EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 663-675 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Block Topological Space and Block Topological Graph induced by Undirected Graphs Justine Bryle C. Macaso1,∗, 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 simple undirected graph. A block of G is a maximal connected subgraph of G that contains no cut-vertices [11]. The family of vertex sets of blocks of G generates a unique topology. In this paper, we formally define the topology generated by the family of blocks in a graph called the block topological space. Moreover, we characterize and describe some special attributes of the block topological space. Finally, we associate a corresponding graph from a given block topological space by defining the block topological graph. 2020 Mathematics Subject Classifications: 37F20, 05C10, 54C35 Key Words and Phrases: Block topological space, Block topological graph, Graph, Block, Hausdorff space, Continuous 1. Introduction There are many ways of associating topology from a graph, as seen in [1], [3], [5], [6], [7], and [8]. The most common method among these is by treating a collection of subsets of a nonempty set (e.g. vertex set or edge set) as a subbase to generate the desired topol- ogy which is reflected in the paper of Hassan and Abed in [7]. This topology is called the independent topology and is generated from the family of independent sets of each of the vertices in the graph. The same method was applied in the study of Abdu and Kilicman in [1] where they associated two topologies on the set of edges from a particular directed graph called edge-compatible topology and edge-incompatible topology. Another fascinat- ing intercrossing of topology and graph theory is establishing an adjacency condition to obtain the desired graph from a given finite topological space. This idea was reflected in the paper of Alsanaia et.al. in [2] as they gave a formal definition of converting the finite discrete topological space using a suitable adjacency condition to obtain the graph called the discrete topological graph. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5135 Email addresses: s.macaso.justinebryle@cmu.edu.ph (JB. Macaso), f.cherrymae.balingit@cmu.edu.ph (CM. Balingit) https://www.ejpam.com 663 © 2024 EJPAM All rights reserved. Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 664 Let G = (V (G), E(G)) be a simple undirected graph. For a nonempty subset S of V (G), the subgraph induced by S, denoted by G[S], has S as its vertex set and two ver- tices u and v are adjacent in G[S] if and only if u and v are adjacent in G. A subgraph H of a graph G is called an induced subgraph if there is a nonempty subset S of V (G) such that H = G[S]. A vertex v of G is an isolated vertex if it is not adjacent to any other vertices of G. Two vertices of G are connected if there is a path that connect them. If every two vertices of G are connected, then the graph G is connected. A component of G is a connected subgraph of G that is not contained in any larger connected subgraph of G. The number of components of G is denoted by κ(G). A vertex v of G is a cut-vertex of G if κ(G − v) > κ(G). If G is a nontrivial graph and v is a cut-vertex of component Cj of G, then the subgraph Cj − v has m components G1, G2, · · · , Gm for m ≥ 2 and the induced subgraphs Brji = G[V (Gi)∪ {v}] are connected and referred to as branches of Cj at v [4]. A block of a graph is a maximal connected subgraph that contains no cut-vertices [11]. The smallest possible block in a graph is a subgraph induced by a single vertex with a degree equal to zero. Moreover, two distinct blocks have at most one vertex in common and if they share the same vertex, then this vertex is a cut-vertex [4]. In addition, if B1, B2, · · · , Bk are the blocks of G, then ⋃k i=1 V (Bi) = V (G). A topology τ on a nonempty set X is a class of subsets of X that is closed under arbi- trary union and finite intersection, and X and ∅ belong to τ . The member of τ is called an open set and the pair (X, τ) is called a topological space. The topology containing all the subsets of X is called the discrete topology on X and the topology containing exactly X and ∅ is called the indiscrete topology on X. A collection Γ of open sets is a base for a topology of X if each nonempty open is a union of sets belonging to Γ. A collection Σ of open sets is called a subbase if the set {A : A = ⋂k i=1Wi, k ∈ Z+,Wi ∈ Σ} is a base for a topology on X[10]. Any class A of subsets of X is a subbase of for a unique topology on X. That is, the finite intersection of sets in A form a base for a topology on X [9]. If (X, τ1) and (Y, τ2) are two topological space, then f : (X, τ1) → (Y, τ2) is continuous if the preimage of any open subset of Y is an open subset of X[10]. As seen in the above discussion, it is possible to exhaust the distinct blocks of a given graph and apply various methods of topologizing the family of the vertex sets of these blocks. It is with this motivation that we aim to introduce a novel approach to topologizing a graph using the blocks in a graph. The generated topology will then be called the block topological space of a graph. Moreover, we examine and investigate some elementary properties of sets in a block topological space. Finally, we introduce the notion of a block topological graph. 2. Steps in enumerating the blocks in a graph General Assumption: Let G be a simple undirected graph with components C1, · · · , Cj for some j ∈ Z+. The following steps are ways on enumerating the blocks in a graph. Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 665 Step 1: For any 1 ≤ i ≤ j, if Ci contains no cut-vertices, then Ci is a block of G. Consequently, if G has no cut-vertex, then C1, C2, · · · , Cj are precisely the blocks of G. Step 2: If Ci contains a cut-vertex vi1 , then obtain the branches of Ci at vi1 . Step 3: If all the branches of Ci at vi1 contains no cut-vertices, then of these branches is a block of G. Otherwise, take the branches of Ci at vi1 with cut-vertices. Step 4: For each branch of Ci with a cut-vertex, choose one cut-vertex vi2 and obtain the sub-branches at vi2 . Step 5: Repeat the steps of separating the branch until all the resulting sub-branches contain no cut-vertices. Step 6: Do these for all the components of G that contains cut-vertices. Step 7: Collect all the components, branches, and sub-branches of G that contain no cut-vertices. These are precisely the blocks of G. For example, consider the graph G in Figure 1. G : Figure 1: The graph G The following are the components of G, each containing cut-vertices as shown in Figure 2. Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 666 C1 : C2 : C3 : Figure 2: Components of G For C1, choose a cut-vertex v11 so that the branches of C1 at v11 are shown in Figure 3. v11 v11 Figure 3: Branches of C1 at v11 Notice that one of the branches of C1 at v11 has cut-vertices. Choose another cut- vertex v12 and obtain the sub-branches at v12 . The sub-branches at v12 are as shown in the Figure 4. v11 v12 v12 Figure 4: Sub-branches at v12 Repeat the steps of separating the branch until the resulting sub-branches contain no Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 667 cut-vertices. Thus, the following are the sub-branches at every cut-vertices of C1 as shown in Figure 5. Furthermore, the blocks of G at C1 are as shown in Figure 5. v11 v11 v12 v12 v14 v14 v13 v13 Figure 5: The blocks of G at C1 Doing the preceding steps for all the components of G gives the distinct blocks of G as shown in the Figure 6. Figure 6: The blocks of G 3. The block topological space Definition 1. Let G be a graph and let B1, B2, · · · , Bk be the distinct blocks of G for some k ∈ Z+. Then the topology on V (G) generated by the family ΣB(G) = {V (B1), V (B2), · · · , V (Bk)} is called the block topology of G, denoted by τB(G) and the pair (V (G), τB(G)) is called the block topological space of G. Denote ΓB(G) to be the family of finite intersections of sets in ΣB(G). In this case, ΓB(G) is a base for τB(G). A subset A of V (G) is τB(G)-open if A belongs to τB(G) and if Ac belongs to τB(G), then A is τB(G)-closed. Example 1. Consider the graph G in Figure 7. Observe that the blocks of G are given by B1, B2, and B3 implying that ΣB(G) = {{v1, v2, v3, v4}, {v4, v5}, {v4, v6}}. By the Def- inition 1, taking the finite intersections of sets in ΣB(G) we obtain the family ΓB(G) = Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 668 {{v1, v2, v3, v4}, {v4, v5}, {v4, v6}, {v4}}. Finally, by taking the arbitrary union of sets in ΓB(G) we obtain the block topology τB(G) = {∅, V (G), {v4}, {v4, v5}, {v4, v6}, {v4, v5, v6}, {v1, v2, v3, v4}, {v1, v2, v3, v4, v5}, {v1, v2, v3, v4, v6}}. v1 v2 v3 v4 v1 v2 v3 v4 v5 v6 v5 v6 v4 v4 G : B2 : B3 : B1 : Figure 7: The blocks of G By observation, two distinct blocks have at most one vertex in common and if they share a vertex, then this vertex is a cut-vertex. This means that the greater the number of cut-vertices in a graph the larger the number of blocks in a graph. However, increasing the number of blocks does not imply an increased number of cut-vertices (e.g. the case where we add an isolated vertex to the graph). Theorem 1. Let G be a graph. For a vertex v of G, {v} is τB(G)-open if and only if v is a cut-vertex of G or v is an isolated vertex of G. Proof. (⇒) Let G be a graph with blocks B1, B2, · · · , Bk. Suppose that {v} ∈ τB(G). Then {v} ∈ ΓB(G). This means that for some nonempty A ⊆ {1, 2, · · · , k}, {v} =⋂ i∈A V (Bi). If |A | = 1, then {v} ∈ ΣB(G), which means that G[{v}] is a block of G implying further that v is an isolated vertex. On the other hand, if |A| > 1, then {v} is the intersection of two or more distinct blocks which further implies that v is a cut-vertex. (⇐) If v is an isolated vertex, then G[{v}] is a block of G and by Definition 1, {v} is a τB(G)-open. Suppose that v is a cut-vertex and let Cj be a component of G containing v. Let Bri, i = 1, 2, · · · , s, be the branches of Cj at v. Note that for each i = 1, 2, · · · , s, Bri is composed of blocks of G which further means that each of the branch of Cj at v is τB(G)-open. Hence, by Defintion 1, ⋂s i=1Bri = {v} is τB(G)-open. Theorem 2. A graph G is an empty graph if and only if τB(G) is the discrete topology on V (G). Proof. (⇒) Let G be an empty graph. Then G[{v}] is a block of G and by Definition 1, any subset of V (G) is τB(G)-open. Hence, τB(G) is discrete. (⇐) Suppose that τB(G) is discrete. Then for all v ∈ V (G), {v} is τB(G)-open. Suppose on the contrary that G is not an empty graph. Then |E(G)| ≥ 1 and so let Cj be a component of G for some j ∈ Z+ such that |V (Cj)| ≥ 2. In this case, Cj has at least two vertices that are not cut-vertices nor isolated vertices, say v and w. By Theorem Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 669 1, {v}, {w} /∈ τB(G). A contradiction since τB(G) is discrete. Hence, G is an empty graph. Corollary 1. Let G be a graph. If A is the collection of cut-vertices of G, then any subset of A is τB(G)-open. Theorem 3. A graph G is connected and contains no cut-vertices if and only if τB(G) is the indiscrete topology on V (G). Proof. (⇒) Let G be a connected graph without cut-vertices. Then V (G) is the only block of G and so ΣB(G) = {V (G)} which further implies thats τB(G) = {∅, V (G)}. (⇐) Suppose that τB(G) is indiscrete. Then any nonempty proper subset of V (G) is not τB(G)-open. Suppose on the contrary that either G disconnected or has cut-vertices. If G is disconnected, then G has two or more components. Let Cj be a component of G for some j ∈ Z+. Then Cj are composed of blocks of G and so V (Cj) is the union of the vertex sets of these blocks implying that V (Cj) is τB(G)-open. But V (Cj) ⊊ V (G), a contradiction since τB(G) is indiscrete. Therefore, G must be connected. On the other hand, if G has a cut-vertices, then let v be a cut-vertex of G. By Theorem 1, {v} ∈ τB(G), a contradiction. Hence G is connected and contains no cut-vertices. Suppose that G is a graph and let B1, B2, · · · , Bk be the blocks of G. Denote C (G) to be the family of all the cut-vertices of G. Recall that two distinct blocks have at most one vertex in common and this vertex is a cut-vertex [4]. By Definition 1, the collection of the vertex set of each of the blocks of G together with all singletons containing the cut-vertices of G is a base for the block topological space of G. By this observation, the following theorem characterizes the τB(G)-open sets. Theorem 4. Let G be a graph and let B1, B2, · · · , Bk be the blocks of G. A set A ⊆ V (G) is τB(G)-open if and only if A is the union of sets in ΣB(G) ∪ {T : T ⊆ C (G)}. Proof. (⇒) Let B1, B2, · · · , Bk be the distinct blocks of a graph G. Suppose A is a τB(G)-open. Then A is the union of finite intersections of sets in ΣB(G). We now note that the intersection of two distinct blocks is at most one vertex which is a cut-vertex. Hence, the conclusion follows. (⇐) Let A ∈ ΣB(G) ∪ {T : T ⊆ C (G). By Definition 1 and Theorem 1, A is τB(G)- open. Remark 1. Let G be a graph with more than one block. Then the union of all the nontrivial τB(G)-open proper subsets of V (G) equals to V (G). Remark 2. Let (V (G), τB(G)) be a non indiscrete block topological space. Then the smallest possible number of τB(G)-open sets is 4. Example 2. Consider the graph G in Figure 8. Here, G[{v1, v2}] and G[{v2, v3}] are the blocks of G and τB(G) = {∅, V (G), {v1, v2}, {v3, v4}}. Moreover, {v1, v2} ∪ {v3, v4} = V (G) and |τB(G)| = 4. Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 670 v4 v2 v1 v3 G : Figure 8: Graph G Theorem 5. Let G be a graph such that |E(G)| ≥ 1. If v is not a cut-vertex adjacent to w, then every τB(G)-open set containing v also contains w. Proof. Let G be a graph such that |E(G)| ≥ 1. Suppose that vertex v is not a cut- vertex adjacent to w. Let O be a τB(G)-open containing v. Suppose on the contrary that w /∈ O. Since v and w are adjacent in G and E(G) is the disjoint union of the edges of Bis, then there exists a block Bj of G such that v, w ∈ V (Bj). On the other hand, since v is not a cut-vertex, choose Bk as a block such that v ∈ V (Bk) ⊆ O. In this case, Bk and Bj are distinct and {v} ∈ V (Bk) ∩ V (Bj). But, two distinct blocks intersect in at most one vertex which is a cut-vertex; hence v is a cut-vertex, a contradiction to our choice of v. Therefore, w ∈ O. A topological space is a Hausdorff space if for every two distinct elements in the mother set can be separated by two disjoint open sets [9]. The following Theorem charac- terizes the block topological space as being a Hausdorff space. Theorem 6. A (V (G), τB(G)) is a Hausdorff space if and only if G is an empty graph. Proof. (⇒) Let (V (G), τB(G)) be a Hausdorff space. Then every two distinct vertices in G can be separated by two disjoint τB(G)-open sets. If G is not an empty graph, then |E(G)| ≥ 1. In this case, choose a component Cj of G having more than one vertices. Here, Cj has a vertex v that is not a cut-vertex of G and v is adjacent to some vertex, say w. By Theorem 5, every τB(G)-open set containing v also contains w. This is a contradiction since by assumption that (V (G), τB(G)) is a Hausdorff space. Hence, G is an empty graph. (⇐) Let G be an empty graph. By Theorem 2, the generated block topology is discrete and thus the conclusion follows. Example 3. Consider the graphs in Figure 9. Observe that vertex v is not a cut-vertex of G that is adjacent to w. By Theorem 5, any τB(G)-open containing v also contains w. Hence, the block topological space of G is not a Hausdorff space. On the other hand, H is an empty graph and by Theorem 2, τB(H) is the discrete topology of V (H). Hence, every singleton of V (H) is τB(H)-open implying further that the block topological space of H is a Hausdorff space. Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 671 v w G : H : Figure 9: Graph of a non-Hausdorff space and Hausdorff space, respectively Theorem 7. Let G be a graph of order n < 4. Then G has no isolated vertices if and only if there exists a one-to-one correspondence from (V (Pn), τB(Pn)) to (V (G), τB(G)) that is continuous. Proof. (⇒) If G is a graph of order n < 4 without isolated vertices, then G is one of P2, P3, and K3. For P2 and P3, use the identity mapping so that we arrive the desired conclusion. For K3, since |V (K3)| = |V (P3)|, put f to be a one-to-one correspondence from (V (P3), τB(P3)) to (V (K3), τB(K3)). Since, τB(K3) = {∅, V (K3)} and f−1(∅) = ∅ and f−1(V (K3)) = V (P3), f is continuous. (⇐) Let f be a continuous one-to-one correspondence from (V (Pn), τB(Pn)) to (V (G), τB(G)). Suppose that G has an isolated vertices. Case 1: If n = 2, then τB(G) is discrete. Let w be an isolated vertex of G. Then f−1({w}) is a singleton subset of V (P2). Since by Theorem 3, τB(P2) is indiscrete, f −1({w}) is not τB(P2)-open. This is a contradiction since f is continuous. Hence, G must not have an isolated vertex. Case 2: Let w be an isolated vertex of G. Then for all v ∈ V (G) ∖ {w}, wv /∈ E(G). Also, f−1(w) must be a cut-vertex of P3 since by assumption f is continuous. Subcase 1: If v is an isolated vertex of G, then G must be an empty graph. This means that τB(G) is the discrete topology. Let u be a vertex in G such that f−1(u) is an end-vertex of P3. Here, {u} is τB(G)-open and f−1({u}) is not a τB(P3)-open set. This is a contradiction since f is continuous. Subcase2: If v is not an isolated vertex, then v must be adjacent to some vertex, say u. In this case, G[{v, u}] is a block of G implying further that {v, u} is τB(G)-open. Since f is bijective, f−1({v, u}) = f−1({v}) ∪ f−1({u}) . Note that f−1(w) is a cut-vertex of P3, and so f−1(v) and f−1(u) are end-vertices of P3. By Theorem 1, f−1({v, u}) is not τB(P3)-open, a contradiction. Remark 3. If G = P1, then the identity map satisfies the above theorem. Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 672 Theorem 8. Let G be a graph of order n ≥ 4. Then G has at least two nonadjacent edges if and only if there is a one-to-one correspondence from (V (Pn), τB(Pn)) to (V (G), τB(G)) that is continuous. Proof. (⇒) Let G be a graph of order n ≥ 4 such that G has at least two non-adjacent edges. Since G has at least 1 nontrivial component, there exists at least two vertices that are not cut-vertices and non-isolated vertices in that component. Denote the vertices of G by {v1, v2, · · · , vn} such that v1 and vn are not cut-vertices nor isolated vertices of G and v1v2, vn−1vn ∈ E(G). Let Pn be a path such that V (Pn) = {a1, a2, · · · , an} and E(Pn) = {aiai+1 : i = 1, 2, · · · , n− 1}. Now, define f : (V (Pn), τB(Pn)) → (V (G), τB(G)) by f(ai) = vi. Obviously, f is a one-to-one correspondence from (V (Pn), τB(Pn)) to (V (G), τB(G)). It remains to show that f is continuous. Let A be τB(G)-open. If A is empty, then f−1(A) = ∅ ∈ τB(Pn). Suppose that A is not empty. Case 1: If v1 ∈ A and vn /∈ A, then by Theorem 5, v2 ∈ A. Observe that, if A = {v1, v2}, then f−1(A) = f−1({v1, v2}) = {a1, a2} ∈ τB(Pn). Suppose that A ∖ {v1, v2} ≠ ∅. Then for all vj ∈ A∖ {v1, v2}, f−1(vj) is a cut-vertex of Pn implying further that f−1(A ∖ {v1, v2}) is a subset of C (Pn). Hence, f−1(A) = f−1({v1, v2}) ∪ f−1(A∖{v1, v2}) is τB(G)-open. The argument follows when vn ∈ A and v1 /∈ A. Case 2: If v1, vn ∈ A, then v2, vn−1 ∈ A. Similarly, if A = {v1, v2, vn−1, vn}, then f−1(A) = f−1({v1, v2, vn−1, vn}) = {a1, a2, an−1, an} ∈ τB(Pn). Also, if A ∖ {v1, v2, vn−1, vn} ̸= ∅, then f−1({v1, v2, vn−1, vn}) ⊆ C (Pn). Hence, f−1(A) = f−1({v1, v2, vn−1, vn}) ∪ f−1(A∖ {v1, v2, vn−1, vn}) is τB(G)-open. Case 3: If v1, vn /∈ A, then f−1(A) ⊆ C (Pn) so that f−1(A) is τB(G)-open. Thus, f is continuous. (⇐) Let f be a one-to-one correspondence from (V (Pn), τB(Pn)) to (V (G), τB(G)) that is continuous. Suppose that G has no nonadjacent edges. Then G is one of the following: (1) G = Kn; (2) |E(G)| = 1; or (3) The edges of G share a common vertex. Case 1: Suppose G = Kn. Let v be a vertex in G such that f−1(v) is an end vertex of Pn. Here, {v} is τB-open and {f−1(v)} is not τB(Pn)-open; a contradiction since f is continuous. Hence, G is not an empty graph. Case 2: Suppose |E(G)| = 1. Let vw ∈ E(G). Then {v, w} is τB(G)-open and so by the continuity of f , f−1({v, w}) is τB(Pn)-open. Note that f−1(v) and f−1(w) cannot be both end-vertices of Pn; otherwise, f −1({v, w}) is not τB(Pn)-open. Now, choose a vertex u in G different from v and w such that f−1(u) is an end- vertex of Pn. Note that {f−1(u)} is not a τB(Pn)-open set. In this case, u is an isolated vertex and thus {u} is τB(G)-open. This is a contradiction since f is continuous. Hence, |E(G)| > 1. Case 3: Suppose the edges of G share a common vertex and let x and y be vertices in G such that f−1(x) and f−1(y) are the end-vertices of Pn. Now, x and y are not Justine Bryle C. Macaso, Cherry Mae R. Balingit / Eur. J. Pure Appl. Math, 17 (2) (2024), 663-675 673 isolated vertices of G nor cut-vertices of G, otherwise f is not continuous since f−1({x, y}) is not τB(Pn)-open. Subcase 1: If xy ∈ E(G), then there is another vertex v that is adjacent to both x and y since x and y are not cut-vertices. In this case, G[{x, v, y}] is a block of G which further implies that {x, v, y} is τB(G)-open. Also, note that f−1(v) is a cut-vertex of Pn. Now, we have f−1({x, v, y}) = {f−1(x)} ∪ {f−1(v)} ∪ {f−1(y)} = f−1({x, v}) ∪ {f−1(y)} = f−1({x, y}) ∪ {f−1(v)} = f−1({v, y}) ∪ {f−1(x)}. Thus, f−1({x, v, y}) is not τB(Pn)-open by Theorem 4. This is a contradic- tion since f is continuous. Subcase 2: If xy /∈ E(G), then there exists a vertex v that is adjacent to x and y. Here, G[{x, v}] and G[{v, y}] are blocks of G and so {x, v, y} is τB(G)-open. Similarly, f−1({x, v, y} cannot be expressed as a union of vertex sets of blocks in Pn and a subset of C (Pn). This means that f−1({x, v, y} is not τB(Pn)-open. But f is continuous; hence a contradiction. 4. The block topological graph Definition 2. Let (V (G), τB(G)) be block topological space where τB(G) is not the indis- crete topology on V (G). A block topological graph of (V (G), τB(G)) is a graph GτB(G) with vertex set V (GτB(G)) = τB(G) ∖ {∅, V (G)} and edge set E(GτB(G)) = {AB : A ⊆ B,A,B ∈ V (GτB(G))}. Example 4. The corresponding block topological graph of G in Figure 7 is shown in Figure 10. GτB(G) : {v4} {v4, v5, v6} {v4, v5} {v4, v6} {v1, v2, v3, v4, v5}{v1, v2, v3, v4, v6} {v1, v2, v3, v4} Figure 10: Block topological graph of G REFERENCES 674 Remark 4. Let GτB(G) be a block topological graph. Then |V (GτB(G))| = |τB(G)| − 2. A connected graph containing no cut-vertices has no corresponding block topological graph. Recall that a block in a graph is not a subgraph to any other block in a graph. Hence, a block topological graph is never trivial nor complete. 5. Concluding remarks The notion of block topological space induced by undirected simple graphs has been successfully introduced in this paper together with some important characterizations and special attributes of the resulting block topological space. Here, the authors presented an initial idea of the corresponding block topological graph. One definite extension of this research is the study of the block topological space and the block topological graph induced by special families of graphs and those graphs resulting from unary and binary operations that the authors had already started working on. Meanwhile, some possible and interesting direction for further study is on extending the idea of the block topology of a graph to various topological structures such as soft bitopological spaces [13], soft topological subspaces [12] and [14], fuzzy topological space [15], regular spaces, normal spaces, and completely regular spaces [9] or perhaps in looking into the block topology of a directed graph using the method of Hassan and Abed in [7]. Acknowledgements The authors are beyond grateful and would love to express their heartfelt apprecia- tion to the people who were behind the success of this paper. Special thanks to Prof. Rolito G. Eballe for the two questions in his exam which were the motivations of this research study. Also, the authors would like to express their profound gratitude to the Department of Science and Technology-Science Education Institute Science and Tech- nology Regional Alliance of Universities for Inclusive National Development (DOST-SEI STRAND) through the Office of Admissions, Scholarship, and Placement (OASP) of Cen- tral Mindanao University for the opportunities and privileges given to them which are a very big factor in this research journey. References [1] K.A. Abdu and A. Kilicman. Topologies on the edges set of directed graphs. Inter- national Journal of Mathematical Analysis, 12(2):71–84, 2018. [2] A Alsinaia, B. V. Dhanaanjayamurthy, A Mohammed, M Abdlhusein, and M Cancan. Topological Space Generated by Edges Neighborhoods of Discrete Topological Graph. [3] S Canoy and R Lemence. Topologies Induced by some special Graphs. Journal of Mathematics, 2(2):45–50, 1999. REFERENCES 675 [4] G Chartrand, L Lesniak, and P Zhang. Graphs and digraphs. 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