(G) = a. Theorem 8. For any regular graph G, γce(G) = γc(G). Proof. Let S ⊆ V (G) be a γc-set of G and u ∈ V (G) \ S. Since S is a dominating set in G, there exists v ∈ S such that uv ∈ E(G). Since G is a regular graph, say n regular, degG(x) = n for all x ∈ V (G). Thus, | degG(u)− degG(v)| = 0 < 1 implying that S is an equitable set in G. Thus S is a connected equitable dominating set in G. Hence, γce(G) ≤ |S| = γc(G). From Remark 4, γce(G) = γc(G). H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 8 of 14 3. The Join of Graphs 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)}. Theorem 9. Let G be a graph such that ∆(G) ≤ |V (G)− 3|. Then S ⊆ V (K1 +G) is a connected equitable dominating set of K1 +G if and only if S = V (K1) ∪ SG where SG is an equitable dominating set of G. Proof. Suppose S is a connected equitable dominating set of K1 + G. Since ∆(G) ≤ |V (G) − 3| and degK1+G(v) = |V (G)|, where v ∈ V (K1), it follows that v ∈ S. Let SG = V (G) ∩ S. Since {v} is not an equitable dominating set of K1 + G, SG ̸= ∅. Let u ∈ V (G) \ SG. Then ∃w ∈ S such that |degK1+G(w) − degK1+G(u)| ≤ 1. Since degK1+G(u) = 1 + degG(u) ≤ 1 + |V (G)| − 3 = |V (G)| − 2 and degK1+G(v) = |V (G)|, w ̸= v. Thus, w ∈ SG and |degG(w)− degG(u)| ≤ 1. This implies that SG is an equitable dominating set of G. The converse is clear. Corollary 1. Let G be a graph such that ∆(G) ≤ |V (G)| − 3. Then γce(K1 +G) = 1 + γe(G). Theorem 10. Let G be a graph such that ∆(G) ≥ n−2, where n = |V (G)|. Then S is an equitable connected dominating set of K1+G if and only if it satisfies one of the following statements: (i.) S ⊆ V (G) and is a connected equitable dominating set of G. (ii.) S = V (K1)∪SG such that SG ⊆ V (G) equitably dominates V (G) \ (DG ∪SG) where DG = {z ∈ V (G) \ SG : degG(z) ≥ n− 2}. Proof. Suppose S is a connected equitable dominating set of K1 +G. Let v ∈ V (K1) and set SG = V (G)∩S. Suppose first that v /∈ S. Then S ⊆ V (G). Since S is a connected equitable dominating set of K1+G, S is a connected equitable dominating set of G. Next, suppose that v ∈ S. Let z ∈ V (G)\DG. Then degG(z) ≤ n−3, that is, degK1+G(z) ≤ n−2. Since degK1+G(v) = n and S is a connected equitable dominating set ofK1+G, there exists y ∈ SG such that | degG(y)− degG(z)| ≤ 1. This implies that every element of V (G) \DG is equitably dominated by SG. For the converse, suppose that (i) holds. Then S contains a vertex u ∈ V (G) with degG(u) ≥ n−2. Hence, | degK1+G(v)−degK1+G(u)| ≤ 1. Further, since S is a connected equitable dominating set of G, S is a connected equitable dominating set of K1 +G. Next, suppose that (ii) holds. Let z ∈ DG. Then v equitably dominates z. Moreover, since SG equitably dominates V (G)\DG, S is a connected equitable dominating set of K1 +G. Corollary 2. Let G be a graph such that ∆(G) ≥ n− 2, where n = |V (G)|. Then γce(K1 +G) = min{γce(G), 1 + γe(⟨V (G) \D∗ G⟩)} where D∗ G = {z ∈ V (G) : degG(z) ≥ n− 2}. H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 9 of 14 Theorem 11. γce(G+K1) = 1 if and only if ∆(G+K1) ≤ degG(v) + 2 for all v in G. Proof. Let G a connected graph of order n. Let v1, v2, . . . , vn ne vertices in G. Suppose that γce(G + K1) = 1. Then there exists a vertex u in G + K1 which is an equitable dominating in G+K1. Consider the following cases: Case 1 : u ∈ V (K1). Then 1 ≥ |degG+K1 (u)− degG+K1 (vi)| = |n− (degG(vi) + 1)| = |(n− 1)− degG(vi)|. Thus, degG(vi) ≥ n − 2 for all i = 1, 2, . . . , n. Hence, ∆(G + K1) = degG+K1 (u) = n ≤ degG(vi) + 2 for all vi in G. Case 2 : If u ∈ V (G)., then |degG+K1 (u)− degG+K1 (v)| ≤ 1 for all v ̸= u in G+K1. This means that degG+K1 (u) ≤ degG+K1 (v) + 1. Thus, ∆(G+K1) = degG+K1 (u) ≤ degG+K1 (v) + 1 , v ∈ V (G+K1) \ {u} = degG(v) + 1 + 1 , v ̸= u in G = degG(v) + 2. For the converse, suppose that ∆(G+K1) ≤ degG(v) + 2 for all v in G. Let u in G+K1 be of maximum degree. If x ∈ V (K1), then degG+K1 (u) = ∆(G+K1) degG+K1 (v) + 1 for all v in G. That is, degG+K1 (v) ≥ degG+K1 (u)− 1 for all v in G. This implies that G is a (t, t+1) bi-regular graph. Take {u}. Observe that u is an equitable dominating in G+K1. Thus, γce(G +K1) = 1. Similarly, if u ∈ V (G) , degG+K1 (v) ≥ degG+K1 (u) − 1 for all v in G. Taking {u} which is a connected equitable dominating set in G + K1. Therefore, γce(G+K1) = 1. Theorem 12. Let G be a complete graph of order n and H be any graph of order m with n−m = k > 0. Then γce(G+H) = 1 + γe(⟨V (H \DH⟩) where V (H) \ DH = {v ∈ V (H) : degH(v) < ∆(G) − (k + 1) = n − k − 2} and DH = {z ∈ V (H) : degH(z) ≥ ∆(G)− (k + 1) = n− k − 2}. Proof. Let u ∈ V (G) and S ⊆ V (H) \ D∗ H be a γe-set of ⟨V (H \ DH⟩ where DH = {z ∈ V (H) : degH(z) ≥ ∆(G) − (k + 1) = n − k − 2}. Put C = {u} ∪ S. We will show that C is a γce-set of G+H. Let v ∈ V (G+H) \C. If v ∈ V (G), then v ∈ NG+H(u) and |degG+H(u)− degG+H(v)| = |degG(u) + |V (H)| − (degG(v) + |V (H)|)| = | degG(u)− degG(v)| = (n− 1)− (n− 1) = 0 < 1. H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 10 of 14 Suppose that v ∈ V (H). If degH(v) ≥ n− k − 2, the v is equitably dominated by u, that is, v ∈ NG+H(u) and |degG+H(u)− degG+H(v)| = | degG(u) + |V (H)| − (degH(v) + |V (G)|)| = | degG(u)− degH(v)− (|V (G)| − |V (H)|)| = | degG(u)− degH(v)− (n−m)| = | degG(u)− degH(v)− k| < | degG(u)− (n− k − 2)− k| = |n− 1− n+ k + 2− k| = 1. Suppose that degH(v) < n− k− 2. Since S is an equitable dominating set in ⟨V (H \DH⟩ there exists y ∈ S such that v ∈ NG+H(y) and | degG+H(y)− degG+H(v)| = | degH(y) + |V (G)| − (degH(v) + |V (G)|)| = | degH(y)− degH(v)| < 1. Thus, C is a connected equitable dominating set in G+H. Hence, γce(G+H) ≤ |C|. Suppose that γce(G+H) < |C|. Then there exists a γce-set in G+H say A with |A| < |C|. Suppose that A = C \ {x} with x ∈ {u} ∪ S with S a γe-set of ⟨V (H \D∗ H⟩ . If x = u, then for all z ∈ V (G+H) whose degree is less than n+ k − 2 is not equitably dominated by A, a contradiction. If x ∈ S, then there exists an element w ∈ ⟨V (H \ DH⟩ which nor equitably dominated by A, a contradiction. Thus, γce(G + H) = |C|. Therefore, γce(G+H) = 1 + γe(⟨V (H \DH⟩). Theorem 13. Let G be a complete graph of order n and H be any graph of order m with m− n = k ≥ 0. Then γce(G+H) = 1 + γe(⟨V (H \D∗ H⟩) where V (H) \D∗ H = {v ∈ V (H) : degH(v) < ∆(G) + (k− 1) = n+ k− 2} and D∗ H = {z ∈ V (H) : degH(z) ≥ ∆(G) + (k − 1) = n+ k − 2} Proof. Let u ∈ V (G) and S ⊆ V (H) \ D∗ H be a γe-set of ⟨V (H \ D∗ H⟩ where D∗ H = {z ∈ V (H) : degH(z) ≥ ∆(G) + (k − 1) = n − k − 2}. Put C = {u} ∪ S. We will show that C is a γce-set of G+H. Let v ∈ V (G+H) \C. If v ∈ V (G), then v ∈ NG+H(u) and |degG+H(u)− degG+H(v)| = |degG(u) + |V (H)| − (degG(v) + |V (H)|)| = | degG(u)− degG(v)| = (n− 1)− (n− 1) = 0 < 1. H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 11 of 14 Suppose that v ∈ V (H). If degH(v) ≥ n+ k − 2, the v is equitably dominated by u, that is, v ∈ NG+H(u) and |degG+H(u)− degG+H(v)| = | degG(u) + |V (H)| − (degH(v) + |V (G)|)| = | degG(u)− degH(v) + (|V (H)| − |V (G)|)| = | degG(u)− degH(v) + (n−m)| = | degG(u)− degH(v) + k| < | degG(u)− (n+ k − 2) + k| = |n− 1− n− k + 2 + k| = 1. Suppose that degH(v) < n+ k− 2. Since S is an equitable dominating set in ⟨V (H \D∗ H⟩ there exists y ∈ S such that v ∈ NG+H(y) and | degG+H(y)− degG+H(v)| = | degH(y) + |V (G)| − (degH(v) + |V (G)|)| = | degH(y)− degH(v)| < 1. Thus, C is a connected equitable dominating set in G+H. Hence, γce(G+H) ≤ |C|. Suppose that γce(G+H) < |C|. Then there exists a γce-set in G+H say A with |A| < |C|. Suppose that A = C \ {x} with x ∈ {u} ∪ S with S a γe-set of ⟨V (H \D∗ H⟩ . If x = u, then for all z ∈ V (G+H) whose degree is less than n+ k − 2 is not equitably dominated by A, a contradiction. If x ∈ S, then there exists an element w ∈ ⟨V (H \ D∗ H⟩ which nor equitably dominated by A, a contradiction. Thus, γce(G + H) = |C|. Therefore, γce(G+H) = 1 + γe(⟨V (H \D∗ H⟩). 4. Corona of Graphs Let G and H be graphs of order m and n, respectively. The corona G ◦H of G and H is the graph obtained by taking one copy of G and m copies of H, and then joining the ith vertex of G to every vertex of the ith copy of H. For everyv ∈ V (G), denote by Hv the copy of H whose vertices are attached one by one to the vertex v. Denote by v +Hv the subgraph of the corona G ◦H corresponding to the join ⟨{v}⟩+Hv. Theorem 14. Let G be a connected non-trivial graph and H be any graph of order n. The C ⊆ V (G ◦ H) is a connected equitable dominating set of G ◦ H if and only if C = V (G) ∪ ( ⋃ v∈V (G) Bv), where for each v ∈ V (G), (i.) Bv is an equitable dominating set of Hv whenever ∆(H) ≤ n − 2 or degG(v) ≥ 2 and (ii.) Bv dominates equitably V (Hv)\Sv, where Sv = Bv∪{x ∈ V (Hv) : degH(x) = n−1} whenever degG(v) = 1 and ∆(H) = n− 1. H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 12 of 14 Proof. Suppose that C is a connected equitable dominating set of G ◦ H. Let A = C ∩ V (G) and let Bv = C ∩ V (Hv) for each v ∈ V (G). Since ⟨C⟩ is connected subgraph of G ◦H, ⟨A⟩ is a connected subgraph of G. If there exists v ∈ V (G) \ A, then Bv ̸= ∅ since C is a dominating set of G ◦ H. This implies that ⟨C⟩ is not connected, contrary to our assumption that C is a connected dominating set of G ◦ H. Thus, A = V (G). Next, let v ∈ V (G) and let q ∈ V (Hv) \ Bv. Then there exists w ∈ C ∩ NG◦H(q) such that | degG◦H(w) − degG◦H(q)| ≤ 1. If ∆(H) ≤ n − 2, then degG◦H(v) ≥ n + 1 and degG◦H(q) ≤ n − 1. If degG(v) ≥ 2, then degG◦H(v) ≥ n + 2 and degG◦H(q) ≤ n. Hence, w ̸= v whenever ∆(H) ≤ n − 2 or degG(v) ≥ 2. It follows that w ∈ Bv ∩ NHv(q) and |degHv(w) − degHv(q)| ≤ 1.This shows that Bv is an equitable dominating set of Hv whenever ∆(H) ≤ n − 2 or degG(v) ≥ 2. Suppose that degG(v) = 1 and ∆(H) = n − 1. Suppose further that degHv(q) ̸= n− 1. Then degG◦H(v) = n+ 1 and degG◦H(q) ≤ n− 1. This implies that w ̸= v. Hence, w ∈ Bv and | degHv(w) − degHv(q)| ≤ 1. Consequently, Bv dominates equitably V (Hv) \ Sv. The converse is easy. The next result is a consequence of the Theorem above. Corollary 3. Let G be a connected graph of order m ≥ 2 and H be any graph of order n. If δ(G) ≥ 2 or ∆(H) ≤ n− 2, then γce(G ◦H) = [1 + γe(H)]m. Proof. Let C be a γce-set of G◦H. By Theorem 14, C = V (G)∪( ⋃ v∈V (G) Bv), where Bv is a connected equitable dominating set of Hv for each v ∈ V (G). Since C is a γce-set , Bv is a γe-set of Hv for each v ∈ V (G). Thus, γce(G◦H) = |C| = m+mγe(H) = m[1+γe(H)]. Corollary 4. For any graph G of order n, γce(G ◦Kp) = n+ |S| where S = {u ∈ V (G) : degG(u) ≥ 2}. Proof. Follows from Theorem 14 and Corollary 3. Corollary 5. For any graph G of order n, and a r-regular graph H with r ̸= |V (H)| − 1, γce(G ◦H) = n+ nγ(H). Proof. Let v ∈ V (G) and Cv ⊆ V (Hv) be a γ-set in Hv. Then Cv is an equitable dominating set in Hv since H is r-regular. Furthermore, since r ̸= |V (H)| − 1, the set⋃ u∈V (G) {v} ∪Cv is a connected equitable dominating set in G ◦H of minimum cardinality. Thus, γce(G ◦H) = n+ n|Cv| = n+ nγ(H). H. Nuenay-Maglanquel / Eur. J. Pure Appl. Math, 18 (3) (2025), 6123 13 of 14 5. Conclusion In this paper, we have investigated the concept of connected equitable domination in graphs focusing on some families of graphs, and on graphs arising from the join and corona of graphs. Specifically, the connected equitable dominating sets in the join and corona of graphs are characterized. We also obtained the exact value of γce(G) for some families of graphs; and graphs formed in the join and corona of graphs. Additionally, a realization problem is established. The significance of this study lie in its extension of domination theory to connected eq- uitable domination through combining both connectivity and equity constraints, reflecting more realistic models of control, influence, and resource distribution in networked systems; and focus on more complex structure of graphs: graphs obtained in the join and corona of graphs. Unlike classical domination parameters that focus solely on coverage or con- nectivity, the connected equitable domination framework ensures balanced distribution of domination responsibilities among vertices while maintaining network cohesion.This study provides a foundation for efficient network design in practical domains such as sensor place- ment, facility location, communication networks, and distributed computing, where both connectivity and fairness are critical. The exact values and characterizations obtained here can also serve as benchmarks for algorithm development, particularly in designing heuristics and approximation algorithms for complex networks. Nonetheless, the work is confined to unweighted, undirected, and simple graphs, which may limit its direct applica- bility to more complex or dynamic networks. 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