EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6951 ISSN 1307-5543 – ejpam.com Published by New York Business Global Vertex-Generator Subgraphs of Complete Bipartite and Tadpole Graphs Gino Derek M. Sepillo1,∗, Ma. Joyce G. Valdez1, Meguilito Y. Eyao1, Neil M. Mame1 1 College of Arts and Sciences, Batangas State University, The National Engineering University, Pablo Borbon Campus, Batangas City, Batangas, Philippines Abstract. Graphs considered in this paper are finite simple undirected graphs. Let G = (V (G), E(G)) be a graph with the vertex set V (G) = {x1, x2, . . . , xn}, for some positive integer n. The vertex space V (G) of G, is a vector space over the field Z2 = {0, 1}. The elements of V (G) are all the subsets of V (G). Vector addition is defined as A+B = A△B, the symmetric difference of sets A and B, for all A,B ∈ V (G). Scalar multiplication is defined as 1 · A = A and 0 · A = ∅, for all A ∈ V (G). The subgraph G⟨S⟩ of G induced by a subset S of V (G), is the largest subgraph whose vertex set is S. The vertex-uniform set VH(G) of a subgraph H with respect to G, is the set of all elements of V (G) that induces a subgraph isomorphic to H. The span of VH(G) shall be denoted by VH(G). If VH(G) is a generating set, that is VH(G) = V (G), then H is called a vertex-generator subgraph of G. This study determines some vertex-generator subgraphs of complete bipartite graph Km,n and tadpole graph Tn,m. 2020 Mathematics Subject Classifications: 05C50 Key Words and Phrases: Vertex space, induced subgraph, vertex-uniform set, generating set, vertex-generator subgraph 1. Introduction Graph theory, a fundamental area of mathematics concerned with the study of rela- tionships through networks, has evolved into a vital tool across numerous scientific and technological disciplines. Its capacity to model complex systems has led to significant advancements in fields ranging from computer science to social network analysis. The integration of algebraic structures has significantly broadened the scope of graph theory. A key development in this direction is the concept of vector spaces, notably edge spaces and vertex spaces [1]. Within this framework, these spaces are treated as vector ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6951 Email addresses: 21-58341@g.batstate-u.edu.ph (G. D. Sepillo), 20-59247@g.batstate-u.edu.ph (M. J. Valdez), 15-52878@g.batstate-u.edu.ph (M. Y. Eyao), neil.mame@g.batstate-u.edu.ph (N. M. Mame) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 2 of 23 spaces over the finite field Z2 = {0, 1}, with vector addition defined as the symmetric difference of sets and scalar multiplication resulting in either the empty set (multiplication by 0) or the original set (multiplication by 1). The only difference is that, the edge spaces contains all the subsets of the edge set of that graph, while the vertex spaces contains all the subsets of the vertex set of that graph. The notion of vertex space has been applied in studies such as that by Butenko, Festa, and Pardalos [2], who utilized it to analyze colored vertex sets with adjacency and color- ing constraints. Building on the concept of vertex spaces and induced subgraphs, Torino and Mame [3] introduced the vertex-generator subgraph, a concept parallel to the edge- generator subgraph (or generator subgraph) first explored by Gervacio [4]; see also [5]. A key difference is that vertex-generator subgraphs can contain isolated vertices, while generator subgraphs do not allow isolated vertices. While generator subgraphs have been studied for various graph classes, research on vertex-generator subgraphs remains less ex- tensive, with Torino and Mame’s work being a significant contribution. A related concept, the even vertex space (elements of the vertex space with even cardinality) offers another perspective for studying graph structure and identifying vertex-generator subgraphs. A graph G is an ordered pair (V (G), E(G)), where the vertex set V (G) is a finite nonempty set of objects called vertices, and the edge set E(G) is a set of unordered pairs of vertices called edges. The order n of G is the number of elements of V (G), and the size m of G is the number of elements of E(G). Let H = (V (H), E(H)) be another graph. A mapping ϕ : V (G) 7→ V (H) is called a graph isomorphism if the following conditions are satisfied: (i) ϕ is bijective; (ii) [a, b] ∈ E(G) implies that [ϕ(a), ϕ(b)] ∈ E(H); and (iii) [c, d] ∈ E(H) implies that [ϕ−1(c), ϕ−1(d)] ∈ E(G). A graph G is isomorphic to a graph H, denoted as G ≃ H, if there is an isomorphism ϕ : V (G) 7→ V (H) between their vertex sets. A graph H is a subgraph of G, denoted by H ⊆ G, if V (H) ⊆ V (G) and E(H) ⊆ E(G). For a non-empty subset S of V (G), the subgraph of G vertex-induced by S, denoted by G⟨S⟩, is the subgraph of G whose vertex set is V (G⟨S⟩) = S and whose edge set is E(G⟨S⟩) = {[x, y] ∈ E(G) | x, y ∈ S}. A subgraph H of G is called a vertex-induced subgraph or simply induced subgraph, if there exists a non-empty subset S ⊆ V (G) such that H = G⟨S⟩. Torino and Mame’s work [3] primarily focused on characterizing vertex-generator sub- graphs for several well-known graph classes, including path graphs, cycle graphs, empty graphs, complete graphs, star graphs, and wheel graphs. However, the study of vertex- generator subgraphs in other graph classes, such as complete bipartite graphs and tadpole graphs, remains an open area of research. A complete bipartite graph Km,n is a graph in which V (G) is partitioned into subsets U and V called partite sets, where the cardinality of U and V are m and n, respectively, and every vertex of U is adjacent to every vertex of V [6]. A tadpole graph Tn,m is the graph obtained by joining a cycle Cn to a path Pm, with a bridge [x, y], where x ∈ V (Cn) and y ∈ V (Pm) and degPm (y) is either 0 or 1 [7]. Other graph classes that have been identified as vertex-generator subgraphs of tadpole and complete bipartite graphs are defined in the pertinent part of this work. Readers may refer to the books written by Bollobás [8], Bondy and Murty [9], Chartrand, Lesniak and Zhang [10], and Harary [6], for other basic G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 3 of 23 concepts in graph theory. Also, readers may refer to the books written by Nering [11], and Larson and Falvo [12], for the linear algebra concepts, particularly the vector spaces. The objective of this research is to determine some vertex-generator subgraphs of complete bipartite and tadpole graphs. The researchers first established a fixed labeling for each of the two graphs, and this labeling was then used to determine some vertex- uniform sets of subgraphs with respect to these two graphs. By applying the symmetric difference to the elements of a vertex-uniform set, it can be shown that a subgraph is a vertex-generator subgraph by applying the existing results of Torino and Mame [3]. The researchers tested several smaller subgraphs to identify patterns and eventually formulated a general result. 2. Preliminaries 2.1. Vertex-Generator Subgraph of a Graph This section gives the definition of the vertex space, the vertex-uniform set, the span of a vertex-uniform set, and the vertex-generator subgraph. This section also provides results relevant to the said concepts. The main reference for this section is [3]. Definition 1. [1] Let G = (V (G), E(G)) be a graph. The vertex space of G, denoted by V (G), is a vector space over a field Z2 = {0, 1}, which composes of all subsets of V (G), where for A,B ∈ V (G), vector addition and scalar multiplication are given by (i) A+B = A△B, the symmetric difference of A and B. (ii) cA = A if c = 1 and cA = ∅ if c = 0. Notably, the basis of the vertex space can be found in the book of Diestel [1], which is presented in the following theorem. Theorem 1. [1] Let G be a graph with V (G) = {x1, x2, x3, ..., xn}. Then the set A = {{x1}, {x2}, {x3}, ..., {xn}} forms a basis for V (G). Hence, dimV (G) = n, the order of G. Knowing the structure of the vertex space through its basis, we now explore specific subsets of this space—particularly those associated with induced subgraphs—and how they contribute to generating the entire vertex space. This leads to the notions of vertex- uniform sets, their span, and the concept of vertex-generator subgraphs. Definition 2. [3] Let H be a subgraph of a graph G. The vertex-uniform set of H with respect to G, denoted by VH(G), is the set of all elements of V (G) that induces a subgraph isomorphic to H. Definition 3. [3] Let VH(G) be a vertex-uniform set of H with respect to G. The span of VH(G), denoted by VH(G), is the set of all linear combinations of the elements of VH(G). That is, if VH(G) = {A1, A2, A3, ..., Ak} where Ai ∈ V (G), then VH(G) = { k∑ i=1 ciAi | ci ∈ {0, 1} } . G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 4 of 23 Definition 4. [3] Let VH(G) be the span of VH(G). If VH(G) = V (G), then H is a vertex-generator subgraph of G. Note that A = {{x1}, {x2}, {x3}, ..., {xn}} forms a basis for V (G) by Theorem 1. Additionally, VH(G) ⊆ V (G). To show that H is a vertex-generator subgraph of G, it is sufficient to show that V (G) ⊆ VH(G). That is, {{x1}, {x2}, {x3}, ..., {xn}} ⊆ VH(G). Thus, the following remark gives a necessary and sufficient condition for a subgraph to be a vertex-generator subgraph of a graph. Remark 1. [3] Let G be a graph with vertex set V (G) = {x1, x2, x3, ..., xn}. Let H be a subgraph of G. Then H is a vertex-generator subgraph of G if and only if {xi} ∈ VH(G) for all 1 ≤ i ≤ n. For example, consider the cycle graph C5 in Figure 1, with V (C5) = {a1, a2, a3, a4, a5} and E(C5) = {[a1, a2], [a2, a3], [a3, a4], [a4, a5], [a1, a5]}. a1 a2 a3 a4 a5 C5 : Figure 1: A cycle graph C5 We show that the path graph P3 is a vertex-generator subgraph of C5. First, we determine the vertex-uniform set of P3 with respect to C5, as follows. A = {{a1, a2, a3}, {a1, a2, a5}, {a1, a4, a5}, {a2, a3, a4}, {a3, a4, a5}}. It can be observed that each element of A induces a subgraph that is isomorphic to P3, hence A ⊆ VP3(C5). Next, we show that each singleton is a linear combination of the elements of A. {a2, a3, a4} △ {a3, a4, a5} △ {a1, a2, a5} = {a1}, {a3, a4, a5} △ {a1, a4, a5} △ {a1, a2, a3} = {a2}, {a1, a4, a5} △ {a1, a2, a5} △ {a2, a3, a4} = {a3}, {a1, a2, a5} △ {a1, a2, a3} △ {a3, a4, a5} = {a4}, and {a1, a2, a3} △ {a2, a3, a4} △ {a1, a4, a5} = {a5}. Hence, {ai} ∈ VP3(C5) for all 1 ≤ i ≤ 5. Therefore, by Remark 1, P3 is a vertex-generator subgraph of C5. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 5 of 23 2.1.1. Some Known Results The first theorem shows that the trivial graph is a vertex-generator subgraph of any graph. Theorem 2. [3] The trivial graph K1 is a vertex-generator subgraph of any graph G. The trivial graph is clearly a vertex-generator subgraph, because it is always isomorphic to each vertex of any graph, so its uniform set always consists of singletons. The next theorem is very useful in finding a vertex-generator subgraph of a certain graph, which is based on the order of that graph. Theorem 3. [3] Let H be a subgraph of G. If H is a vertex-generator subgraph of G, then |V (H)| is odd. The next theorem tells us the relationship of the cardinality of the sets V (G) and VH(G), where H is a subgraph of any graph G. Theorem 4. [3] Let H be a subgraph of the graph G. If H is a vertex-generator subgraph of G, then |VH(G)| ≥ |V (G)|. The next theorem gives a necessary and sufficient condition for a vertex-generator subgraph of any graph G, where |V (G)| ≤ 3. Theorem 5. [3] Let H be a subgraph of G, where |V (G)| ≤ 3. Then H is a vertex- generator subgraph of G if and only if H ≃ K1. Let V ∗(G) be the set of all elements of V (G) with even cardinality. Torino and Mame called V ∗(G) as the even vertex space of G [3]. The following theorem presents the relevance of the even vertex space of a graph, to the vertex space of the graph. Theorem 6. [3] Let G be a graph of order n. Then, V ∗(G) is a subspace of V (G). Moreover, dimV ∗(G) = n− 1. A basis formed from the even vertex space of any graph is presented in the theorem below, which is parallel to Theorem 1. Theorem 7. [3] Let G be a graph with V (G) = {x1, x2, x3, ...xn}. Then the set B = {{x1, x2}, {x1, x3}, {x1, x4}, ..., {x1, xn}} forms a basis for V ∗(G). The next theorem is also important in the concept of the even vertex space, which is presented below. Theorem 8. [3] Let G and H be graphs such that H ⊆ G, and |V (H)| is odd. If V ∗(G) ⊆ VH(G), then H is vertex-generator subgraph of G. Theorem 8 tells us that we only need to show the basis for the even vertex space of the graph is a subset of the span of the the vertex-uniform set, in order to show that a subgraph is a vertex-generator subgraph of a graph. This follows a useful remark, which is parallel to Remark 1. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 6 of 23 Remark 2. [3] Let G be a graph with vertex set V (G) = {x1, x2, x3, ..., xn}. Let H be a subgraph of G. Then H is a vertex-generator subgraph of G if and only if {x1, xi} ∈ VH(G) for all 2 ≤ i ≤ n. The following theorem provides the necessary and sufficient conditions for a subgraph H to be a vertex generator of an empty graph Kn. This particular result will be used in following discussion. Theorem 9. [3] Let n and t be postive integers. Let H be a subgraph of Kn, where |V (H)| = t. Then H is a vertex-generator subgraph of Kn if and only if the following conditions are satisfied: (i) t is odd; (ii) 1 ≤ t ≤ n− 1; and (iii) H ≃ Kt. 3. Main Results 3.1. Vertex-Generator Subgraph of Complete Bipartite Graph Km,n This section provides some vertex-generator subgraphs of complete bipartite graph Km,n. Let Km,n be the complete bipartite graph with vertex set V (Km,n) = M ∪N , where M = {x1, x2, x3, ..., xm−1, xm} and N = {y1, y2, y3, ..., yn−1, yn} are the partite sets, and edge set E(Km,n) = {[xi, yj ]} for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. Presented in Figure 2 is the labeling of a complete bipartite graph, which will be considered in the discussion of this section. x1 x2 x3 xm−1 xm y1 y2 yn−1 yn Km,n : Figure 2: The Labeling of Km,n A complete bipartite graph Km,n has order m + n and size mn for all posi- tive integers m and n. By Definition 1, the vertex space of Km,n is given by V (Km,n) = {S | S ⊆ V (Km,n)}. Given the vertex set V (Km,n), the set A = {{x1}, {x2}, {x3}, ..., {xm−1}, {xm}, {y1}, {y2}, {y3}, . . . , {yn−1}, {yn}} G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 7 of 23 forms a basis for V (Km,n), thus dimV (Km,n) = m+ n by Theorem 1. Furthermore, the even vertex space of Km,n is given by V ∗(Km,n) = {S ∈ V (Km,n) | |S| is even}. In view of Theorem 7, the set B = {{x1, x2}, {x1, x3}, ..., {x1, xm}, {x1, y1}, {x1, y2}, ..., {x1, yn}} forms a basis for V ∗(Km,n). Hence, dimV ∗(Km,n) = m+ n− 1 by Theorem 6. By Theorem 2, we know that trivial subgraph K1 is a vertex-generator subgraph of Km,n. Additionally, We can observe that Km,n exhibits some well-known isomorphisms for small values of m and n, such as K1,1 ≃ P2, K2,1 ≃ S2 ≃ P3, and K2,2 ≃ C4. By Theorem 5, the vertex-generator subgraph of P2 and P3 is the trivial graph. The vertex-generator subgraph of the cycle graph C4 was also studied. Hence, will now focus our investigation on determining the vertex-generator subgraphs of Km,n where min{m,n} ≥ 2 and m+n ≥ 5. The following theorem presents a necessary condition for an empty graph Kt to be a vertex-generator subgraph of Km,n. Theorem 10. Let m, n and t be positive integers such that min{m,n} ≥ 2, m + n ≥ 5, and t is odd. If t < min{m,n}, then Kt is a vertex-generator subgraph of Km,n. Proof. From the labeling of complete bipartite graph, it can be observed that the subgraphs of Km,n induced by the partite sets M and N , denoted by Km,n⟨M⟩ and Km,n⟨N⟩, are isomorphic to the empty graphs Km and Km, respectively. This is shown in Figure 3. x1 x2 x3 xm−1 xm y1 y2 yn−1 yn Km,n⟨M⟩ : x1 x2 x3 xm−1 xm y1 y2 yn−1 yn Km,n⟨N⟩ : Figure 3: Illustrating the subgraphs of Km,n induced by M and N Let Kt be a subgraph of both Km and Kn. It was given that t is odd, so condition (i) of Theorem 9 is sastified. Next, since t < min{m,n}, it follows that t < m and t < n, or t ≤ m − 1 and t ≤ n − 1, which satisfies condition (ii) of Theorem 9. Lastly, since Kt is isomorphic to itself, condition (iii) of Theorem 9 is also satisfied. All the conditions have met, hence Kt is a vertex-generator subgraph of both Km and Kn. It follows that {xi} ∈ VKt (Km) for 1 ≤ i ≤ m, and {yj} ∈ VKt (Kn) for 1 ≤ j ≤ n, by Remark 1. Now, we know that VKt (Km) ∪ VKt (Kn) = VKt (Km,n⟨M⟩) ∪ VKt (Km,n⟨N⟩) = VKt (Km,n), this means that {xi}, {yj} ∈ VKt (Km,n). Therefore, by Remark 1, Kt is a vertex-generator subgraph of Km,n. The following theorem will give a necessary condition for the path graph Pt to be a vertex-generator subgraph of Km,n. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 8 of 23 Theorem 11. Let m and n be positive integers such that min{m,n} ≥ 2 and m+ n ≥ 5. If t = 1 or t = 3, then Pt is a vertex-generator subgraph of Km,n. Proof. Suppose t = 1 or t = 3. By Theorem 2, P1 is a trivial vertex-generator subgraph of Km,n. Now, consider the labeling of Km,n. For any 1 ≤ i ≤ m, let xi be an arbitrary vertex of partite set M in the complete bipartite graph Km,n, and let A, B, and C be defined as follows: A = {x1, x2, y1}, B = {x1, x2, y2}, and C = {xi, y1, y2} where 1 ≤ i ≤ m. It can be verified that A,B,C ∈ VP3(Km,n), as shown in Figure 4. x1 x2 x3 xm−1 xm y1 y2 yn−1 yn Km,n⟨A⟩ : x1 x2 x3 xm−1 xm y1 y2 yn−1 yn Km,n⟨B⟩ : x1 x2 xi xm−1 xm y1 y2 yn−1 yn Km,n⟨C⟩ : Figure 4: Illustrating the subgraphs of Km,n induced by A, B, and C Hence, for any i = 1, 2, 3, ...,m, we get A△B △ C = {x1, x2, y1} △ {x1, x2, y2} △ {xi, y1, y2} = {y1, y2} △ {xi, y1, y2} = {xi}. Hence, {xi} ∈ VP3(Km,n) for all 1 ≤ i ≤ m. Similarly, for any 1 ≤ j ≤ n, let yj be an arbitrary vertex of partite set N in the complete bipartite graph Km,n, and let D, E, and F be defined as follows: D = {x1, y1, y2}, E = {x2, y1, y2}, and F = {x1, x2, yj} where 1 ≤ j ≤ n. It can be verified that D,E, F ∈ VP3(Km,n), as shown in Figure 5. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 9 of 23 x1 x2 x3 xm−1 xm y1 y2 yn−1 yn Km,n⟨D⟩ : x1 x2 x3 xm−1 xm y1 y2 yn−1 yn Km,n⟨E⟩ : x1 x2 x3 xm−1 xm y1 y2 yj yn Km,n⟨F ⟩ : Figure 5: Illustrating the subgraphs of Km,n induced by D, E, and F Consequently, for any j = 1, 2, 3, ..., n, we obtain D△ E △ F = {x1, y1, y2} △ {x2, y1, y2} △ {x1, x2, yj} = {x1, x2} △ {x1, x2, yj} = {yj}. Hence, {yj} ∈ VP3(Km,n) for all 1 ≤ j ≤ n. Therefore, P3 is a vertex-generator subgraph of Km,n by Remark 1. It is notable that the path graph P3, as a vertex-generator subgraph of Km,n, is iso- morphic to a star graph S2. We can extend this up to St of order t + 1, hence t must be even. With this, the following theorem gives us a necessary condition for St to be a vertex-generator subgraph of Km,n. Theorem 12. Let m, n, and t be positive integers such that min{m,n} ≥ 2, m+ n ≥ 5, and t is even. If t ≤ min{m,n}, then St is a vertex-generator subgraph of Km,n. Proof. Let t ≤ min{m,n}. For any 1 ≤ i ≤ m, let xi be an arbitrary vertex of partite set M in the complete bipartite Km,n, and for any 1 ≤ p ≤ t, let Ap and A be defined as follows: Ap = {x1, x2, x3, ..., xt, yp} where 1 ≤ p ≤ t, and A = {xi, y1, y2, y3, ..., yt} where 1 ≤ i ≤ m. It can be verified that Ap, A ∈ VSt(Km,n), as shown in Figure 6. Hence, for any i = 1, 2, 3, ...,m, we have t∑ p=1 Ap +A = (A1 +A2 +A3 + · · ·+At) +A G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 10 of 23 x1 x2 xt−1 xt xt+1 xm−1 xm y1 y2 yt−1 yt yn−1 yn Km,n⟨Ap⟩ : x1 x2 x3 xi xm−2 xm−1 xm y1 y2 yt−1 yt yn−1 yn Km,n⟨A⟩ : Figure 6: Illustrating the subgraphs of Km,n induced by Ap and A = (A1 △A2 △A3 △ · · · △At)△A = {y1, y2, y3, ..., yt} △ {xi, y1, y2, y3, ..., yt} = {xi}. Thus, {xi} ∈ VSt(Km,n) for all 1 ≤ i ≤ m. In a similar manner, for any 1 ≤ j ≤ n, let yj be an arbitrary vertex of partite set N in the complete bipartite graph Km,n, and for any 1 ≤ q ≤ t, let Bq and B be defined as follows: Bq = {xq, y1, y2, y3, ..., yt} where 1 ≤ q ≤ t, and B = {x1, x2, x3, ..., xt, yj} where 1 ≤ j ≤ n. It can be verified that Bq, B ∈ VSt(Km,n), as shown in Figure 7. x1 x2 xt−1 xt xt+1 xm−1 xm y1 y2 yt−1 yt yn−1 yn Km,n⟨Bq⟩ : x1 x2 xt−1 xt xt+1 xm−1 xm y1 y2 y3 yj yn−1 yn Km,n⟨B⟩ : Figure 7: Illustrating the subgraphs of Km,n induced by Bq and B G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 11 of 23 Hence, for any j = 1, 2, 3, ..., n, we get t∑ q=1 Bq +B = (B1 +B2 +B3 + · · ·+Bt) +B = (B1 △B2 △B3 △ · · · △Bt)△B = {x1, x2, x3, ..., xt} △ {x1, x2, x3, ..., xt, yj} = {yj}. Thus, {yj} ∈ VSt(Km,n) for all 1 ≤ j ≤ n. Therefore, by Remark 1, St is a vertex-generator subgraph of Km,n. The following theorem provides a necessary condition for the complete bipartite graph Kr,r+1 to be a vertex-generator subgraph of Km,n. Theorem 13. Let m, n, and r be positive integers such that min{m,n} ≥ 2 and m+n ≥ 5. If r + 1 ≤ min{m,n}, then Kr,r+1 is a vertex-generator subgraph of Km,n. Proof. Let r + 1 ≤ min{m,n}. Then, we have the following cases: Case 1. If r = 1, then we have a subgraph K1,2 of Km,n. It can be observed that K1,2 is isomorphic to the path graph P3. Since we have shown in Theorem 11 that P3 is a vertex-generator subgraph of Km,n, it follows that K1,2 is a vertex-generator subgraph of Km,n. Case 2. We let r ≥ 2. Then, for any 2 ≤ p ≤ m − r and for any m − r + 2 ≤ q ≤ m, let xp and xq be arbitrary vertices of partite set M in the complete bipartite graph Km,n, and let sets A, B, C, and D be defined as follows: A = {x1, r−1 vertices︷ ︸︸ ︷ xm−r+2, xm−r+3, ..., xm, r+1 vertices︷ ︸︸ ︷ y1, y2, ..., yr, yr+1}, B = A \ {x1} ∪ {xp} where 2 ≤ p ≤ m− r, C = { r vertices︷ ︸︸ ︷ xm−r+1, xm−r+2, xm−r+3, ..., xm, r+1 vertices︷ ︸︸ ︷ y1, y2, ..., yr, yr+1}, and D = C \ {xq} ∪ {x1} where m− r + 2 ≤ q ≤ m. It can be verified that A,B,C,D ∈ VKr,r+1(Km,n), as shown in Figure 8. Consequently, for p = 2, 3, 4, ...,m− r and for q = m− r + 2,m− r + 3, ...,m, we obtain A△B = {x1, xp}, A△ C = {x1, xm−r+1}, and C △D = {x1, xq}. Hence, {x1, xp}, {x1, xm−r+1}, {x1, xq} ∈ VKr,r+1(Km,n) for all 2 ≤ p ≤ m − r and for all m− r + 2 ≤ q ≤ m, or equivalently, {x1, xi} ∈ VKr,r+1(Km,n) for all 2 ≤ i ≤ m. By similar argument, for any 1 ≤ s ≤ n − r − 1 and for any n − r + 1 ≤ t ≤ n, let ys and yt be G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 12 of 23 x1 x2 xm−r xm−r+1xm−r+2xm−r+3 xm y1 y2 yr yr+1 yn−1 yn Km,n⟨A⟩ : x1 x2 xm−r xm−r+1xm−r+2xm−r+3 xm y1 y2 yr yr+1 yn−1 yn Km,n⟨B⟩ : x1 x2 xm−r xm−r+1xm−r+2xm−r+3 xm y1 y2 yr yr+1 yn−1 yn Km,n⟨C⟩ : x1 x2 xm−r xm−r+1xm−r+2xm−r+3 xm y1 y2 yr yr+1 yn−1 yn Km,n⟨D⟩ : Figure 8: Illustrating the subgraphs of Km,n induced by A, B, C, and D arbitrary vertices of partite set N in the complete bipartite graph Km,n, and let sets W , X, Y , and Z be defined as follows: W = { r+1 vertices︷ ︸︸ ︷ x1, x2, x3, ..., xr, xr+1, r vertices︷ ︸︸ ︷ yn−r+1, yn−r+2, ..., yn}, X = W \ {x1} ∪ {ys} where 1 ≤ s ≤ n− r − 1, Y = { r vertices︷ ︸︸ ︷ x2, x3, ..., xr, xr+1, r+1 vertices︷ ︸︸ ︷ yn−r, yn−r+1, yn−r+2, ..., yn}, and Z = Y \ {yt} ∪ {x1} where n− r + 1 ≤ t ≤ n. It can be verified that W,X, Y, Z ∈ VKr,r+1(Km,n), as shown in Figure 9. As a consequence, for s = 1, 2, 3, ..., n− r − 1 and for t = n− r+1, n− r+2, ..., n, we get W △X = {x1, ys}, W △ Y = {x1, yn−r}, and Y △ Z = {x1, yt}. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 13 of 23 x1 x2 x3 xr xr+1 xm−1 xm y1 yn−r−1 yn−r yn−r+1 yn−r+2 yn Km,n⟨W ⟩ : x1 x2 x3 xr xr+1 xm−1 xm y1 yn−r−1 yn−r yn−r+1 yn−r+2 yn Km,n⟨X⟩ : x1 x2 x3 xr xr+1 xm−1 xm y1 yn−r−1 yn−r yn−r+1 yn−r+2 yn Km,n⟨Y ⟩ : x1 x2 x3 xr xr+1 xm−1 xm y1 yn−r−1 yn−r yn−r+1 yn−r+2 yn Km,n⟨Z⟩ : Figure 9: Illustrating the subgraphs of Km,n induced by W , X, Y , and Z Hence, {x1, ys}, {x1, yn−r}, {x1, yt} ∈ VKr,r+1(Km,n) for all 1 ≤ s ≤ n − r − 1 and for all n − r + 1 ≤ t ≤ n, or equivalently, {x1, yj} ∈ VKr,r+1(Km,n) for all 1 ≤ j ≤ n. Thus, by Remark 2, Kr,r+1 is a vertex-generator subgraph of Km,n if r ≥ 2. Therefore, in all cases, Kr,r+1 is a vertex-generator subgraph of Km,n. 3.2. Vertex-Generator Subgraph of Tadpole Graph Tn,m This section provides some vertex-generator subgraphs of tadpole graph Tn,m. Let Tn,m be a tadpole graph whose vertex set is given by V (Tn,m) = V (Cn) ∪ V (Pm), where V (Cn) = {x1, x2, x3, ..., xn−1, xn} and V (Pm) = {y1, y2, y3, ..., ym−1, ym}, and the edge set is given by E(Tn,m) = E(Cn) ∪ E(Pm) ∪ {[x1, y1]} where [x1, y1] is a bridge. Presented in Figure 10 is the labeling of a tadpole graph, which will be considered in the discussion of this section. A tadpole graph Tn,m has order n+m and size n+m for all positive integers n ≥ 3 and m. By Definition 1, the vertex space of Tn,m is given by V (Tn,m) = {S | S ⊆ V (Tn,m)}. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 14 of 23 x2 x3 x4 x5 x6 xn−2 xn−1 xn x1 y1 y2 ym−1 ym Tn,m : Figure 10: The Labeling of Tn,m Given the vertex set V (Tn,m), the set A = {{x1}, {x2}, {x3}, . . . , {xn−1}, {xn}, {y1}, {y2}, {y3}, . . . , {ym−1}, {ym}} forms a basis for V (Tn,m). It follows that dimV (Tn,m) = n+m by Theorem 1. By Theorem 2, the trivial graph is a vertex-generator subgraph of Tn,m, so we are interested in the finding the nontrivial vertex-generator subgraph of Tn,m. The following theorem gives us a necessary condition for the disjoint union of a path graph Pt of order t, so t should be even, and a trivial graph K1, denoted by Pt ⊔K1, to be a vertex-generator subgraph of Tn,m. Theorem 14. Let n,m ≥ 3 and t ≥ 2 be positive integers such that t is even. If t+ 1 ≤ min{n,m}, then Pt ⊔K1 is a vertex-generator subgraph of Tn,m. Proof. Let t < min{n,m}. For any 1 ≤ i ≤ m, let xi be an arbitrary vertex of Cn in the tadpole graph Tn,m, and for any 1 ≤ p ≤ t, let Ap and A be defined as follows: Ap = { t vertices︷ ︸︸ ︷ x2, x3, x4, ..., xt+1, yp+1} where 1 ≤ p ≤ t, and A = {xi, t vertices︷ ︸︸ ︷ y2, y3, y4, ..., yt+1} where 1 ≤ i ≤ n. It can be verified that Ap, A ∈ VPt⊔K1(Tn,m), as shown in Figure 11. Thus, for i = 1, 2, 3, ..., n, we obtain t∑ p=1 Ap +A = (A1 +A2 +A3 + · · ·+At) +A = (A1 △A2 △A3 △ · · · △At)△A = {y2, y3, y4, ..., yt+1} △ {xi, y2, y3, y4, ..., yt+1} = {xi}. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 15 of 23 x2 x3 x4 xt+1 xn−2 xn−1 xn x1 y1 y2 y3 y4 yt+1 ym−1 ym Tn,m⟨Ap⟩ : x2 x3 x4 xn−3 xn−2 xn−1 xn x1 y1 y2 y3 y4 yt+1 ym−1 ym Tn,m⟨A⟩ : Figure 11: Illustrating the subgraphs of Tn,m induced by Ap and A Hence, {xi} ∈ VPt⊔K1(Tn,m) for all 1 ≤ i ≤ n. Similarly, for any 1 ≤ j ≤ m, let yj be an arbitrary vertex of Pm in the tadpole graph Tn,m, and for any 1 ≤ q ≤ t, let Bq and B be defined as follows: Bq = {xq+1, t vertices︷ ︸︸ ︷ y2, y3, y4, ..., yt+1} where 1 ≤ q ≤ t, and B = { t vertices︷ ︸︸ ︷ x2, x3, x4, ..., xt+1, yj} where 1 ≤ j ≤ m. It can be verified that Bq, B ∈ VPt⊔K1(Tn,m), as shown in Figure 12. x2 x3 x4 xt+1 xn−2 xn−1 xn x1 y1 y2 y3 y4 yt+1 ym−1 ym Tm,n⟨Bq⟩ : x2 x3 x4 xt+1 xn−2 xn−1 xn x1 y1 y2 y3 y4 ym−2 ym−1 ym Tm,n⟨B⟩ : Figure 12: Illustrating the subgraphs of Tn,m induced by Bq and B Hence, for j = 1, 2, 3, ...,m, we get t∑ q=1 Bq +B = (B1 +B2 +B3 + · · ·+Bt) +B G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 16 of 23 = (B1 △B2 △B3 △ · · · △Bt)△B = {x2, x3, x4, ..., xt+1} △ {x2, x3, x4, ..., xt+1, yj} = {yj}. Thus, {yj} ∈ VPt⊔K1(Tn,m) for all 1 ≤ j ≤ m. Therefore, by Remark 1, Pt ⊔ K1 is a vertex-generator subgraph of Tn,m. The next theorem gives us a necessary condition for the empty graph Kt of order t, so t should be odd, to be a vertex-generator subgraph of Tn,m. Theorem 15. Let n,m ≥ 4 and t ≥ 3 be positive integers where t is odd. If 2t− 2 ≤ min{n,m}, then Kt is a vertex-generator subgraph of Tn,m. Proof. Let 2t− 2 ≤ min{n,m}. For any 1 ≤ i ≤ n, let xi be an arbitrary vertex of Cn in the tadpole graph Tn,m, and for any 1 ≤ p ≤ t− 1, we define Ap and A as follows: Ap = { t−1 vertices︷ ︸︸ ︷ x2, x4, x6, ..., x2t−2, y2p} where 1 ≤ p ≤ t− 1, and A = {xi, t−1 vertices︷ ︸︸ ︷ y2, y4, y6, ..., y2t−2} where 1 ≤ i ≤ n. It can be verified that Ap, A ∈ VKt (Tn,m), as shown in Figure 13. x2 x3 x4x5 x6 x2t−3 x2t−2 x2t−1 x2t xn−1 xn x1 y1 y2 y3 y4 y5 y6 y2t−3 y2t−2 y2t−1 ym−1 ym Tn,m⟨Ap⟩ : x2 x3 x4x5 x6 xn−5 xn−4 xn−3xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y2t−3 y2t−2 y2t−1 ym−1 ym Tn,m⟨A⟩ : Figure 13: Illustrating the subgraphs of Tn,m induced by Ap and A Hence, for i = 1, 2, 3, ..., n, we have t−1∑ p=1 Ap +A = (A1 +A2 +A3 + · · ·+At−1) +A G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 17 of 23 = (A1 △A2 △A3 △ · · · △At−1)△A = {y2, y4, y6, ..., y2t−2} △ {xi, y2, y4, y6, ..., y2t−2} = {xi}. Thus, {xi} ∈ VKt (Tn,m) for all 1 ≤ i ≤ n. In similar manner, for any 1 ≤ j ≤ m, let yj be an arbitrary vertex of Pm in the tadpole graph Tn,m, and for any 1 ≤ q ≤ t− 1, we define Bq and B as follows: Bq = {x2q, t−1 vertices︷ ︸︸ ︷ y2, y4, y6, ..., y2t−2} where 1 ≤ q ≤ t− 1, and B = { t−1 vertices︷ ︸︸ ︷ x2, x4, x6, ..., x2t−2, yj} where 1 ≤ j ≤ m. It can be verified that Bq, B ∈ VKt (Tn,m), as shown in Figure 14. x2 x3 x4x5 x6 x2t−3 x2t−2 x2t−1 x2t xn−1 xn x1 y1 y2 y3 y4 y5 y6 y2t−3 y2t−2 y2t−1 ym−1 ym Tn,m⟨Bq⟩ : x2 x3 x4x5 x6 x2t−3 x2t−2 x2t−1 x2t xn−1 xn x1 y1 y2 y3 y4 y5 y6 ym−4 ym−3 ym−2 ym−1 ym Tn,m⟨B⟩ : Figure 14: Illustrating the subgraphs of Tn,m induced by Bq and B Thus, for j = 1, 2, 3, ...,m, we get t−1∑ q=1 Bq +B = (B1 +B2 +B3 + · · ·+Bt−1) +B = (B1 △B2 △B3 △ · · · △Bt−1)△B = {x2, x4, x6, ..., x2t−2} △ {x2, x4, x6, ..., x2t−2, yj} = {yj}. Hence, {yj} ∈ VKt (Tn,m) for all 1 ≤ j ≤ m. Therefore, by Remark 1, Kt is a vertex- generator subgraph of Tn,m. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 18 of 23 The next theorem gives us a necessary condition for the graph kP2 ⊔K1, which is the disjoint union of the graph kP2 and a trivial graph K1, to be a vertex-generator subgraph of Tn,m. Theorem 16. Let n,m ≥ 3 and k be positive integers. If 3k ≤ min{n,m}, then kP2 ⊔K1 is a vertex-generator subgraph of Tn,m. Proof. Let 3k ≤ min{n,m}. Then, for any 1 ≤ i ≤ n, let xi be an arbitrary vertex of Cn in the tadpole graph Tn,m, and for any 1 ≤ p ≤ k, let A2p−1, A2p and A be defined as follows: A2p−1 = { k copies of P2︷ ︸︸ ︷ P2︷ ︸︸ ︷ x2, x3, P2︷ ︸︸ ︷ x5, x6, ..., P2︷ ︸︸ ︷ x3k−1, x3k, y3p−1} where 1 ≤ p ≤ k, A2p = A2p−1 \ {y3p−1} ∪ {y3p} where 1 ≤ p ≤ k, and A = {xi, k copies of P2︷ ︸︸ ︷ P2︷ ︸︸ ︷ y2, y3, P2︷ ︸︸ ︷ y5, y6, ..., P2︷ ︸︸ ︷ y3k−1, y3k} where 1 ≤ i ≤ n. It can be verified that that A2p−1, A2p, A ∈ VkP2⊔K1(Tn,m), as shown in Figure 15. Thus, for i = 1, 2, 3, ..., n, we obtain k∑ p=1 A2p−1 + k∑ p=1 A2p +A = (A1 +A3 + · · ·+A2k−1) + (A2 +A4 + · · ·+A2k) +A = (A1 +A2 +A3 +A4 + · · ·+A2k−1 +A2k) +A = (A1 △A2 △A3 △A4 △ · · · △A2k−1 △A2k)△A = {y2, y3, y5, y6, ..., y3k−1, y3k} △ {xi, y2, y3, y5, y6, ..., y3k−1, y3k} = {xi}. Hence, {xi} ∈ VkP2⊔K1(Tn,m) for all 1 ≤ i ≤ n. Similarly, for any 1 ≤ j ≤ m, let yj be an arbitrary vertex of Pm in the tadpole graph Tn,m, and for any j where 1 ≤ q ≤ k, let B2q−1, B2q and B be defined as follows: B2q−1 = {x3q−1, k copies of P2︷ ︸︸ ︷ P2︷ ︸︸ ︷ y2, y3, P2︷ ︸︸ ︷ y5, y6, ..., P2︷ ︸︸ ︷ y3k−1, y3k} where 1 ≤ q ≤ k, B2q = B2q−1 \ {x3q−1} ∪ {x3q} where 1 ≤ q ≤ k, and B = { k copies of P2︷ ︸︸ ︷ P2︷ ︸︸ ︷ x2, x3, P2︷ ︸︸ ︷ x5, x6, ..., P2︷ ︸︸ ︷ x3k−1, x3k, yj} where 1 ≤ j ≤ m. It can be verified that B2q−1, B2q, B ∈ VkP2⊔K1(Tn,m), as shown in Figure 16. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 19 of 23 x2 x3 x4x5 x6 x3k−2 x3k−1 x3k x3k+1 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y3k−2 y3k−1 y3k ym−1 ym Tn,m⟨A2p−1⟩ : x2 x3 x4x5 x6 x3k−2 x3k−1 x3k x3k+1 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y3k−2 y3k−1 y3k ym−1 ym Tn,m⟨A2p⟩ : x2 x3 x4x5 x6 xn−5 xn−4 xn−3xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y3k−2 y3k−1 y3k ym−1 ym Tn,m⟨A⟩ : Figure 15: Illustrating the subgraphs of Tn,m induced by A2p−1, A2p, and A Thus, for j = 1, 2, 3, ...,m, we have k∑ q=1 B2q−1 + k∑ q=1 B2q +B = (B1 +B3 + · · ·+B2k−1) + (B2 +B4 + · · ·+B2k) +B = (B1 +B2 +B3 +B4 + · · ·+B2k−1 +B2k) +B = (B1 △B2 △B3 △B4 △ · · · △B2k−1 △B2k)△B = {x2, x3, x5, x6, ..., x3k−1, x3k} △ {x2, x3, x5, x6, ..., x3k−1, x3k, yj} = {yj}. Hence, {yj} ∈ VkP2⊔K1(Tn,m) for all 1 ≤ j ≤ m. Therefore, by Remark 1, kP2 ⊔K1 is a vertex-generator subgraph of Tn,m. The following theorem gives us a necessary condition for the graph P2 ⊔Kt, which is the disjoint union of a path graph P2, and an empty graph Kt of order t, so t should be odd, to be a vertex-generator subgraph of Tn,m. Theorem 17. Let n,m ≥ 3 and t be positive integers where t is odd. If 2t+1 ≤ min{n,m}, then P2 ⊔Kt is a vertex-generator subgraph of Tn,m. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 20 of 23 x2 x3 x4x5 x6 x3k−2 x3k−1 x3k x3k+1 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y3k−2 y3k−1 y3k ym−1 ym Tn,m⟨B2q−1⟩ : x2 x3 x4x5 x6 x3k−2 x3k−1 x3k x3k+1 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y3k−2 y3k−1 y3k ym−1 ym Tn,m⟨B2q⟩ : x2 x3 x4x5 x6 x3k−2 x3k−1 x3k x3k+1 xn−1 xn x1 y1 y2 y3 y4 y5 y6 ym−4 ym−3 ym−2 ym−1 ym Tn,m⟨B⟩ : Figure 16: Illustrating the subgraphs of Tn,m induced by B2q−1, B2q, and B Proof. Let 2t+ 1 ≤ min{n,m}. Then, for any 1 ≤ i ≤ n, let xi be an arbitrary vertex of Cn in the tadpole graph Tn,m, and for any 1 ≤ p ≤ t, let A1, Ap+1, and A be defined as follows: A1 = { P2︷ ︸︸ ︷ x2, x3, t−1 vertices︷ ︸︸ ︷ x5, x7, ..., x2t+1, y2}, Ap+1 = A1 \ {y2} ∪ {y2p+1} where 1 ≤ p ≤ t, and A = {xi, P2︷ ︸︸ ︷ y2, y3, t−1 vertices︷ ︸︸ ︷ y5, y7, ..., y2t+1} where 1 ≤ i ≤ n. It can be verified that that A1, Ap+1, A ∈ VP2⊔Kt (Tn,m), as shown in Figure 17. Hence, for i = 1, 2, 3, ..., n, we get A1 + t∑ p=1 Ap+1 +A = A1 + (A2 +A3 +A4 + · · ·+At+1) +A = (A1 +A2 +A3 +A4 + · · ·+At+1) +A = (A1 △A2 △A3 △A4 △ · · · △At+1)△A G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 21 of 23 x2 x3 x4x5 x6 x7 x2t x2t+1xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y7 y2t y2t+1 ym−1 ym Tn,m⟨A1⟩ : x2 x3 x4x5 x6 x7 x2t x2t+1xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y7 y2t y2t+1 ym−1 ym Tn,m⟨Ap+1⟩ : x2 x3 x4x5 x6 xn−5 xn−4 xn−3xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y7 y2t y2t+1 ym−1 ym Tn,m⟨A⟩ : Figure 17: Illustrating the subgraphs of Tn,m induced by A1, Ap+1, and A = {y2, y3, y5, y7, ..., y2t+1} △ {xi, y2, y3, y5, y7, ..., y2t+1} = {xi}. Consequently, {xi} ∈ VP2⊔Kt (Tn,m) for all 1 ≤ i ≤ n. Similarly, for any 1 ≤ j ≤ m, let yj be an arbitrary vertex of Pm in the tadpole graph Tn,m, and for any 1 ≤ q ≤ t, let B1, Bq+1 and B be defined as follows: B1 = {x2, P2︷ ︸︸ ︷ y2, y3, t−1 vertices︷ ︸︸ ︷ y5, y7, ..., y2t+1}, Bq+1 = B1 \ {x2} ∪ {x2q+1} where 1 ≤ q ≤ t, and B = { P2︷ ︸︸ ︷ x2, x3, t−1 vertices︷ ︸︸ ︷ x5, x7, ..., x2t+1, yj} where 1 ≤ j ≤ m. It can be verified that B1, Bq+1, B ∈ VP2⊔Kt (Tn,m), as shown in Figure 18. Thus, for j = 1, 2, 3, ...,m, we get B1 + t∑ q=1 Bq+1 +B = B1 + (B2 +B3 +B4 + · · ·+Bt+1) +B G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 22 of 23 x2 x3 x4x5 x6 x7 x2t x2t+1xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y7 y2t y2t+1 ym−1 ym Tn,m⟨B1⟩ : x2 x3 x4x5 x6 x7 x2t x2t+1xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 y7 y2t y2t+1 ym−1 ym Tn,m⟨Bq+1⟩ : x2 x3 x4x5 x6 x7 x2t x2t+1xn−2 xn−1 xn x1 y1 y2 y3 y4 y5 y6 ym−4 ym−3 ym−2 ym−1 ym Tn,m⟨B⟩ : Figure 18: Illustrating the subgraphs of Tn,m induced by B1, Bq+1, and B = (B1 +B2 +B3 +B4 + · · ·+Bt+1) +B = (B1 △B2 △B3 △B4 △ · · · △Bt+1)△B = {x2, x3, x5, x7, ..., x2t+1} △ {x2, x3, x5, x7, ..., x2t+1, yj} = {yj}. Hence, {yj} ∈ VP2⊔Kt (Tn,m) for all 1 ≤ j ≤ m. Therefore, by Remark 1, P2 ⊔ Kt is a vertex-generator subgraph of Tn,m. 4. Conclusions This paper provides some vertex-generator subgraphs of Km,n, such as the empty graph, path graph, star graph, and complete bipartite graph Kr,r+1. This study also provides some vertex-generator subgraphs of Tn,m, such as the empty graph and the dis- joint union of graphs Pt ⊔ K1, kP2 ⊔ K1, and P2 ⊔ Kt. It is recommended to find the other vertex-generator subgraphs of Km,n and Tn,m. Furthermore, characterization for the vertex-generator subgraphs of the two graphs remains open for research. G. D. Sepillo et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6951 23 of 23 Acknowledgements Part of this research was done while the authors were taking bachelor’s degrees at the Batangas State University, The National Engineering University (BatStateU The NEU). Additionally, the first author wishes to thank the Department of Science and Technology - Science Education Institute (DOST-SEI) for extending financial support, which helped him pursue his career in the field of mathematics. 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