EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6046 ISSN 1307-5543 – ejpam.com Published by New York Business Global Independent Edge Domination Topology of Some Graph Families Jhon Neceir S. Ontulan1,∗, Cherry Mae R. Balingit1 1 Department of Mathematics, College of Arts and Sciences, Central Mindanao University, University Town, Musuan, 8710 Maramag, Bukidnon, Philippines Abstract. Let G = (V (G), E(G)) be a nonempty graph. An independent edge dominating set is an independent set of edges of G which is also an edge dominating set of G. The family of independent edge dominating sets of G generates a unique topology. In this paper, we formally define the new notion of this topology generated by the family of independent edge dominating sets (IEDS) in a graph called the independent edge domination topology on E(G), herein denoted as τEID(G). Moreover, we characterize the subbasic sets generated from the family of independent edge dominating sets and the independent edge domination topology of some graph families. 2020 Mathematics Subject Classifications: 05C70, 54A05, 54A25, 05C75 Key Words and Phrases: Independent edge domination topology, Independent edge dominating sets, Complete graph, Friendship graph, Complete bipartite graph 1. Introduction Topology and graph theory are two practically integrated branches of mathematics that overlap in the structural analysis of spaces and the study of networks and connections. From the point of view of topological ideas connected with various graph models, it could look at network properties like continuity, connectedness, and space-containing networks. This lets researchers in both directions make new tools or programs [1]. There are many ways of constructing a topological space from a given graph − undirected or directed graphs − or devising a method of generating a graph from a given (finite) topological space. Some construct models solely based on the set of vertices, while others base their models on the set of edges. However, the most common approach relies solely on the set of vertices. In the study by Macaso and Balingit [2], they introduced a new topology called the block topology, generated by the family of the vertex sets of the blocks of the graph. In 2018, Abdu and Kilicman introduced new topologies generated by edges. These two types of topologies are called edge-compatible topology and edge-incompatible topology. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6046 Email addresses: ontulanjhonneceir@gmail.com (J. N. Ontulan), f.cherrymae.balingit@cmu.edu.ph (C. M. Balingit) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 2 of 14 They are generated by the edge-compatible sets of directed graphs and edge-incompatible sets of directed graphs [3]. The same association of edge sets is the study of Alsinaia et al. [4], where they associated a new method of topologizing, in which the topology is generated by the edge neighborhoods of the discrete topological graphs. In a vertex dominating set of a graph, every vertex in the graph is either part of the subset or adjacent to at least one vertex, ensuring that the set dominates or covers every vertex through an edge. This idea is crucial in network design since the nodes in the dominant set function as control points or access nodes that efficiently govern or interact with the whole network [1]. Meanwhile, an independent set is a collection of vertices in which no two vertices are adjacent, indicating the absence of common edges and direct connections. Independent sets are important because they address issues requiring non-interference, such as the allocation of shared resources [5]. In 2020, Hassan and Abed presented another intriguing way of constructing a topological space from the graph that permits isolated vertices. This topology is called the independent topology and is generated by the family of independent sets of vertices of the graph [6]. On the other hand, an edge-dominating set is a subset of edges that covers all edges in a graph, either by inclusion in the subset or by adjacency, thereby ensuring complete domination of the edges. This concept is especially relevant in situations requiring the management or observation of relationships among nodes with a limited number of connections, such as enhancing network coverage or ensuring efficient oversight of infrastructure systems [7]. The notion of an independent set of edges, referred to as matching, is closely associated. A matching is a set of edges that do not share any common vertices. Matchings are fundamental in solving a wide variety of real-world problems, such as job assignments where tasks need to be assigned to workers, ensuring that no worker is assigned more than one task at a time [5]. Moreover, an independent edge-dominating set exemplifies a noteworthy integration of edge domination and independence characteristics. One set of edges dominates the graph and maintains its independence, meaning no two edges are adjacent. This idea makes sure that the chosen edges cover or dominate the graph efficiently and do not overlap. This notion is useful when the need arises to monitor or control edges without having any extra or direct links between them, such as in power grid optimization or routing protocols in telecommunications networks [8]. 2. Preliminaries Definition 1. [9] A graph G = (V (G), E(G)) is a finite nonempty set V (G) of objects called vertices (the singular is vertex) together with a possibly empty set E(G) of 2- element subsets of V (G) called edges. Here, V (G) is the vertex set of a graph G while E(G) is the edge set of the graph G. The order of a graph G refers to the number of vertices in G, while the size (or length) of a graph G refers to the number of edges in G. Two distinct vertices v1 and v2 are adjacent if v1, v2 ∈ G and two edges are adjacent if they have a common vertex. A graph of size 0 is called an empty graph. In any empty graph, no two vertices are adjacent. A nonempty graph then has one or more edges. A graph G is connected if every two vertices of G are adjacent, that is, if G contains a J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 3 of 14 u − v path for every pair u, v of vertices of G. A graph G that is not connected is called disconnected. Notation: Let G = (V (G), E(G)) be a simple graph of order n ∈ N, where V (G) = {vα1 , vα2 , . . . , vαn} for some indices α1, . . . , αn. Henceforth, as a convention for n ≥ 1, we denote [n] = {1, 2, . . . , n} and the edge vαivαj = ei,j , for i, j ∈ [n]. Observe that the two edges are adjacent if they have a common vertex. With the above notation, the following remark is immediate. Remark 1. Let G = (V (G), E(G)) be a simple graph. Two edges ei1,j1 and ei2,j2 of G are adjacent if and only if {i1, j1} ∩ {i2, j2} ≠ ∅. Illustration: The graph G in Figure 1 is labeled considering the convention for denot- ing the vertices and edges. Observably, two edges are adjacent if they share a common subscript. v1 v2 v3 v4 v5 v6G : e1,2 e2,3e3,4 e1,4 e4,5 e1,5 e3,5 e2,6 Figure 1: Graph G Definition 2. [9] A connected subgraph H of a graph G is a component of G if H is not a proper subgraph of any connected subgraph of G. The number of components in a graph G is denoted by k(G). Therefore, G is connected if and only if k(G) = 1. Illustration: The graph H in Figure 2 has a total of 3 components; hence, k(H) = 3. v3 v1 v2 v4 v5 v6 H : Figure 2: A graph H with k(H) = 3 Definition 3. [9] Two graphs G and H are isomorphic, denoted by G ∼= H, if there exists a bijective function ϕ : V (G) → V (H) such that two vertices u and v are adjacent in G if J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 4 of 14 and only if ϕ(u) and ϕ(v) are adjacent in H. The function ϕ is called an isomorphism from G to H. Illustration: The graphs G and H shown in Figure 3 are isomorphic, via the isomorphism ϕ : V (G) → V (H) defined by, ϕ(u1) = v2, ϕ(u2) = v3, ϕ(u3) = v1, ϕ(u4) = v4, ϕ(u5) = v5, ϕ(u6) = v6, and ϕ(u7) = v7. u1 u2 u3 u4u5 u6 u7 G : v1 v2 v3 v4 v5 v6 v7 H : Figure 3: The graph G is isomorphic to graph H Definition 4. [10] A set F of edges in a graph G is an edge dominating set if every edge not in F is adjacent to some edge in F . A set F of edges is an independent edge set if no two edges in F are adjacent. Consequently, an independent edge dominating set (IEDS) of G is an independent set of edges which is also an edge dominating set. (The family of all IEDS of G is denoted by IDE G). Definition 5. [11] Let X be a set. A topology on a point set X is a collection τ of subsets of X having the following properties: i. ∅ and X are in τ . ii. The union of the elements of any subcollection of τ is in τ ; that is, if {Uα}α∈A ⊂ τ then ⋃ α∈A Uα ∈ τ . iii. The intersection of the elements of any finite subcollection of τ is in τ ; that is, if U1, U2, . . . , Un ∈ τ then ⋂n i=1 Ui ∈ τ . A set X for which a topology τ has been specified is a topological space, denoted as the pair (X, τ). A subset of X which is in τ is called a τ-open set. If X is any set and τ1 is the collection of all subsets of X (that is, τ1 is the power set of X, τ1 = P(X)) then this is a topological space. τ1 is called the discrete topology on X. At the other extreme is the topology τ2 = {∅, X}, called the indiscrete topology or trivial topology on X. Theorem 1. [11] In a discrete topological space (X,P(X)), every singleton set {x} is both open and closed. J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 5 of 14 Definition 6. [12] Given any family Σ = {Aα : α ∈ A} of subsets of X, there always exists a unique, smallest topology τ(Σ) ⊃ Σ. The family τ(Σ) can be described as follows: It consists of ∅, X, all finite intersections of the Aα and all arbitrary unions of these finite intersections. Σ is called a subbasis for τ(Σ), and τ(Σ) is said to be generated by Σ. Example 1. Let X = {a, b, c, d} and Σ = {{a, b}, {a, c}, {b, d}, {c, d}}. By direct ap- plication of Definition 6, τ(Σ) = {∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, X} = P(X). This means that τ(Σ) is the dis- crete topology on X. 3. Independent Edge Domination Topology Definition 7. Let G be a nonempty graph. The independent edge domination topol- ogy of G, denoted by τEID(G) is the topology generated by the family IDE G of all indepen- dent edge dominating sets of G. The pair (E(G), τEID(G)) is called the independent edge domination topological space of G. Example 2. Consider the graph G in Figure 4 with E(G) = {e1,2, e1,3, e1,4, e2,3, e3,4}. The family of all independent edge dominating sets of G is given by IDE G = {{e1,3}, {e1,2, e3,4}, {e1,4, e2,3}}, and so, by Definition 6 and Definition 7, τEID(G) = {∅, {e1,3}, {e1,2, e3,4}, {e1,4, e2,3}, {e1,2, e1,3, e3,4}, {e1,3, e1,4, e2,3}, {e1,2, e1,4, e2,3, e3,4}, E(G)}. v1 v2 v3 v4G : e 1,2 e 2, 3 e 3,4 e 1, 4 e 1 ,3 Figure 4: Graph G Theorem 2. Let G be a nonempty graph. The topology τEID(G) is the indiscrete topology on E(G) if and only if G has k ≥ 1 components, where each component is isomorphic to either P1 or P2. Proof. Suppose G has a component H that has at least 2 adjacent edges ei1,j and ei2,j . Then for any IEDS S of G either ei1,j , ei2,j /∈ S or exactly one of them belongs to S. Both cases imply that there exists S ∈ IDE G ⊆ τEID(G) such that S ̸= E(G). Therefore, τEID(G) is not the indiscrete topology on E(G). J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 6 of 14 Conversely, it is easy to see that if G is nonempty and its components are either P1 or P2, then E(G) is an IEDS of G, and any proper subset of E(G) is not an edge dominating set of G. Thus, IDE G contains only E(G). Consequently, τEID(G) is the indiscrete topology on E(G). ■ Example 3. Consider the graph H in Figure 5 with E(H) = {e1,2, e4,5}. Observe that IDE H = {{e1,2, e4,5}}, and so, by Theorem 2, τEID(H) = {∅, E(H)} is the indiscrete topol- ogy on E(H). v1 v2 v3 v4 v5 H : e 1 ,2 e 4 ,5 Figure 5: The component of graph H 4. Independent Edge Domination Topology of Complete Graphs Definition 8. [9] A graph G is complete if every two distinct vertices of G are adjacent. A complete graph of order n is denoted by Kn. Therefore, Kn has the maximum possible size of a graph with n vertices. Since every two distinct vertices of Kn are joined by an edge, the number of pairs of vertices in Kn is n(n− 1) 2 . Notation: Let Kn be a complete graph of order n. We use the following notations: i. V (Kn) = {v1, v2, . . . , vn}; and ii. E(Kn) = {ei,j = vivj : vi, vj ∈ V (Kn)}. Illustration: The complete graph K5 in Figure 6 is labeled using the abovementioned notation convention. v1 v2 v3v4 v5 K5 : e 1,2 e 1 ,3e 1 ,4 e1,5 e 2 ,3 e 2, 4 e2,5 e3,4 e 3,5 e 4,5 Figure 6: The complete graph K5 J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 7 of 14 Theorem 3. Let Kn be a complete graph with n ≥ 2 and S ⊆ E(Kn). S ∈ IDE Kn if and only if S = {ei1,j1 , . . . , eik,jk} such that k = |S| = ⌊ n 2 ⌋ and i1 ̸= . . . ̸= ik ̸= j1 ̸= . . . ̸= jk. Proof. If S = {ei1,j1 , . . . , eik,jk} is as described, then the condition i1 ̸= . . . ̸= ik ̸= j1 ̸= . . . ̸= jk implies that S is independent, by Remark 1. Now, let ep,q ∈ E(Kn) \ S. If n is odd, then k = n− 1 2 and so |{i1, . . . , ik, j1, . . . , jk}| = 2k = n − 1. This means that, if p ∈ {i1, . . . , ik, j1, . . . , jk}, then ep,q is adjacent to one of the edges in S. Also, if p /∈ {i1, . . . , ik, j1, . . . , jk}, then p is the remaining element of [n] not in {i1, . . . , ik, j1, . . . , jk}, implying that q ∈ {i1, . . . , ik, j1, . . . , jk} since p ̸= q. Therefore, by Remark 1 ep,q is adjacent to an edge in S. If n is even, then k = n 2 and |{i1, . . . , ik, j1, . . . , jk}| = 2k = n this imply that |{i1, . . . , ik, j1, . . . , jk}| = [n]. Thus, if p, q ∈ {i1, . . . , ik, j1, . . . , jk}. By Remark 1, ep,q is adjacent to one of the edges in S. Hence, S is an edge dominating set of Kn. Therefore, S ∈ IDE Kn . Conversely, suppose S is not as described. If |S| > ⌊ n 2 ⌋ , then |{i1, . . . , ik, j1, . . . , jk}| = 2k ≥ n+1 implying that two of the subscripts in S are equal. By Remark 1, there exists an edge in S that share a common vertex, and so S is not independent. Also, if |S| < ⌊ n 2 ⌋ , then |{i1, . . . , ik, j1, . . . , jk}| = 2k ≤ n − 2. This means that there exist p, q ∈ [n] not appearing in {ei1,j1 , . . . , eik,jk}. So, there exists an edge ep,q in E(Kn) \ S which is not adjacent to any of the edges in S. Therefore, S is not an edge dominating set of Kn. Suppose that S contains elements ei1,j1 and ei2,j2 such that {i1, j1} ∩ {i2, j2} ̸= ∅. Then S is not an independent set. ■ Theorem 4. For the complete graph Kn with n ≥ 2, |IDE Kn | = n! 2k · k! , where k = ⌊ n 2 ⌋ . Proof. In view of Theorem 3, an IEDS of Kn is formed by choosing the k pairwise disjoint 2-element subsets {i1, j1}, {i2, j2}, . . . , {ik, jk} of [n] where k = ⌊ n 2 ⌋ . There are( n 2 ) ways to choose 2 elements from n objects, ( n− 2 2 ) ways to choose 2 elements from n−2 objects, . . . , and ( n− 2k + 2 2 ) ways to select the last 2 elements from the remaining n− 2k + 2 objects. In total, given that n− 2k is either 0 or 1,( n 2 )( n− 2 2 )( n− 2− 2 2 ) · · · ( n− 2k + 2 2 ) ︸ ︷︷ ︸ k factors = n! 2k(n− 2k)! = n! 2k . J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 8 of 14 Since the order of the elements in S does not matter, we divide this quantity by k!. Hence, |IDE Kn | = n! 2k · k! . ■ Example 4. Consider the complete graph K5 in Figure 6 with E(K5) = {e1,2, e1,3, e1,4, e1,5, e2,3, e2,4, e2,5, e3,4, e3,5, e4,5}. Observe that IDE K5 = {{e1,2, e3,4}, {e1,2, e3,5}, {e1,2, e4,5}, {e1,3, e2,4}, {e1,3, e2,5}, {e1,3, e4,5}, {e1,4, e2,3}, {e1,4, e2,5}, {e1,4, e3,5}, {e1,5, e2,3}, {e1,5, e2,4}, {e1,5, e3,4}, {e2,3, e4,5}, {e2,4, e3,5}, {e2,5, e3,4}}. Indeed, with k = ⌊5 2 ⌋ = 2, |IDE K4 | = 5! 22 · 2! = 15. Remark 2. In view of Theorem 2, τEID(K2) is the indiscrete topology on E(K2). Fur- thermore, τEID(K3) is the discrete topology on E(K3), given that IDE K3 = {{e1,2}, {e1,3}, {e2,3}} ⊆ τEID(K3), by Theorem 1. However, τEID(K4) is neither the discrete nor the in- discrete topology on E(K4). To see this, observe that E(G) = {e1,2, e1,4, e1,3, e2,3, e2,4, e3,4}, IDE K4 = {{e1,2, e3,4}, {e1,4, e2,3}, {e1,3, e2,4}} and so, τEID(K4) = {∅, {e1,2, e3,4}, {e1,4, e2,3}, {e1,3, e2,4}, {e1,2, e3,4, e1,4, e2,3}, {e1,2, e3,4, e1,3, e2,4}, {e1,4, e2,3, e1,3, e2,4}, E(K4)}. Theorem 5. Let Kn be a complete graph where n ≥ 5. Then τEID(Kn) is the discrete topology on E(Kn). Proof. Let ep,q ∈ E(Kn) where n > 5. Then, by Theorem 3, and since k = ⌊ n 2 ⌋ ≥ 3, S = {ep,q, ei1,j1 , . . . , eik−1,jk−1 } where k = ⌊ n 2 ⌋ and i1 ̸= ik−1 ̸= j1 ̸= . . . ̸= jk−1 ̸= p ̸= q is an IEDS of Kn. Observe that S′ = {ep,q, ei1,jk−1 , ei1,jk−2 , . . . , eik−1,j1} is an IEDS of Kn such that S ∩ S′ = {ep,q} ∈ τEID(Kn). If n = 5, S = {ep,q, ei1,j2} is an IEDS, for any distinct i1, j1 ∈ [5] \ {p, q}. Putting r = [5] \ {i1, j1, p, q}, S′ = {ep,q, ei1,r} is an IEDS of K5 with S ∩ S′ = {ep,q} ∈ τEID(K5), by Theorem 1. In both cases, {ep,q} ∈ τEID(Kn) for all ep,q ∈ E(Kn). Thus, by Theorem 1, τEID(Kn) is the discrete topology on E(Kn) for all n ≥ 5. ■ Corollary 1. For a complete graph Kn of order n ≥ 2, |τEID(Kn)| =  2 , if n = 2 8 , if n = 3, 4 2 n(n−1) 2 , if n ≥ 5 5. Independent Edge Domination Topology of Friendship Graphs Definition 9. [13] The friendship graph of order n ≥ 2, denoted by Frn, is a set of n copies of cycle C3 having a common vertex v0. J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 9 of 14 Notation: For the friendship graph Frn of order n ≥ 2, we use the following notations: i. V (Frn) = {v0, v1a , v1b , v2a , v2b , . . . , vna , vnb }, where via is the first vertex of the ith copy of C3; vib is the second vertex of the ith copy of C3; and v0 is the common vertex of all copies of C3 ii. E(Frn) = {e0,1a , e0,1b , e1a,1b , e0,2a , e0,2b , e2a,2b , . . . , e0,na , e0,nb , ena,nb }, where e0,ia = v0via , e0,ib = v0vib , and eia,ib = viavib . Illustration: The friendship graph Fr3 in Figure 7 is labeled using the notation conven- tion. v0 v1a v1b v3b v3a v4b v4a v2a v2b Fr4 : e0,1a e0,1b e0,2a e0,2b e0,3ae0,3b e0,4a e0,4b e1a,1b e2a,2b e3a,3b e4a,4b Figure 7: The friendship graph Fr4 Theorem 6. Let Frn be a friendship graph with n ≥ 2 and S ⊆ E(Frn). S ∈ IDE Frn if and only if S takes one of the following forms: i. S1 = {e1a,1b , e2a,2b , . . . , ena,nb } ii. Sk a = [{e1a,1b , e2a,2b , . . . , ena,nb } \ {eka,kb}] ∪ {e0,ka} for some k ∈ [n] iii. Sk b = [{e1a,1b , e2a,2b , . . . , ena,nb } \ {eka,kb}] ∪ {e0,kb} for some k ∈ [n]. Proof. By Remark 1, S1 = {e1a,1b , e2a,2b , . . . , ena,nb } is an independent edge set. Let ep,q ∈ E(Frn) \ S1. Then p = 0 and q is either ka or kb for some k ∈ [n]. Now, observe that, eka,kb ∈ S1 and is adjacent to ep,q, by Remark 1. Since ep,q is arbitrary, S1 is an edge dominating set of E(Frn), and consequently, S1 ∈ IDE Frn . In Sk a = {e1a,1b , e2a,2b , . . . , ena,nb } \ {eka,kb}, note that by removing eka,kb and replacing it with e0,ka , which is adjacent only to eka,kb , S k a is an independent edge set. Let ep,q ∈ E(Frn) \ Sk a . Then either p = 0 and q ∈ {ra, rb} for some r ∈ [n] \ {k}, or ep,q = eka,kb , and so ep,q is either adjacent to era,rb or to e0,ka which are in Sk a . Hence, Sk a is an edge dominating set. Therefore, Sk a ∈ IDE Frn . J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 10 of 14 Similarly, Sk b ∈ IDE Frn . Conversely, let S ⊆ E(Frn) such that S is not one of the given forms. If |S| > n, then there exists k ∈ [n] such that two of eka,kb , e0,ka , or e0,kb are in S, so that S is not independent. If |S| < n, then there exists k ∈ [n] such that eka,kb , e0,ka , e0,kb /∈ S. This means that eka,kb is not dominated by S, and so S is not an edge dominating set of Frn. Now, if |S| = n and there exist distinct k, k′ ∈ [n] such that e0,kx , e0,k′y ∈ S with x, y ∈ {a, b}, S is not an independent edge set by Remark 1. ■ Corollary 2. For the friendship graph Frn, n ≥ 2, |IDE Frn | = 2n+ 1. Example 5. Consider the friendship graph Fr4 in Figure 7 with E(Fr4) = {e0,1a , e0,1b , e1a,1b , e0,2a , e0,2b , e2a,2b , e0,3a , e0,3b , e3a,3b , e0,4a , e0,4b , e4a,4b , }. By Theorem 6, observe that IDE Fr4 = {{e1a,1b , e2a,2b , e3a,3b , e4a,4b}, {e2a,2b , e3a,3b , e4a,4b , e0,1a}, {e1a,1b , e3a,3b , e4a,4b , e0,2a}, {e1a,1b , e2a,2b , e4a,4b , e0,3a}, {e1a,1b , e2a,2b , e3a,3b , e0,4a}, {e2a,2b , e3a,3b , e4a,4b , e0,1b}, {e1a,1b , e3a,3b , e4a,4b , e0,2b}, {e1a,1b , e2a,2b , e4a,4b , e0,3b}, {e1a,1b , e2a,2b , e3a,3b , e0,4b}}. Indeed, |IDE Fr4 | = 2(4) + 1 = 9. Theorem 7. Let Frn be a friendship graph and S1 = {e1a,1b , e2a,2b , . . . , ena,nb }. A set S ⊆ E(Frn) is τEID(Frn)-open if and only if S satisfies any of the following forms: i. S ⊆ S1; ii. S = S1 ∪ {e0,ia : i ∈ A} ∪ {e0,ib : i ∈ B} such that A,B ∈ [n]; and iii. S = [S1 \ {eka,kb}] ∪ S∗ such that S∗ ⊆ {e0,ka , e0,kb} for all k ∈ [n]. Proof. Let S ⊆ E(Frn). If S ⊆ S1, then for some A ⊆ [n], S = {eia,ib : i ∈ A} =⋂ i/∈A Sk a ∈ τEID(Frn). Let A,B ⊆ [n]. Then, S = S1 ∪ {e0,ia : i ∈ A} ∪ {e0,ib : i ∈ B} = S1 ∪ [ ⋃ i∈A Si a] ∪ [ ⋃ i∈B Si b] ∈ τEID(Frn). If k ∈ [n] and S = [S1 \ {eka,kb}] ∪ S∗ where S∗ = ∅, then S ⊆ S1. If S∗ is either {e0,ka} or {e0,kb}, then S is also either Sk a or Sk b . If S ∗ = {e0,ka , e0,kb}, then S = Sk a ∪ Sk b . In all cases, S is τEID(Frn)-open, by Definition 6. Conversely, suppose S ⊆ E(Frn) does not take any of the given forms. Then there exist k, k′, k∗ ∈ [n] such that eka,kb , ek′a,k′b /∈ S and either e0,k∗a ∈ S or e0,k∗b ∈ S. If S is a τEID(Frn)-open set, S can be generated out of the sets in IDE Frn given in the Theorem 6. Necessarily, either Sk∗ a ⊆ S ⊆ Sk x ∩ Sk′ y or Sk∗ b ⊆ S ⊆ Sk x ∩ Sk′ y where x, y ∈ {a, b}, which are both impossible. Thus, S cannot be τEID(Frn)-open. ■ Remark 3. Let Frn be a friendship graph and S ⊆ E(Frn) with n ≥ 2. Then τEID(Frn) is not the indiscrete nor the discrete topology. Corollary 3. For the friendship graph Frn with n ≥ 2, |τEID(Frn)| = 22n + 2n + 3n− 1. J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 11 of 14 Proof. The proof immediately follows by counting the numbers of τEID(Frn)-open sets in Theorem 7. ■ 6. Independent Edge Domination Topology of Complete Bipartite Graphs Definition 10. [14] A complete bipartite is a graph whose vertex set can be partitioned into two disjoint nonempty sets Va and Vb such that two vertices via and vib are adjacent if and only if via ∈ Va and vib ∈ Vb. If |Va| = m and |Vb| = n, then such a graph is denoted Km,n. Notation: For the complete bipartite graph Km,n, the two partite sets are labeled as i. Va(Km,n) = {v1a , v2a , . . . , vma}; and ii. Vb(Km,n) = {v1b , v2b , . . . , vnb }, so that E(Km,n) = {ei,j = viavjb : via ∈ Va(Km,n), vjb ∈ Vb(Km,n)}. Illustration: The complete bipartite graph K2,3 in Figure 8 is labeled using the notation convention. v1a v2a v1b v2b v3b K2,3 : e1a,1b e 2a ,1 b e1a ,2b e 1 a ,3 b e2a,2b e2a ,3b Figure 8: The complete bipartite graph K2,3 Theorem 8. Let Km,n be a complete bipartite graph and S ⊆ E(Km,n). S ∈ IDE Km,n if and only if S = {ei1,j1 , . . . , eik,jk} such that |S| = k = min{m,n} and i1 ̸= . . . ̸= ik, j1 ̸= . . . ̸= jk where i1, . . . , ik ∈ {1a, . . . ,ma} and j1, . . . , jk ∈ {1b, . . . , nb}. Proof. Independence follows immediately by Remark 1 and by the definition of S. Let ep,q ∈ E(Km,n) \ S, and assume that k = min{m,n} = m. Since |S| = k and S is independent, all the subscripts 1a, . . . ,ma appear in S. Hence, p ∈ {1a, . . . ,ma}. This means that there exist c ∈ [m] and d ∈ [n] such that eca,db ∈ S and ca = p. By Remark 1, eca,db is adjacent to ep,q. Since ep,q is arbitrary, S is an edge dominating set of Km,n. J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 12 of 14 Conversely, assume WLOG that k = min{m,n} = m. If |S| > k = m, then by the Pigeonhole Principle applied to the subscript p of the edges ep,q in S, there exist at least two edges in S sharing the same first subscripts. By Remark 1, S is not independent. If |S| < k = m < n, then there exists an edge ep,q ∈ E(Km,n) \ S such that p and q do not appear as subscripts of the edges in S. Hence, ep,q is not adjacent to any element of S, by Remark 1. Therefore, S is not an edge dominating set of Km,n. If |S| = k but fails to satisfy the condition i1 ̸= . . . ̸= ik, j1 ̸= . . . ̸= jk, then, by Remark 1, S is not an independent nor an edge dominating set of Km,n. ■ Theorem 9. For the complete bipartite graph Km,n with m ≤ n |IDE Km,n | = n! (n−m)! . Proof. Assume that m ≤ n. Observe that by Theorem 8, all subscripts 1a, 2a, . . . ,ma appear in every independent edge dominating set S of Km,n. Each of these subscripts is paired to one of the subscripts 1b, . . . , nb, where the chosen subscripts are all distinct. There are n choices for the subscript paired to 1a, n − 1 choices for the subscript paired to 2a, . . . , and n −m + 1 choices for the subscript paired to ma. This means that there are n(n− 1)(n− 2) . . . (n−m+1) = n! (n−m)! such pairs and so |IDE Km,n | = n! (n−m)! . ■ Example 6. Consider the complete bipartite graph K2,3 in Figure 8 with E(K2,3) = {e1a,1b , e1a,2b , e1a,3b , e2a,1b , e2a,2b , e2a,3b}. Observe that IDE K2,3 = {{e1a,1b , e2a,2b}, {e1a,1b , e2a,3b}, {e1a,2b , e2a,1b}, {e1a,2b , e2a,3b}, {e1a,3b , e2a,1a}, {e1a,3b , e2a,2b}}. Indeed, |IDE Km,n | = 6. Remark 4. i. τEID(K1,1) is the indiscrete topology on E(K1,1), by Theorem 2. ii. τEID(K2,2) is neither the discrete nor the indiscrete topology on E(K2,2). To see this, observe that IDE K2,2 = {{e1a,1b , e2a,2b}, {e1a,2b , e2a,1b}} and so τEID(K2,2) = {∅, {e1a,1b , e2a,2b}, {e1a,2b , e2a,1b}, E(K2,2)}. Theorem 10. For the complete bipartite graph K1,n (star graph) with n ≥ 2, τEID(K1,n) is the discrete topology on E(K1,n). Proof. Let S ⊆ E(K1,n) where n ≥ 2. By Theorem 8, |S| = min{1, n} = 1, implying that every singleton subsets of E(K1,n) is an independent edge dominating set. Conse- quently, every subset of E(Km,n) is τEID(Km,n)-open, thus, by Theorem 1, τEID(Km,n) is the discrete topology. ■ Theorem 11. Let Km,n be a complete bipartite graph with m ≤ n. If m ≥ 2 and n ≥ 3, then τEID(Km,n) is the discrete topology on E(Km,n). J. N. Ontulan, C. M. Balingit / Eur. J. Pure Appl. Math, 18 (2) (2025), 6046 13 of 14 Proof. Consider the complete bipartite graph Km,n such that m ≤ n, m ≥ 2 and n ≥ 3. Let ep,q ∈ E(Km,n) and S1 = {ep,q, ei1,j1 , ei2,j2 , . . . , eim−1,jm−1} such that p, i1, . . . , im−1 ∈ {1a, 2a, . . . ,ma}, q, j1, . . . , jm−1 ∈ {1b, 2b, . . . , nb} where p ̸= i1 ̸= i2 ̸= · · · ≠ im−1 and q ̸= j1 ̸= j2 ̸= · · · ̸= jm−1. By the preceding theorem, S1 ∈ IDE Km,n . Now, if |S1| = m = 2, let S2 = {ep,q, ei1,j2} where j2 ̸= q ̸= j1; otherwise, let S2 = {ep,q, ei1,jm−1 , ei2,jm−2 , . . . , eim−1,j1}. Then S2 ∈ IDE Km,n such that S1 ∩ S2 = {ep,q} ∈ τEID(Km,n). By Theorem 1, τEID(Km,n) is the discrete topology on E(Km,n). ■ Corollary 4. 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