EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6326 ISSN 1307-5543 – ejpam.com Published by New York Business Global Friendly Domination in Graph Isagani S. Cabahug, Jr.1,∗, Rolito G. Eballe1, Reynard T. Fernandez1 1 Department of Mathematics, College of Arts and Sciences, Central Mindanao University, 8714 Bukidnon, Philippines Abstract. Friendly domination combines two ideas in graph theory: domination, which captures influence, and friendly sets, which balance that influence. We study the friendly dominating set and the friendly domination number, the smallest size of a set that is both dominating and friendly. First, we prove that for every graph the usual domination number is never larger than its friendly domination number, and we identify all graphs whose friendly domination number equals one or two. We then show that the gap between these two parameters can be made arbitrarily large. Exact formulas are derived for paths and cycles. Sharp Nordhaus–Gaddum bounds are obtained for the sum and product of the friendly domination number of a graph and its complement. Finally, we give complete structural characterizations of friendly dominating sets in the join and corona of two graphs. 2020 Mathematics Subject Classifications: 05C69 Key Words and Phrases: Friendly sets, friendly domination, domination 1. Introduction Graph theory has been around since the eighteenth century and has grown into an important part of mathematics. It helps us understand communication networks, chemical structures, disease spread, and social connections [1, 2]. One of the key ideas studied is domination, where a small group of vertices can control or influence the entire graph. Many researchers, like Henning and Yeo [3], have expanded these ideas and connected them to other topics like hypergraphs, graphs with special distance properties, and computer algorithms. Recently, the idea of a friendly set, where each vertex outside the set has at least as many neighbors outside as inside, was introduced by Haynes, Hedetniemi, and Henning [4]. Building on this concept, we define friendly domination as a dominating set that is also a friendly set. This new concept reflects situations where influence needs to be applied carefully without overwhelming any single vertex. Friendly domination is closely related to other ideas like offensive alliance [5–8] and defensive alliances [9–12]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6326 Email addresses: isaganicabahugjr@cmu.edu.ph (I. S. Cabahug, Jr.), rgeballe@cmu.edu.ph (R. G. Eballe), f.reynard.fernandez@cmu.edu.ph (R. T. Fernandez) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 2 of 15 At the same time, there has been growing interest in other fairness conditions on dom- inating sets. One example is fair domination, where each vertex outside the set must have the same number of connections to the set [13]. These ideas show a trend toward studying domination with more detailed and realistic rules compared to traditional domination. Even with these developments, there are still many open questions about the friendly domination. Basic properties that were solved long ago for ordinary domination, such as Nordhaus–Gaddum-type inequalities [14], are still open for friendly domination. Also, exact values and sharp bounds are only known for a few simple types of graphs, and there is little known about how friendly domination behaves under graph operations like graph products. In this paper, we prove that the domination number is always less than or equal to the friendly domination number, and we describe graphs where the friendly domination num- ber equals one or two. We also show that the difference between the friendly domination number and the domination number can be made as large as desired by constructing ap- propriate graphs. Furthermore, we find exact formulas for the friendly domination number in paths, cycles, and graphs with maximum degree two. In addition, we establish sharp Nordhaus–Gaddum bounds for the sum and product of the friendly domination number of a graph and its complement. Finally, we characterize friendly dominating sets in the join and corona of graphs. 2. Terminology and Notation Given a graph G = (V (G), E(G)), the open neighborhood of a vertex v ∈ V , denoted N(v), is the set of all vertices adjacent to v, that is,N(v) = {u ∈ V | {u, v} ∈ E(G)}. Its closed neighborhood, denoted N [v], is obtained by adding the vertex itself: N [v] = N(v) ∪ {v}. More generally, for any subset S ⊆ V , the open neighborhood of S is the union of the open neighborhoods of its elements, N(S) = ⋃ v∈S N(v), and the closed neighborhood of S is the union of their closed neighborhoods (equivalently N(S) ∪ S), N [S] = ⋃ v∈S N [v] = N(S) ∪ S. The complement of G, denoted G, is the simple graph on the same vertex set V whose edge set is E(G) = { {u, v} ⊆ V : u ̸= v and {u, v} /∈ E(G) } . The degree of a vertex v ∈ V , denoted deg(v), is defined to be the number of vertices adjacent to v. Equivalently, deg(v) = ∣∣N(v) ∣∣, N(v) = {u ∈ V | {u, v} ∈ E(G)}. A vertex v for which deg(v) = 1 is called a leaf. The maximum degree of G is ∆(G) = maxv∈V deg(v), and the minimum degree of G is δ(G) = minv∈V deg(v). For any subset S ⊆ V , the degree of v restricted to S is degS(v) = ∣∣N(v) ∩ S ∣∣. A subset D ⊆ V is called a dominating set if every vertex not in D is adjacent to at least one vertex in D. Equivalently, the closed neighborhood of D, defined by N [D] = ⋃ v∈D ( {v} ∪N(v) ) , satisfies N [D] = V , so that every vertex of G lies either in D or has a neighbor in D. The domination number of G, denoted γ(G), is the smallest size of any dominating set in G: γ(G) = min { |D| : D ⊆ V, N [D] = V } . [1, 15] A subset F ⊆ V is called a friendly set if each vertex outside F has at least as many neighbors outside F as inside F . Equivalently, for every v ∈ V (G) \ F , degF (v) ≤ degV (G)\F (v). This notion was introduced by Haynes, Hedetniemi, and Henning in their I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 3 of 15 study of structures of domination in graphs. [4]. Standard graph-theoretic terminology not defined herein may be found in classic texts such as [1, 2, 15]. For more advanced notions of domination and friendly sets, the reader is referred to Haynes et al. [4]. The join of two graphs G and H, written G ∨ H, is formed by first taking disjoint copies of G and H (so no vertex is shared and no edge connects the two graphs), and then adding every possible edge between each vertex of G and each vertex of H. Thus the vertex set of G∨H is V (G)∪V (H), its edge set contains E(G) and E(H), and in addition it contains the set of edges { {u, v} | u ∈ V (G), v ∈ V (H) }. Intuitively, the join “glues” the two graphs together by making every vertex of one graph adjacent to every vertex of the other. [1, 15] The corona of two graphs G and H, written G ◦H, is obtained by taking one copy of G and, for every vertex x of G, adjoining an entire disjoint copy Hx of H and then adding edges from x to every vertex of that copy. Thus each vertex of G becomes the centre of a “star” whose leaves induce a copy of H, while all internal adjacencies of G and of each Hx are retained. [16]. Given any statement or property A, the indicator (or characteristic) function 1{A} is defined to be 1 if the property A is true, and 0 if the property A is false. In particular, for a subset S of vertices, the indicator 1{x∈S} equals 1 when the vertex x belongs to S and 0 otherwise, while 1{x/∈S} equals 1 when x does not belong to S and 0 otherwise. For any fixed vertex x, exactly one of these two indicators equals 1 and the other equals 0. This convenient notation allows case distinctions to be written compactly inside a single unified formula, simplifying many expressions and arguments. For non–negative real numbers a1, a2, . . . , an the Arithmetic–Geometric Mean Inequal- ity asserts that the arithmetic mean a1+···+an n is never smaller than the geometric mean (a1 · · · an)1/n; in symbols, a1 + a2 + · · ·+ an n ≥ (a1a2 · · · an)1/n, with equality precisely when all the ai are equal [17]. 3. Results Definition 1. Let G be a nontrivial connected graph. A nonempty set F ⊆ V (G) is a friendly dominating set if N [F ] = V (G) and for every v ∈ V (G) \ F , degF (v) ≤ degV (G)\F (v). The minimum cardinality of a friendly dominating set in G is called the friendly domination number of G, denoted by γf (G). A friendly dominating set whose cardinality is the γf (G), is called the γf -set of G. Example 1. Consider the path graph P4 with vertices v1, v2, v3, v4 in order, and let F = {v1, v4}. I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 4 of 15 v1 v2 v3 v4 vertex in F vertex in V \ F Clearly N [F ] = V (P4), since v1 dominates v2 and v4 dominates v3. For every vertex outside F we have degF (v2) = 1 ≤ degV \F (v2) = 1, degF (v3) = 1 ≤ degV \F (v3) = 1. Thus F is a friendly dominating set of P4. In particular, γf (P4) = 2. 3.1. Some Realizations and the Friendly Dominating Sets of Some Known Graphs Given nontrivial connected graph G = (V (G), E(G)). Every friendly dominating set of G is a dominating set of G. Thus we have the following remark: Remark 1. Let G be a graph. Then, γ(G) ≤ γf (G). Let F be a nonempty subset of V (G). If F = V (G), then F is trivially a dominating set because it contains every vertex of G. Moreover, the friendly condition only needs to be verified for vertices in V (G) \ F . Since V (G) \ F = ∅, there are no vertices to check the inequality, and hence the condition is satisfied vacuously. Remark 2. Let G be a graph. Then the vertex set, V (G), is a friendly dominating set of G. Theorem 1. Let G be a nontrivial connected graph and let F ⊆ V (G) be a friendly dominating set of G. If G contains any leaf vertices, then every leaf vertex of G must belong to F . Proof. Let G be a nontrivial connected graph and let F ⊆ V (G) be a friendly domi- nating set of G. Suppose that there exists a leaf vertex u ∈ V (G) such that u /∈ F . Since F is a dominating set, u must be adjacent to some vertex in F . However, as u is a leaf, it has exactly one neighbor in G, say w. Thus, w ∈ F and consequently degF (u) = 1. On the other hand, u has no other neighbors, so degV (G)\F (u) = 0. This yields to 1 = degF (u) > degV (G)\F (u) = 0, which is a contradiction. Hence, it follows that every leaf vertex of G belongs to F . Theorem 2. Let G be a nontrivial connected graph and F ⊆ V (G) be a friendly domi- nating set of G. Then the induced subgraph of V (G) \ F in G, has no isolated vertices. I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 5 of 15 Proof. Suppose that the induced subgraph G[V (G) \ F ] contains an isolated vertex w. Then w ∈ V (G) \ F and degV (G)\F (w) = 0. Since F is a dominating set, w must be adjacent to some vertex in F , implying that degF (w) ≥ 1. This yields the inequality 1 ≤ degF (w) ≤ degV (G)\F (w) = 0, which is a contradiction. Therefore, the induced subgraph G[V (G) \F ] cannot contain an isolated vertex. Theorem 3. Let G be a nontrivial connected graph of order n > 2. Then, γf (G) = 1 if and only if there exist a vertex v ∈ V (G) such that deg(v) = n − 1 and for every other vertex u ̸= v has at least one neighbor besides v. Proof. Suppose γf (G) = 1. Then there is a friendly dominating set F ⊆ V (G) with |F | = 1. Let F = {v}. Since F is a dominating set, v must be adjacent to every other vertex. Hence, deg(v) = n− 1. Now, for each u ∈ V (G) \ {v}, we have degF (u) = 1. Since F is a friendly set, degF (u) ≤ degV (G)\{v}(u) for all u /∈ F . Thus, degV (G)\{v}(u) ≥ 1, meaning each u ̸= v must have at least one neighbor other than v. Conversely, suppose there exists a vertex v ∈ V (G) with deg(v) = n− 1 and for every u ̸= v, degV (G)\F (u) ≥ 1. Consider F = {v}. Then, F is a dominating set of G. Also, for each u ̸= v, degF (u) = 1. Thus, degF (u) = 1 ≤ degV (G)\F (u). Hence, F is a friendly set in G. Therefore, F = {v} is a friendly dominating set of G and that γf (G) ≤ 1. Clearly, γf (G) ≥ 1. Thus, γf (G) = 1. Corollary 1. For the complete graph Kn, fan graph F1,n, and wheel graph W1,m with n > 1 and m > 2, γf (Km) = γf (F1,n) = γf (W1,m) = 1. Proof. This immediately follows from Theorem 3. Theorem 4. Let G be a non-trivial connected graph of order n ≥ 3. Then γf (G) = 2 if and only if all three conditions hold. (i) There is no vertex u ∈ V (G) such that N [u] = V (G) and the induced subgraph of G− {u} has minimum degree δ ( G− {u} ) ≥ 1. (ii) There exist two distinct vertices x, y ∈ V (G) with the property that every other vertex v ∈ V (G) \ {x, y} is adjacent to at least one of x or y. (iii) For every vertex v ∈ V (G) \ {x, y} we have deg{x,y}(v) ≤ deg V (G)\{x,y}(v). Proof. Assume that the friendly domination number of G is γf (G) = 2. Then there exists a friendly dominating set F = {x, y} ⊆ V (G) of cardinality two. Since F is domi- nating, every vertex outside the pair is adjacent to at least one of x or y; this is exactly the content of condition (ii). Moreover, since F is friendly, degF (v) ≤ degV (G)\F (v) for every v ∈ V (G) \ F , is identical to condition (iii) when written for the set {x, y}. Now, I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 6 of 15 suppose (i) is not true, then exist a single vertex u that is itself a friendly dominating set, which would force γf (G) = 1, contradicting our assumption that γf (G) = 2. Hence all three conditions (i)− (iii) must hold. Conversely, suppose that conditions (i), (ii) and (iii) are satisfied. By condition (ii) the pair {x, y} dominates the graph, while condition (iii) shows that the set {x, y} is friendly . Thus {x, y} is a friendly dominating set, so γf (G) ≤ 2. Condition (i) rules out the possibility that a single vertex can act as a friendly dominating set. Thus, γf (G) > 1, hence, γf (G) ≥ 2. Therefore, γf (G) = 2. Corollary 2. Let G and H be graphs with |V (G)|, |V (H)| ≥ 2 such that ∆(G) < |V (G)|−1 and ∆(H) < |V (H)| − 1. Then, γf (G∨H) = 2. In particular, for every complete bipartite graph Km,n with m,n ≥ 2 we have γf (Km,n) = 2. Proof. This immediately follows from Theorem 4. Theorem 5. Let a and b be positive integers such that a ≤ b. Then there exist a nontrivial connected graph G such that γ(G) = a and γf (G) = b. Proof. Let a and b be positive integers with a ≤ b. Consider the following cases: Case 1: a = b Consider the graph G shown in Figure 1. Clearly, the set F = {xi|i = 1, 2, 3, . . . , a − 1, a} is both a γ-set and a γf -set of G. Thus, γ(G) = γf (G) = |F | = a = b. This completes the proof for the case a = b. x1 x2 x3 · · · xa−1 xa Figure 1: A graph with γ(G) = γf (G) Case 2: a < b LetG be the graph shown in Figure 2. Observe that the setD = {x1, x2, x3, . . . , xa−1, xa} is a γ-set and the set F = D∪{y1, y2, y3, . . . , yb−a−1, yb−a} is a γf -set. Thus, γ(G) = |D| = a and γf (G) = |F | = |D|+ |{y1, y2, y3, . . . , yb−a−1, yb−a}| = a+ b− a = b. This proves the assertion. Corollary 3. For each positive integer n, there exist a connected graph G such that γf (G)− γ(G) = n, that is, the difference between γf (G) and γ(G) can be made arbitrarily large. Proof. By Theorem 5, for any pair of positive integers a and b with a < b, there exists a connected nontrivial graph H such that γ(H) = a and γf (H) = b. Taking a = 1 and I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 7 of 15 x1 x2 x3 · · · xa−1 xa y1 y2 y3 ... yb−a−1 yb−a Figure 2: A graph with γ(G) < γf (G) b = n + 1. Thus, we have γf (G) − γ(G) = (n + 1) − 1 = n. Since n is arbitrary, the difference γf (G)− γ(G) can be made as large as desired. Theorem 6. Let G be a connected nontrivial graph with maximum degree at most 2, and let F ⊆ V (G) be a nonempty subset. Then F is a friendly dominating set of G if and only if the following conditions hold: (i) Every vertex v ∈ V (G) \ F satisfies degF (v) = 1; that is, every vertex outside F is adjacent to exactly one vertex in F . (ii) Every vertex v ∈ V (G) with deg(v) = 1 belongs to F . Proof. Let F ⊆ V (G) be a friendly dominating set of G. By Theorem 1, every vertex u ∈ V (G) with deg(u) = 1 belongs to F . Since the maximum degree in G is 2, each vertex v satisfies deg(v) ≤ 2. If a vertex v /∈ F were adjacent to two vertices in F , then degF (v) = 2 and consequently degV (G)\F (v) = 0, which is a contradiction to the assumption of F . Therefore, every vertex v /∈ F must satisfy degF (v) = 1. Conversely, assume that for every vertex v /∈ F we have degF (v) = 1 and that every leaf of G is contained in F . Then F is clearly a dominating set since every vertex not in F has exactly one neighbor in F . Moreover, for every vertex v /∈ F with deg(v) ≤ 2, if v is not a leaf it must have degree 2, and the only possibility is that it has one neighbor in F and one neighbor in V (G) \ F ; hence degV (G)\F (v) = 1 and the inequality degF (v) = 1 ≤ 1 = degV (G)\F (v) holds. Thus, F is a friendly dominating set of G. Corollary 4. Let Pn be the path graph on n ≥ 2 vertices. Then the friendly domination I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 8 of 15 number of Pn, is given by γf (Pn) =  n+ 6 3 , if n ≡ 0 (mod 3), n+ 2 3 , if n ≡ 1 (mod 3), n+ 4 3 , if n ≡ 2 (mod 3). Proof. Let Pn = [v1, v2, . . . , vn−1, vn]. Consider the following cases for n: Case 1: n ≡ 0 (mod 3) If n = 3, then the γf -set of P3 is V (P3). Thus, γf (P3) = n 3 + 2 = n+ 6 3 = 3 + 6 3 = 3 = V (P3). Now, for n > 3. Choose F = {v1, vn} ∪ {v4, v7, . . . , vn−2, , vn−1}, then |F | = 2 + n 3 = n+ 6 3 . By Theorem 6, F is a friendly dominating set in Pn. Thus, γf (G) ≤ n+ 6 3 . Since there can be no another friendly dominating sets whose cardinality is strictly less than n+ 6 3 , we can have γf (Pn) ≥ n+ 6 3 . Thus, γf (Pn) = n+ 6 3 . Case 2: n ≡ 1 (mod 3) and n ≡ 2 (mod 3) An entirely analogous argument as Case 1. Choosing F = {v1, vn} ∪ {v4, v7, . . . , vn−3} with |F | = 2 + n− 4 3 = n+ 2 3 for n ≡ 1 (mod 3) and F = {v1, vn} ∪ {v4, v7, . . . , vn−2} with |F | = 2 + n− 2 3 = n+ 4 3 for n ≡ 2 (mod 3). Corollary 5. Let Cn be the cycle graph on n ≥ 3 vertices. Then the friendly domination number of Cn is γf (Cn) = ⌈n 3 ⌉ . Proof. By Theorem 6 (i), finding the friendly dominating set in Cn is equivalent to finding a set F such that every vertex is either in F or adjacent to exactly one vertex of F . Thus, in Cn, the minimum number of vertices needed so that every vertex in Cn is dominated by exactly one member of the set is ⌈n 3 ⌉ and no smaller set can satisfy this condition in Cn. Thus, γf (Cn) = ⌈n 3 ⌉ . 3.2. Nordhaus-Gaddum Type Inequalities for γf -set. Theorem 7. Let G be a connected nontrivial graph on n vertices. Then, 0 ≤ ∣∣γf (G)− γf (G) ∣∣ ≤ n− 1. Proof. Let G be a connected nontrivial graph on n vertices, and let γf (G) and γf ( G ) be the friendly domination numbers of G and its complement G, respectively. Now, γf (G) I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 9 of 15 and γf ( G ) are each at least 1 since a friendly dominating set must be nonempty, and they are each at most n since taking the entire vertex set V (G) and V (G) always yields a friendly dominating set. It follows that γf (G) and γf ( G ) both lie in the integer interval [1, n], so their absolute difference satisfies 0 ≤ ∣∣γf (G) − γf ( G )∣∣ ≤ n− 1. To see that these bounds are sharp, consider the complete graph Kn. By Corollary 1, γf (Kn) = 1. However, the complement of Kn is the empty graph on n vertices, Kn, in which no vertices are adjacent. In Kn, every vertex must be in the friendly dominating set to dominate itself, so γf (Kn) = n. Hence ∣∣γf (Kn) − γf ( Kn )∣∣ = ∣∣ 1 − n ∣∣ = n− 1, matching the upper bound exactly. On the other hand, when G is self-complementary, γf (G) = γf (G), implying that, ∣∣γf (G)− γf (G) ∣∣ = 0. Consequently, for general connected nontrivial graphs G of order n, the difference ∣∣γf (G) − γf ( G )∣∣ is always between 0 and n− 1, and both extremes can be attained. Let G be a graph of order n. For the empty graph we clearly have both the domination number and the friendly-domination number equal to n. When G is connected and non- trivial one might expect these parameters to drop, and indeed the ordinary domination number always does. In contrast, the friendly-domination number can remain as large as n, even in connected graphs, as the next lemma shows. Lemma 1. Let G be a connected graph on at least two vertices. Then γf (G) ≤ n, and the bound is best possible. Proof. Since G is connected, By Remark 2, the whole vertex set V (G) is a non-empty friendly dominating set. Hence γf (G) ≤ |V (G)| = n. To show that the bound is tight. Consider the star K1,n−1 with center c and leaves L = {ℓ1, . . . , ℓn−1}. Let F ⊆ V (K1,n−1) be a friendly dominating set in K1,n−1. If c /∈ F , then c must be dominated by some leaf ℓi ∈ F . Vertex c has no neighbor outside F while degF (c) ≥ 1, contradicting degF (c) ≤ degV (K1,n−1)\F (c). If c ∈ F but some leaf ℓ /∈ F , then degF (ℓ) = 1 (its neighbour c) and degV (K1,n−1)\F (ℓ) = 0, again violating the friendly condition. Therefore every friendly dominating set of the star must be the entire vertex set, giving γf ( K1,n−1 ) = n. Thus, the upper bound n is tight. Theorem 8. Let G be a graph on n ≥ 4 vertices such that both G and its complement G are connected. Then γf (G) + γf (G) ≤ 2n. Proof. By Lemma 1, γf (G) ≤ n. Thus, γf(G) + γf (G) ≤ n+ n = 2n. Corollary 6. For nontrivial connected graph G on n ≥ 4 vertices, we have γf (G) γf ( G ) ≤ n2. I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 10 of 15 Proof. By Theorem 8, γf (G) + γf ( G ) ≤ 2n. Since both γf (G) and γf (G) are nonneg- ative numbers, the Arithmetic Mean–Geometric Mean (AM–GM) Inequality yields γf (G) γf ( G ) ≤ ( γf (G) + γf ( G ) 2 )2 . Substituting the upper bound on the sum, we obtain, γf (G) γf ( G ) ≤ ( 2n 2 )2 = n2. 3.3. Structure of Friendly Dominating Sets in the Join of Graphs Remark 3. Let G and H be empty graphs and let F ⊆ V (G ∨H) be a nonempty set. If F ⊆ V (G) or F ⊆ V (H), then F cannot be a friendly dominating set in G ∨H. Theorem 9. Let G and H be empty graphs with |V (G)| = n1 and |V (H)| = n2. Let F = F1 ∪F2, where F1 ⊆ V (G) and F2 ⊆ V (H) are both nonempty and assume F ̸= V (G∨H) (equivalently, V (G∨H) \F ̸= ∅). Then F is a friendly dominating set in the join G∨H if and only if |F1| ≤ ⌊n1 2 ⌋ and |F2| ≤ ⌊n2 2 ⌋ . Proof. Assume that F is a friendly dominating set with F ̸= V (G ∨H). Because F1 and F2 are both nonempty, every vertex is either in F or adjacent to some vertex of F . Hence, F is a dominating set. Now, pick any vertex y ∈ V (H)\F2. All of y’s neighbors lie in V (G); specifically, degF (y) = |F1| and degV (G∨H)\F (y) = n1 − |F1|. Since F is friendly, we have, |F1| ≤ n1−|F1|. Thus, |F1| ≤ ⌊n1 2 ⌋ . Repeating the same argument with a vertex x ∈ V (G) \ F1 gives |F2| ≤ ⌊n2 2 ⌋ . Conversely, suppose that |F1| ≤ ⌊n1 2 ⌋ and |F2| ≤ ⌊n2 2 ⌋ with F1 and F2 both nonempty. Because F1 ̸= ∅, every vertex in V (H) is adjacent to some vertex of F ; similarly, F2 ̸= ∅ guarantees that every vertex in V (G) is adjacent to some vertex of F . Hence, N [F ] = V (G∨H). Thus, F is a dominating set in G∨H. Now, let v ∈ V (G)\F1. Its neighbors are precisely the n2 vertices of V (H), of which |F2| lie in F and n2−|F2| lie outside F . Hence, degF (v) = |F2| and degV (G∨H)\F (v) = n2 − |F2|. By assumption, |F2| ≤ ⌊n2 2 ⌋ , implying that,|F2| ≤ n2− |F2|. Thus, degF (v) ≤ degV (G∨H)\F (v). An analogous calculation applies for w ∈ V (H) \ F2, using |F1| ≤ n1 − |F1| gives degF (w) ≤ degV (G∨H)\F (w). Thus, F is friendly. Therefore, F is a friendly dominating set in G ∨H. Theorem 10. Let G and H be non-trivial connected graphs and let F ⊆ V (G). Then F is a friendly dominating set of G ∨H if and only if the following three conditions hold: (i) F dominates G; (ii) |F | ≤ |V (G) \ F |+ δ(H); I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 11 of 15 (iii) degGF (x) ≤ |V (H)|+ degGV (G)\F (x) for every x ∈ V (G) \ F Proof. Assume F is a friendly dominating set of G ∨ H. Since every vertex of H is adjacent to all vertices of G in the join, F already dominates G ∨H, hence in particular G; this is (i). Choose y ∈ V (H). Then degF (y) = |F | and degV (G∨H)\F (y) = |V (G)\F |+ degH(y). Since F is friendly and degH(y) ≥ δ(H), we have, |F | ≤ |V (G) \ F | + δ(H). Thus (ii) holds. Now take x ∈ V (G)\F . Then, degF (x) = degGF (x) and degV (G∨H)\F (x) = |V (H)| + degGV (G)\F (x). Again, since F is friendly, degGF (x) ≤ |V (H)| + degGV (G)\F (x). Thus, (iii) holds. Conversely, suppose (i)–(iii) hold. Because F dominates G and every vertex of H is adjacent to all vertices of G in the join, F also dominates H; hence it dominates G ∨H. To show that F is friendly, we consider two cases. Case 1: Let y ∈ V (H). We have degF (y) = |F | and degV (G∨H)\F (y) = |V (G) \ F | + degH(y). With degH(y) ≥ δ(H) and (ii) we get degF (y) ≤ degV (G∨H)\F (y). Case 2: Let x ∈ V (G) \ F . Here degF (x) = degGF (x) and degV (G∨H)\F (x) = |V (H)|+ degGV (G)\F (x). By condition (iii), again, degF (y) ≤ degV (G∨H)\F (y). Thus F is friendly. Hence, F is a friendly dominating set in G ∨H. Theorem 11. Let G and H be non-trivial connected graphs and let F ⊆ V (H). Then F is a friendly dominating set of G ∨H if and only if the following three conditions hold: (i) F dominates H; (ii) |F | ≤ |V (H) \ F |+ δ(G); (iii) degHF (x) ≤ |V (G)|+ degHV (H)\F (x) for every x ∈ V (H) \ F Proof. The same analogous proof as Theorem 10. Theorem 12. Let G and H be non-trivial connected graphs on |V (G)| = n1 and |V (H)| = n2 vertices, respectively. Let F1 ⊆ V (G) and F2 ⊆ V (H) and put F = F1 ∪ F2. Then F is a friendly dominating set of G ∨ H if and only if the following two inequalities are satisfied: (i) degGF1 (x) ≤ degGV (G)\F1 (x) + n2 − 2|F2|, for every x ∈ V (G) \ F1, (ii) degHF2 (y) ≤ degHV (H)\F2 (y) + n1 − 2|F1|, for every y ∈ V (H) \ F2. Proof. Assume F is friendly dominating set in G ∨H. Pick x ∈ V (G) \ F1 and count separately the neighbors of x inside and outside F . Then, degF (x) = degGF1 (x) + |F2| and degV (G∨H)\F (x) = degGV (G)\F1 (x) + |V (H) \ F2| = degGV (G)\F1 (x) + n2 − |F2|. Since F is friendly, we have, degGF1 (x) ≤ degGV (G)\F1 (x) + n2 − 2|F2|. An analogous statement for y ∈ V (G) \ F2 gives degHF2 (y) ≤ degHV (H)\F2 (y) + n1 − 2|F1|. Conversely, suppose (i) and (ii) hold. Since F1 (resp. F2) contains at least one vertex of G (resp. H), every vertex of H (resp. G) is adjacent (through the join) to a vertex of F ; I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 12 of 15 hence F is a dominating set in G∨H. To show that F is friendly. Let x ∈ V (G)\F1. By (i), degGF1 (x) ≤ degGV (G)\F1 (x)+n2−2|F2|,. This implies that, degGF1 (x)+|F2| ≤ degGV (G)\F1 (x)+ n2 − |F2|. Now, degF (x) = degGF1 (x) + |F2| and degV (G∨H)\F (x) = degGV (G)\F1 (x) + n2 − |F2|. Thus, degF (x) ≤ degV (G∨H)\F (x). Similarly for y ∈ V (H) \ F2 and by (ii) gives, degF (y) ≤ degV (G∨H)\F (y). Hence, F is friendly. Therefore, F is a friendly dominating set in G ∨H. 3.4. Friendly Dominating Sets in the Corona of Graphs Theorem 13. Let G and H be connected non-trivial graphs. Suppose F ⊆ V (G ◦H) for which F ⊆ V (G). Then F is a friendly dominating set of G ◦H if and only if F = V (G). Proof. Assume first that F is a friendly dominating set of G◦H. If F ̸= V (G) choose a vertex y ∈ V (G)\F . Since F contains no vertices of the copy Hy, every vertex h ∈ V (Hy) is adjacent only to its root y and to vertices of Hy itself. Since neither y nor any vertex of Hy lies in F , the vertex h has no neighbor in F ; hence h /∈ N [F ], contradicting the assumption that F is dominating. Therefore domination forces F = V (G). Conversely, suppose F = V (G). Clearly, F is a dominating set in G◦H. Fix x ∈ V (G) and let h ∈ Hx \ F . Exactly one edge, namely {h, x}, joins h to F , so degF (h) = 1. On the other hand, h is adjacent inside Hx to degH(h) vertices, all of which lie in V (G ◦ H) \ F . Since H is connected and non–trivial, degH(h) ≥ 1, whence, degF (h) = 1 ≤ degV (G◦H)\F (h) = degH(h). Thus the inequality holds for every vertex outside F , so F is friendly. Therefore F = V (G) is indeed a friendly dominating set. Theorem 14. Let G and H be connected non–trivial graphs and let F ⊆ V (G ◦ H). Suppose F ⊆ ⋃ x∈V (G) V (Hx) and Px = F ∩ V (Hx) ̸= ∅ for every x ∈ V (G). Then F is a friendly dominating set of G ◦H if and only if the following hold for every x ∈ V (G): (i) Px is a dominating set of Hx; (ii) |Px| ≤ ⌊ |V (H)|+ degG(x) 2 ⌋ . Proof. Assume F is a friendly dominating set. Since x /∈ F for each x ∈ V (G), domination of xmust occur via an edge xp with p ∈ Px, so Px ̸= ∅. If some h ∈ Hx\Px had no neighbour in Px, it would not be dominated; hence Px dominates Hx, establishing (i). For (ii), observe that, degF (x) = |Px| and degV (G◦H)\F (x) = degG(x)+|V (H)|−|Px|, so the friendly inequality degF (x) ≤ degV (G◦H)\F (x) is equivalent to |Px| ≤ |V (H)|+ degG(x) 2 . Conversely, suppose that (i) and (ii) hold. Every x ∈ V (G) is adjacent to Px ̸= ∅, and (i) ensures Px dominates Hx; hence NG◦H [F ] = V (G◦H). To show that F is friendly. Let v ∈ V (G ◦ H) \ F . If v = x ∈ V (G), then degF (v) = |Px| ≤ |V (H)|+degG(x) 2 = degV (G◦H)\F (v) by (ii). Now, if v = h ∈ Hx \ Px, put a = degPx (h). Then degF (h) = a and degV (G◦H)\F (h) = degH(h)− a+ 1. Because a ≤ |Px| and 2|Px| ≤ |V (H)|+ degG(x) by (ii), we have 2a ≤ |V (H)|+degG(x). Since degH(h)+1 ≤ |V (H)|+degG(x), it follows I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 13 of 15 that 2 degF (h) ≤ degH(h) + 1 = degG◦H(h), i.e. degF (h) ≤ degV (G◦H)\F (h). Hence, F is friendly. Therefore, F is a friendly dominating set of G ◦H. Theorem 15. Let G and H be connected non-trivial graphs and let F ⊆ V (G◦H). Choose a non-empty subset F0 ⊆ V (G) and, for every x ∈ V (G), a subset Px ⊆ V (Hx) such that Px ̸= ∅ for at least one vertex x. Suppose F = F0 ∪ (⋃ x∈V (G) Px ) . Then the set F is a friendly dominating set of G ◦H if and only if the following conditions are satisfied. (i) For every x ∈ V (G): (a) x ∈ F0, or NG(x) ∩ F0 ̸= ∅, or Px ̸= ∅; (b) if x /∈ F0 then every vertex of Hx is adjacent to a vertex of Px; equivalently NHx [Px] = V (Hx). (ii) For every vertex x ∈ V (G) \ F0, |Px|+ degF0 (x) ≤ ( |V (H)| − |Px| ) + ( degG(x)− degF0 (x) ) . (iii) For every vertex h ∈ V (Hx) \ Px, degPx (h) + 1{x∈F0} ≤ ( degH(h)− degPx (h) ) + 1{x/∈F0}. Proof. Assume that F is a friendly dominating set of G ◦H. Let x ∈ V (G). If x ∈ F0, then we are done. Otherwise x /∈ F , so x must be dominated by a neighbor belonging to F . Its neighbors are exactly the vertices of NG(x)∪Hx; hence either NG(x)∩F0 ̸= ∅ or Px ̸= ∅, which is condition (i.a). For condition (i.b) keep x /∈ F0 and choose an arbitrary h ∈ Hx \ Px. Since x /∈ F , the vertex h can be dominated only through a neighbor in Px, so NHx(h) ∩ Px ̸= ∅. Thus, Px dominates Hx. Now take x ∈ V (G) \ F0. The number of neighbors of x that lie in F equals degF (x) = degF0 (x) + |Px|, while the number that lie outside F is degV (G◦H)\F (x) = ( |V (H)|−|Px| ) + ( degG(x)−degF0 (x) ) . Hence the inequality degF (x) ≤ degV (G◦H)\F (x) reproduces condition (ii). Finally, fix h ∈ Hx \ Px and write a = degPx (h). Then, degF (h) = a+ 1{x∈F0}, degV (G◦H)\F (h) = ( degH(h)− a ) + 1{x/∈F0}, so the friendly inequality for h is exactly condition (iii). Conversely, suppose that conditions (i) − (iii) hold. For every x ∈ V (G), by (i.a), guarantees a neighbor of x that lies in F ; hence x is dominated. Now fix h ∈ Hx. If x ∈ F0, the edge xh dominates h. If x /∈ F0, by (i.b), guarantees a neighbor of h in Px ⊆ F . Thus every vertex of Hx is dominated, and we obtain NG◦H [F ] = V (G ◦ H). Thus, F is a dominating set. To show that F is friendly. Let v /∈ F . If v = x ∈ V (G)\F0, by (ii), we have degF (v) ≤ degV (G◦H)\F (v). If v = h ∈ Hx \Px, by (iii), again, degF (v) ≤ degV (G◦H)\F (v). Hence, F is friendly. Therefore, F is a friendly dominating set of G ◦H. Corollary 7. Let G and H be connected, non-trivial graphs and let G ◦ H denote their corona. Then γf (G ◦H) = |V (G)|. I. S. Cabahug, Jr., R. G. Eballe, R. T. Fernandez / Eur. J. Pure Appl. Math, 18 (4) (2025), 6326 14 of 15 Proof. Theorem 13 shows that the set F = V (G) is a friendly dominating set of G◦H, so γf (G ◦ H) ≤ |V (G)|. Now, Theorem 14 and Theorem 15 together implies that every friendly dominating set which omits at least one vertex of V (G) must contain strictly more than |V (G)| vertices. Hence no friendly dominating set of smaller size than |V (G)| exists, so γf (G ◦H) ≥ |V (G)|. Therefore, γf (G ◦H) = |V (G)|. 4. Conclusion This paper demonstrated that the friendly domination number γf , the smallest size of a dominating set that is also friendly—always lies above or on the ordinary domination number γ. We gave complete characterizations of graphs with γf = 1 or 2, construct families in which the gap γf − γ is arbitrarily large, and derive exact expression for paths and cycles. Moreover, we established sharp Nordhaus–Gaddum bounds for the difference, the sum, and the product of γf of a graph and its complement, and we identified all friendly dominating sets in the join and the corona of two graphs, proving that in every corona γf equals the order of the host graph. The present work is restricted to simple, undirected graphs and to two basic graph operations. It does not treat weighted or directed variants, computational complexity, or behaviour under richer products such as Cartesian or strong products. Future research should therefore address several open directions: determining γf for additional graph families and products; analysing the complexity of computing γf and designing efficient approximation schemes; extending the concept to weighted, directed, or temporal graphs; and studying probabilistic thresholds for friendly domination in random graph models. 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