On maximal block topological space

1 Introduction

A graph G = (U (G) , E (G)) is a finite, nonempty set U (G), together with a (possibly empty) set E (G) of element subsets of U (G). The elements of U are called vertices, while those of E are called edges. A subgraph H of a graph G is a graph having all of its vertices and edges in G. Thus, H is a subgraph of G, if and only if U (H) ⊆ U (G) and E (H) ⊆ E (G). A graph G is connected if for each pair of its vertices there is a path connecting them, otherwise, it is disconnected. A component of G is a maximal connected subgraph of G. A vertex u of a graph G is a cut vertex if the deletion of u increases the number of components in G. A nontrivial connected graph G is said to be a non-separable graph if it has no cut vertices. A block of a graph is a maximal non-separable subgraph of G. If G has no cut vertex, G itself is a block. A block that has exactly one cut vertex of G is called an end block. A graph G is acyclic if it has no cycles. A graph G is acyclic if and only if any block in G is K₂ or if G is acyclic graph, then one of any two vertices is in different blocks. For more details about concepts in graph theory, we refer to [1, 2].

Let X be a non-empty set. A topology on X is a collection τ of subsets of X, such that: ∅ and X belong to τ, an arbitrary union of members in τ belongs to τ and a finite intersection of members of τ belongs to τ, the pair (X, τ) is called topological space. Let H be a subset of a topological space (X, τ), the interior of H is the union of all open sets contained in H and is denoted by int (H). The closure of the set H is the intersection of all closed subsets containing H and is denoted by Cl (H). A topological space (X, τ) is said to be a T₀-space, if for any distinct points in X, then there exists an open set containing one of them but not the other. A topological space (X, τ) is said to be a T_1/2-space if every singleton set in X is either open or closed see [3, 9].

Recently, some researchers have generated topologies from graphs by using different methods. In [8] Kozae et al, introduced new types of topological structures by graphs and constructed a computer program that builds graphs and its topological graphs also they presented some examples of different kinds of graphs and their topological graphs and studied some of their topological and algebraic properties. In [5] Jafarian et al, associated a topology to a graph G, called graphic topology of G and they showed that it is an Alexandroff topology, also investigated some properties of this type of topology. Kilicman and Abdulkalek, in [7] presented a new type of topology induced by set of vertices for any simple graph (finite or infinite), called incidence topology. Some properties of this type of topology were investigated and introduced a comparison between two different subbases to generate a topology. Nianga and Canoy in [11], introduced a way of constructing a topology from a simple undirected graph by utilizing the hop neighborhoods of the graph and described the topologies induced by the complete, path and the fan graphs. In [12] Sari and Kopuzlu, studied topologies generated by simple undirected graphs without isolated vertices and their properties. Also,they presented necessary and sufficient condition for the topological spaces generated by two different graphs to be homeomorphic. Nada et al in [10], structured a new type of topological space generating by graphs, also investigated an algorithm to generate the topological structures from different graphs. In [4], the authors defined new types of topological spaces generated by sets of coneighbor vertices and coneighbor edges of a graph and they proved that every coneighbor topological space is an Alexandroff space. Moreover, a necessary and sufficient condition is provided to a coneighbor topological space of a graph G to be compact. The coneighbor topological space of a graph G does not satisfied the weak separation axioms while it is both regular and normal. In [6] Khalaf and Padma defined neutrosophic fuzzy graphs and they obtained some properties of such graphs. Tran et al in [13] introduced some other types of graph topological spaces called neutrosophic soft graph topological spaces which are constructed from vertices and edges of the graph.

The main purpose of this paper is to define a new type of topology induced by cut vertices and blocks in a graph G called maximal block topology. Some results and properties of T₀-space, T_1/2-space, irreducible graph space, completely irreducible graph space, topologically independent graph space and weak topologically independent graph space via maximal block topological space are present.

2 Maximal block topological space

In this section, we use a set of cut vertices and blocks in a connected undirected graph G to generate a new type of topological space called maximal block topological space, also introduce some properties and results of maximal block topological space.

Definition 2.1. Let G = (U (G) , E (G)) be a connected simple undirected graph, and let S be the family of all blocks of G, the set of all cut vertices in G is denoted by C (G). Then the collection B which consists of any members S₁ of S together with their cut vertices C (S₁) forms a base for a topology of the graph G and it is called block base, by taking the union of members in B we get the topology and this topology is called block topological space denoted by ( G , τ B ) . If B consists of all members of S together with all cut vertices C (G), then the block topology generated by B is called the maximal block topology, we denote it by τ _M . And (G, τ _M ) is called maximal block topological space of the graph G. The members in τ _M are called open subgraphs and the complement of open subgraphs in τ _M are called closed subgraphs. The dual maximal block topology of τ _M is denoted by (τ _M )  ^c is the members of closed subgraphs in τ _M .

Example 2.2. Consider the graph G given in the following diagram.

OPEN IN VIEWER

Fig. 1

Simple connected undirected graph G with their blocks.

We have blocks B₁ = {u₁, u₂}, B₂ = {u₂, u₃, u₄}, B₃ = {u₄, u₅}, B₄ = {u₅, u₆, u₇, u₈, u₉}. Cut vertices of the graph G in Fig. 1 are: {u₂}, {u₄} and {u₅}. So, we have

S = {B₁, B₂, B₃, B₄} = {{u₁, u₂} , {u₂, u₃, u₄} , {u₄, u₅} , {u₅, u₆, u₇, u₈, u₉}}. Hence, the maximal block base induced by the graph G in Fig. 1 is given by B M = { ∅ , { u 2 } , { u 4 } , { u 5 } , { u 1 , u 2 } , { u 2 , u 3 , u 4 } , { u 4 , u 5 } , { u 5 , u 6 , u 7 , u 8 , u 9 } } .

Therefore, the maximal block topology of the graph G in Fig. 1 is given by

τ _M  = {∅ , U (G) , {u₂} , {u₄} , {u₅, u₁, u₂} , {u₂, u₃, u₄} , {u₄, u₅} , {u₅, u₆, u₇, u₈, u₉} , {u₂, u₄} , {u₂, u₅} , {u₂, u₄, u₅} , {u₂, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₄} , {u₄, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₅} , {u₂, u₃, u₄, u₅} , {u₁, u₂, u₃, u₄} , {u₁, u₂, u₄, u₅} , {u₁, u₂, u₅, u₆, u₇, u₈, u₉} , {u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₄, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₃, u₄, u₅}}.

And the dual maximal block topology (τ _M )  ^c is given by

(τ _M )  ^c  = {∅ , U (G) , {u₁, u₃, u₄, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₃, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₃, u₄, u₆, u₇, u₈, u₉} , {u₃, u₄, u₅, u₆, u₇, u₈, u₉} , {u₁, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₃, u₆, u₇, u₈, u₉} , {u₁, u₂, u₃, u₄} , {u₁, u₃, u₅, u₆, u₇, u₈, u₉} , {u₁, u₃, u₆, u₇, u₈, u₉} , {u₁, u₃, u₆, u₇, u₈, u₉} , {u₁, u₃, u₄} , {u₃, u₅, u₆, u₇, u₈, u₉} , {u₁, u₂, u₃} , {u₃, u₄, u₆, u₇, u₈, u₉} , {u₁, u₆, u₇, u₈, u₉} , {u₅, u₆, u₇, u₈, u₉} , {u₃, u₆, u₇, u₈, u₉} , {u₃, u₄} , {u₁} , {u₃} , {u₆, u₇, u₈, u₉}}.

The following results are true if G is a simple connected graph.

Lemma 2.3. Let (G, τ _M ) be the maximal block topological space. Then the following statements are true:

Each block of G is an open subgraph in τ _M .

Proof. Obvious by Definition 2.1. □

The converse of case (3) need not be true in general.

In Example 2.2, u₃ in G is closed subgraph in τ _M , but it is not pendant vertex in G.

Remark 2.4. If G is not a simple graph in above lemma, the converse of case (3) need not be true in general, as it is shown in the following example:

Example 2.5. Consider the graph G in the figure below

OPEN IN VIEWER

Fig. 2

Non-simple connected undirected graph G.

The maximal block topology of the graph G in Fig. 2 is given by

τ _M  = {∅ , U (G) , {u₂} , {u₁, u₂} , {u₂, u₃, u₄}}. Hence, {u₁} is closed subgraph but it is not pendant vertex.

Lemma 2.6. Let (G, τ _M ) be the maximal block topological space and S = {B _i } be a family of blocks of G, then

∩B _i is open in τ _M .

Proof.

The intersection of any family of blocks is either empty or a cut vertex which is open in τ _M . Hence ∩B _i is open in τ _M .

Corollary 2.7. The following statements are true for any block topological space (G, τ _B ):

(G, τ _B ) is coarser than the (G, τ _M ) if and only if G has cut vertex.

Proof. Straightforward. □

Proposition 2.8. If G is either acyclic graph or each cycle block C _p in G has p or p - 1 cut vertices, then every non-cut vertex in G is a closed subgraph in (G, τ _M ).

Proof. If G is acyclic graph, then each block contains only two vertices at least one of them is a cut vertex.

On the other hand, if each cycle block C _p in G has p or p - 1 cut vertices, then every non-cut vertex in G lies in a block in which the other vertices are cut vertices. Hence, by Lemma 2.3, non-cut vertices are closed in (G, τ _M ). □

In this section, we define derived subgraph, closure subgraph and interior subgraph and study their properties in maximal block topological space. separated subgraphs and some results are introduce.

In the following, we redefine some of the topological concepts over this type of maximal block topological spaces.

Definition 3.1. Let (G, τ _M ) be a maximal block topological space. Let H be a subgraph from G, then u ∈ U (G) is called a limit vertex or accumulation vertex of H if each open subgraph S of τ _M containing u, H∩ S \ {u} ≠ ∅.

The set of all limit vertex of H is called the derived subgraph of H and it is denoted by d _M  (H).

Proposition 3.2. Let (G, τ _M ) be a maximal block topological space and H be a subgraph from G. A non-cut vertex u ∈ U (G) is a limit vertex of H if and only if each block B of G containing u, H∩ B \ {u} ≠ ∅.

Proof. If u ∈ d _M  (H) and since each block in G is an open subgraph in τ _M , then H∩ B \ {u} ≠ ∅.

Conversely, if u is a cut vertex, then u ∉ d _M  (H). Hence, each open subgraph S such that H∩ S \ {u} ≠ ∅ is true only when S is a block containing a non-cut vertex. □

Definition 3.3. Let (G, τ _M ) be a maximal block topological space and let H be a subgraph from G, then

The closure subgraph of H, is defined as the intersection of all closed subgraphs of G containing H. That is

Cl _M  (H) = ⋂ {S ∈ (τ _M )  ^c  : H ⊆ S}.

Lemma 3.4. Let (G, τ _M ) be a maximal block topological space and let H be a subgraph from G, then

Cl _M  (H) is the smallest closed subgraph containing H.

Remark 3.5. If u is a cut vertex in a block B, then the set of all cut vertices of B except u is denoted by C^* (B).

Lemma 3.6. For any vertex u ∈ G, then the following statements are true.

u is non-cut vertex in a block B if and only if Cl _M  ({u}) = B \ C (B).

Proof. Follows from Lemma 2.3 and Definition 3.3. □

Proposition 3.7. Let B be any block in G. If (N _u  \ B) ⊆ C (G) for all u ∈ C (B), then B is closed.

Proof. If v ∈ G \ B, then either v is a cut vertex or there is a block B₁ disjoint from the block B such that v ∈ B₁. In both cases we obtain that there is an open set U such that v ∈ U ⊆ G \ B implies that G \ B is open. Hence, B is closed. □

The following corollary follows directly from Proposition 3.7.

Corollary 3.8 The number of closed blocks in P _n is n - 2 for all n ≥ 4.

Theorem 3.9. Let (G, τ _M ) be a maximal block topological space and let u, v ∈ G, the following results are valid.

If u is a cut vertex in a graph G, then u ∉ Cl _M  ({v}) for all v ∈ G and hence, Cl _M  ({u}) ≠ Cl _M  ({v}).

Proof.

Follows from Lemma 3.6.

Theorem 3.10. Let (G, τ _M ) be a maximal block topological space, and let u, v ∈ G, the following statements are true.

u and v are non-cut vertices and they are in the same cycle block if and only if Cl _M  ({u}) = Cl _M  ({v}).

Proof.

Since u and v are non-cut vertices and they are in the same cycle block B in G containing u and v, then by Definition 3.3(1), we have Cl _M  ({u}) = {u} ∪ (B \ C (B)) = B \ C (B) and Cl _M  ({v}) = {v} ∪ (B \ C (B)) = B \ C (B). Hence, Cl _M  ({u}) = Cl _M  ({v}). Conversely, if Cl _M  ({u}) = Cl _M  ({v}), that is u and v are in the same cycle block in G, so u and v are non-cut vertices in G.

Corollary 3.11. Let (G, τ _M ) be a maximal block topological space, if Cl _M  ({u}) = Cl _M  ({v}), then u and v may not be adjacent in G. From Example 2.2, let u₆ and u₇ in G have the same block closure subgraph, but they are non-adjacent vertices.

Corollary 3.12. Let (G, τ _M ) be a maximal block topological space, if two vertices u and v are adjacent, then Cl _M  ({u}) need not be equal to Cl _M  ({v}). From Example 2.2, let u₁ and u₂ are adjacent in G, but Cl _M  ({u₁}) = {u₁} ≠ {u₁, u₂, u₃} = Cl _M  ({u₂}).

4 T₀ and T_1/2-spaces

In this section, we define T₀-space and T_1/2-spaces induced by a graph G and we give some of their properties in the maximal block topological space.

Definition 4.1. A topological space (G, τ) induced by a graph G is said to be T₀-space, if for any distinct vertices in G, there exists an open subgraph in (G, τ) containing one of them but not the other.

Theorem 4.2. Let (G, τ _M ) be a maximal block topological space. If every block in G is K₂, then Cl _M  (u) ≠ Cl _M  (v) for all distinct vertices u and v in G.

Proof. Since every block in G is K₂ and each block has only two vertices u and v, to show that Cl _M  (u) ≠ Cl _M  (v), we have the following cases:

If u and v are in the same block, u is a pendant vertex adjacent to v and v is cut vertex, then by Definition 3.3(1), Cl _M  ({u}) = {u} and Cl _M  ({v}) = {u, v}, then Cl _M  ({u}) ≠ Cl _M  ({v}).

The converse of the above theorem need not be true in general.

From Example 2.2, we have Cl _M  ({u₄}) ≠ Cl _M  ({u₅}), but every block in G is not K₂.

Theorem 4.3. A maximal block topological space (G, τ _M ) is T₀-space if and only if Cl _M  ({u}) ≠ Cl _M  ({v}) for all distinct vertices u and v in G.

Proof. Since (G, τ _M ) is a T₀-space, then there exists an open subgraph H, such that (u ∈ H and v ∉ H) or (v ∈ H and u ∉ H) this implies that ({v} ∩ H = ∅) or ({u} ∩ H = ∅), therefore v ∉ Cl _M  ({u}) or u ∉ Cl _M  ({v}), hence Cl _M  ({u}) ≠ Cl _M  ({v}). Conversely, let u and v are distinct vertices in G and Cl _M  ({u}) ≠ Cl _M  ({v}), so there exists another vertex w in G, such that (w ∈ Cl _M  ({u}) \ Cl _M  ({v})) or (w ∈ Cl _M  ({v} \ Cl _M  ({u})), this implies that (w ∉ Cl _M  ({v}), then {v} ∩ H _w  = ∅) or (w ∉ Cl _M  ({u}), then {u} ∩ H _w  = ∅), but ({u}∩ H _w  ≠ ∅ then u ∈ H _w ) or ({v}∩ H _w  ≠ ∅ then v ∈ H _w ), for some open subgraph H _w containing w. Hence (G, τ _M ) is T₀-space. □

Theorem 4.4. A maximal block topological space (G, τ _M ) is T₀-space, if G is acyclic graph.

Proof. If G is acyclic graph, then the cardinality of each block is 2, that is |B|=2 for all block B ∈ S and the cardinality of each set of cut vertices is 1 or 2 (that is, |C|=1 or 2) for every C ⊆ B, where C is the set of all cut vertices in B, therefore, for any distinct vertices in G, there exists an open subgraph containing one of them but not the other, then (G, τ _M ) is T₀-space. □

The converse of Theorem 4.4 need not be true in general.

Example 4.5. Consider the graph G as shown in Fig. 3. The maximal block base induced by the graph G in Fig. 3, is given by

OPEN IN VIEWER

Fig. 3

Simple connected undirected non-acyclic graph G.

B M = { ∅ , { u 2 } , { u 4 } , { u 6 } , { u 1 , u 2 } , { u 2 , u 3 , u 4 , u 6 } , { u 4 , u 5 } , { u 6 , u 7 } } .

Therefore, the maximal block topology of a graph G in Fig. 3 is given by

τ _M  = {∅ , G, {u₂} , {u₄} , {u₆} , {u₁, u₂} , {u₂, u₃, u₄, u₆} , {u₄, u₅} , {u₆, u₇} , {u₂, u₄} , {u₂, u₆} , {u₄, u₆} , {u₂, u₄, u₅} , {u₂, u₄, u₆} , {u₂, u₆, u₇} , {u₁, u₂, u₄} , {u₄, u₆, u₇} , {u₁, u₂, u₆} , {u₄, u₅, u₆} , {u₁, u₂, u₄, u₅} , {u₂, u₃, u₄, u₆} , {u₁, u₂, u₆, u₇} , {u₄, u₅, u₆, u₇} , {u₁, u₂, u₃, u₄, u₆} , {u₂, u₃, u₄, u₅, u₆} , {u₂, u₃, u₄, u₆, u₇} , {u₁, u₂, u₃, u₄, u₅, u₆} , {u₁, u₂, u₃, u₄, u₆, u₇} , {u₁, u₂, u₄, u₅, u₆, u₇} , {u₂, u₃, u₄, u₅, u₆, u₇}}.

It is easy to show that τ _M of G in Fig. 3 is T₀-space, because for every two distinct vertices in G, there exist an open subgraph in τ _M containing one of them but not the other.

Remark 4.6. A maximal block topological space of a graph in Example 2.2 is not T₀-space, but we can select a subgraph from a graph G a maximal block topological space of that subgraph is T₀-space. See the subgraph in Fig. 4, a maximal block topological space of it is T₀-space.

OPEN IN VIEWER

Fig. 4

Simple connected subgraph from a graph G in Fig. 1.

Theorem 4.7. A maximal block topological space (G, τ _M ) is T₀-space if and only if |B| ≤ |C (B) |+1 for every block B ∈ S.

Proof. Since a maximal block topological space (G, τ _M ) is T₀-space, then for any two distinct vertices u and v, there exists an open subgraph V in (G, τ _M ) such that u ∈ V, v ∉ V or u ∉ V, v ∈ V and any vertex in G containing in one block or in two blocks in G, a vertex containing in one block is not cut vertex and a vertex containing in two blocks is cut vertex, so by above assumption, then |B| ≤ |C (B) |+1 holds for every C ⊆ B.

Conversely, if |B| ≤ |C (B) |+1 for every block B ∈ S, then each block B in G has cut vertices of order p or p - 1 (p order of each block B in G), that is, each vertex in G containing in one block or in two blocks in G, then for any two distinct vertices u and v, there exists an open subgraph V in (G, τ _M ) such that u ∈ V, v ∉ V or u ∉ V, v ∈ V. Hence (G, τ _M ) is a T₀-space. □

Theorem 4.8. A maximal block topological space (G, τ _M ) is T₀-space if and only if each cycle block C _p in G has at least p - 1 cut vertices.

Proof. Since a maximal block topological space (G, τ _M ) is T₀-space, by Theorem 4.7, we have |B| ≤ |C (B) |+1 for every block B ∈ S, suppose each block B in G is cycle block C _p , then |C _p | ≤ |C (C _p ) |+1, hence C _p has p or p - 1 cut vertices.

Conversely, if every cycle block C _p in G has p or p - 1 cut vertices, then the cardinality of each set of cut vertices is p or p - 1 and cardinality of each cycle block C _p is p, then |C _p | ≤ p + 1 or |C _p | ≤ (p - 1) +1 holds for every block B ∈ S, by Theorem 4.7, (G, τ _M ) is a T₀-space. □

Lemma 4.9. If each block in G is K₂, then a maximal block topological space (G, τ _M ) is T₀-space.

Proof. If each block in G is K₂, then by Theorem 4.2, we have Cl _M  ({u}) ≠ Cl _M  ({v}) for all distinct vertices u and v in G and by Theorem 4.3, (G, τ _M ) is T₀-space. □

The converse of the above lemma need not be true in general. See Example 4.5.

Lemma 4.10. A maximal block topological space (G, τ _M ) is T₀-space if and only if each block in G is K₂ and C _p has p or p - 1 cut vertices.

Proof. Follows from Theorem 4.8 and Lemma 4.9. □

Corollary 4.11. If G is acyclic graph and has one cut-vertex, then the maximal block topology τ _M is a T₀-space.

Lemma 4.12. If G is a tree graph, then maximal block topological space (G, τ _M ) is T₀-space.

Proof. Since tree is acyclic graph, then by Theorem 4.4, (G, τ _M ) is T₀-space. □

The Converse of above lemma need not be true in general. See Example 4.5.

Proposition 4.13. If there exists a block B in G such that |B| > |C (B) |+1, then the maximal block topology τ _M is not T₀-space.

Proof. Since |B| > |C (B) |+1 for some block B, so there exists at least two distinct non-cut vertices u and v in B. Hence, by Theorem 3.10, u ∈ Cl _M  ({v}) or v ∈ Cl _M  ({u}). Therefore, G is non-acyclic graph, hence by Theorem 4.4 (G, τ _M ) is not T₀-space. □

In Example 2.2, the maximal block topological space is not T₀-space because |B₄|=5 and |C (B₄) |=1 which implies that |B₄| > |C (B₄) |+1.

The star graph S of order n is a graph in which n - 1 vertices have degree 1 and a single vertex have degree n - 1, this single vertex is the only cut vertex in S.

Remark 4.14. In a star graph, we have |B|=2 for all B ∈ S and we have one cut vertex. Again, in a path graph, we have |B|=2 for all B ∈ S while |K|=1 or 2. Hence, star and path graphs are T₀-spaces.

Corollary 4.15. If a maximal block topological space (G, τ _M ) is T₀-space, then G has at least one cut-vertex.

The converse of the above corollary need not be true in general.

In Example 2.2, G has more than one cut vertex, let u₆ and u₇ be two vertices in G, but there exist no open subgraphs in τ _M containing one of them but not the other. Hence τ _M is not T₀-space.

Corollary 4.16. A maximal block topological space (G, τ _M ) is not T₀-space if G contains a cycle block C _p of order p > 3 and |C (C _p ) |≤2.

Proof. Follows from Proposition 4.13. □

Definition 4.17. A topological space (G, τ) induced by a graph G is said to be T_1/2-space if for every vertex u in G is either open subgraph or closed subgraph in (G, τ).

It is obvious that every T_1/2-space is T₀. However, the converse may not be true in general. But we have the following result.

Theorem 4.18. Every maximal block topological space (G, τ _M ) is T₀ if and only if it is T_1/2.

Proof. Suppose that (G, τ _M ) is T₀ and let u be any vertex in G. If u is a cut vertex, then u is open. Suppose that there is a block B such that u ∈ B and u is a non-cut vertex. Since (G, τ _M ) is a T₀-space, so every vertex in B must be a cut vertex because otherwise, by Proposition 4.13, we get a contradiction. Hence, by Lemma 2.3, we obtain that {u} is closed. Therefore, (G, τ _M ) is T_1/2. □

As an example of a T₀-space which is not T_1/2, we can consider a graph G with set of vertices V (G) = {u₁, u₂, u₃} and a topology τ = {φ, G, {u₁} , {u₁, u₂}}. It is clear that τ is not a maximal block topology for any such graph G. Moreover, this space is not T_1/2 because {u₂} is neither open nor closed but obviously it is T₀.

In this section, we define the surrounding subgraph and comparable subgraphs in a topological space (G, τ) induced by a graph G. Also some properties and results of surrounding subgraph and comparable subgraphs in a maximal block topological space (G, τ _M ) are present.

Definition 5.1. Let (G, τ) be any topological space induced by a graph G and let u be a vertex in a graph G, the surrounding subgraph of u is denoted by M _u , is the intersection of all open subgraphs in (G, τ) containing a vertex u. The surrounding subgraph of a subgraph H in G is denoted by M _H .

Definition 5.2. Let (G, τ) be a topological space induced by a graph G. Two subgraphs H₁ and H₂ in G (resp. two vertices u and v in G) are said to be comparable subgraphs (resp. comparable vertices) if M _{H 1}  ⊆ M _{H 2} or M _{H 2}  ⊆ M _{H 1} (resp. if M _u  ⊆ M _v or M _v  ⊆ M _u ). Otherwise are incomparable subgraphs (resp. are incomparable vertices).

From Example 2.2, let H₁ = {u₂} and H₂ = {u₂, u₃, u₄} are comparable subgraphs. And let H₁ = {u₁, u₂} and H₂ = {u₂, u₃, u₄} are not comparable subgraphs.

In Example 2.2, H₁ = {u₂, u₃} and H₂ = {u₁, u₃, u₄}, then M _{H 1}  = {u₂, u₃, u₄} and M _{H 2}  = {u₁, u₂, u₃, u₄}, therefore M _{H 1}  ⊆ M _{H 2} , hence H₁ and H₂ are comparable subgraphs in G.

Definition 5.3. Let (G, τ _M ) be a maximal block topological space. We define a relation ≤ on the set U (G) by u ≤ v if u ∈ M _v , or equivalently, M _u  ⊆ M _v . We write u < v if the inclusion is proper.

Theorem 5.4. Let (G, τ _M ) be a maximal block topological space. The relation ≤ on U (G) is reflexive and transitive, so that the relation ≤ is a preorder. Moreover, the relation ≤ is antisymmetric if and only if G is acyclic or contains blocks B with |B| ≤ |C (B) |+1.

Proof. For reflexivity, it obvious because M _u  ⊆ M _u .

Transitivity, if u ≤ v and v ≤ w, then M _u  ⊆ M _v and M _v  ⊆ M _w implies that M _u  ⊆ M _w . Hence, u ≤ w.

If G is acyclic or contains blocks B with |B| ≤ |C (B) |+1, then either u and v are incomparable whenever they are both cut vertices or they are in distinct blocks. If u and v are in the same block with one of them cut vertex (say u), then u ∈ M _v but not the converse. Hence, the relation is antisymmetric.

Conversely, suppose that the relation ≤ is antisymmetric and G contains a block with |B| > |C (B) |+1, then there is two vertices u and v are in the same block which are not cut vertices, this implies that u ∈ M _v and v ∈ M _u which is contradiction. Hence G is acyclic graph or |B| ≤ |C (B) |+1. □

Corollary 5.5. Let (G, τ _M ) be a maximal block topological space. The relation ≤ on U (G) is reflexive and transitive, so that the relation ≤ is a preorder. The relation ≤ is antisymmetric if and only if (G, τ _M ) is T₀-space.

Proof. Follows from Theorems 4.4 and 5.4. □

Remark 5.6. From Definition 5.1 and Example 2.2, we conclude that ≤ is not antisymmetric because u₆ ∈ M _{u 7} which means that u₆ ≤ u₇. On the other hand, u₇ ∈ M _{u 6} which means that u₇ ≤ u₆ but u₇,u₆ are distinct. Also, we have M _{u 6}  = M _{u 7} but u₇,u₆ are distinct. Hence, by above corollary, a maximal block topological space (G, τ _M ) in Example 2.2 is not T₀-space.

In Example 2.2, M _{u 6}  = {u₅, u₆, u₇, u₈, u₉} = M _{u 7} , but N (u₆) = {u₅, u₆, u₈} and N (u₇) = {u₅, u₇, u₉}, hence N (u₆) ≠ N (u₇).

Lemma 5.7. Let (G, τ _M ) be a maximal block topological space and let u and v be two vertices in G. M _u  = M _v if and only if Cl _M  ({u}) = Cl _M  ({v}).

Proof. Since M _u  = M _v , that is u and v are non-cut vertices and they are in the same cycle block, so by Theorem 3.10(1), Cl _M  ({u}) = Cl _M  ({v}). Conversely, if Cl _M  ({u}) = Cl _M  ({v}), then by Theorem 3.10(1), u and v are non-cut vertices and they are in the same cycle block, hence M _u  = M _v . □

Theorem 5.8. Let H be a block subspace of a maximal block topological space (G, τ _M ), then M _H  (u) = H ∩ M _G  (u) for all u in G.

Proof. Let u be any vertex in H and V be an open subgraph containing u and it is a subgraph in H, then V = H ∩ L, where L be an open subgraph in τ _M , that is M _G  (u) is a subgraph in L, so H ∩ M _G  (u) ⊆ H ∩ L = V. Hence M _H  (u) = H ∩ M _G  (u) for all u in G. □

Definition 5.9. Let (G, τ) be any topological space induced by a graph G. We say that:

A vertex u in a graph G is irreducible if u ≤ v for all comparable vertices v ∈ G with u.

Theorem 5.10. Let (G, τ _M ) be a maximal block topological space. A vertex u in a graph G is completely irreducible vertex of if and only if u is a cut vertex in G.

Proof. Since u is completely irreducible vertex of G, then {u} = M _u , that is, the intersection of all open subgraphs containing u is u itself, so by Definition 2.1, u is cut vertex in G. Conversely, if u is cut vertex in G, then by Definition 2.1, the intersection of all open subgraphs containing u is u itself, therefore, {u} = M _u , hence u is completely irreducible vertex in G. □

Theorem 5.11. Let (G, τ _M ) be a maximal block topological space. A subgraph H is an open subgraph in (G, τ _M ) if and only if H is completely irreducible subgraph in G.

Proof. Since H is an open subgraph in (G, τ _M ), then the intersection of all open subgraphs containing H is H itself, then H = M _H , hence H is completely irreducible subgraph in G. Conversely, since H is completely irreducible subgraph in G, then H = M _H , that is M _H is the smallest open subgraph containing H, hence H is an open subgraph in G. □

Theorem 5.12. Let (G, τ _M ) be a maximal block topological space, every completely irreducible subgraph in G is irreducible subgraph in G.

Proof. It is obvious by Definition 5.9. □

The converse of the above theorem is not be true in general.

From Example 2.2, let H₁ = {u₁} and H₂ = {u₁, u₂, u₃}, then M _{H 1}  = {u₁, u₂} and M _{H 2}  = {u₁, u₂, u₃, u₄} and M _{H 1}  ⊆ M _{H 2} , hence H₁ is irreducible subgraph, but it is not completely irreducible subgraph in G, because H₁ ≠ M _{H 1} .

Remark 5.13. Every open subgraph in (G, τ _M ) is irreducible subgraph in G, but the converse need not true in general.

In Example 2.2, let H₁ = {u₁} and H₂ = {u₁, u₂, u₃}, then M _{H 1}  = {u₁, u₂} and M _{H 2}  = {u₁, u₂, u₃, u₄}, then M _{H 1}  ⊆ M _{H 2} , so H₁ is irreducible subgraphs in G, but it is not open subgraph in (G, τ _M ).

Corollary 5.14. In a maximal block topological space (G, τ _M ), every pendant vertex is not completely irreducible vertex in G.

Proof. Straightforward. □

Theorem 5.15 Let (G, τ _M ) be a maximal block topological space, if u and v are completely irreducible vertices in G, then M _u ∩ M _v  = ∅.

Proof. Straightforward. □

The converse of above theorem need not be true in general.

In Example 2.2, M _{u 3}  = {u₂, u₃, u₄} and M _{u 6}  = {u₅, u₆, u₇, u₈, u₉}, then M _{u 3} ∩ M _{u 6}  = ∅, but u₃ and u₆ are not completely irreducible vertices.

Remark 5.16. The above theorem is true for completely irreducible subgraphs in G, but it is not true for irreducible subgraphs in G.

In Example 2.2, let H₁ = {u₁}, H₂ = {u₂, u₃} and H₃ = {u₁, u₄} then M _{H 1}  = {u₁, u₂}, M _{H 2}  = {u₁, u₂, u₃, u₄} and M _{H 2}  = {u₁, u₂, u₃, u₄}, then M _{H 1}  ⊆ M _{H 2} and M _{H 2}  ⊆ M _{H 3} , hence H₁ and H₂ are distinct irreducible subgraphs in G, but M _{H 1} ∩ M _{H 2}  = {u₁, u₂} ≠ ∅.

Also, in Example 2.2, we have int _M  ({u₁}) =∅ and int _M  ({u₇}) =∅, then int _M  ({u₁})∩ int _M  ({u₇}) = ∅, but u₁ and u₇ are not completely irreducible vertices.

Theorem 5.17. Let (G, τ _M ) be a maximal block topological space, then every completely irreducible open subgraph in (G, τ _M ) is irreducible subgraph in G.

Proof. Obvious. □ The converse of the above theorem need not be true in general.

In Example 2.2, let H₁ = {u₁, u₃, u₄} and H₂ = {u₁, u₂, u₃}, then M _{H 1}  = {u₁, u₂, u₃, u₄} and M _{H 2}  = {u₁, u₂, u₃, u₄}, then M _{H 1}  ⊆ M _{H 2} , hence H₁ is a subgrph in G, but it is neither irreducible open subgraph nor completely irreducible open subgraph in (G, τ _M ).

Theorem 5.18. Let (G, τ _M ) be a maximal block topological space and H be a subgraph in G, H is completely irreducible subgraph if and only if int _M  (H) = H.

Proof. If H is completely irreducible subgraph in G, then H = M _H , this implies that M _H is the smallest open subgraph containing H, then int _M  (H) = int _M  (M _H ) = H. Conversely, if int _M  (H) = H, that is H is an open subgraph in (G, τ _M ) and by Theorem 5.11, H is completely irreducible subgraph in G. □

Corollary 5.19. Let (G, τ _M ) be a maximal block topological space. Then the following statements are true.

Union of two completely irreducible subgraphs is completely irreducible subgraph.

Proof. Obvious. □

Remark 5.20. Infinite union and infinite intersection of completely irreducible subgraphs is completely irreducible subgraph.

Definition 5.21. Let (G, τ) be topological space induced by a graph G. We say that

Two vertices u and v in a graph G are topologically independent vertices if there exists an open subgraph in (G, τ) containing u but not containing v and also there exists an open subgraph in (G, τ) containing v but not containing u.

Lemma 5.22. In a maximal block topological space (G, τ _M ), the following statements are true for any two vertices u, v ∈ G.

If u, v are cut vertices or they are in different blocks, then they are topologically independent.

Theorem 5.23. Every stable open subgraph in (G, τ _M ) is weak stable open subgraph in (G, τ _M ).

Proof. Follows from Definition 5.21. □ The converse of the above theorem need not be true in general.

From Example 2.2, let H = {u₂, u₃, u₄}, then u₂ and u₃ are not topologically independent vertices, then H is not stable open subgraph, but H is weak stable open subgraph. And let H = {u₄, u₅}, then u₄ and u₅ are topologically independent vertices, then H is stable open subgraph and also is weak stable open subgraph.

Remark 5.24. From above example, we can see that every open subgraph in (G, τ _M ) my not be stable open subgraph in (G, τ _M ).

Theorem 5.25. Every open subgraph in (G, τ _M ) is weak stable open subgraph if and only if (G, τ _M ) is T₀-space.

Proof. Since (G, τ _M ) is T₀-space, then for any two distinct vertices there exists an open subgraph containing one of them but not the other, that is the vertices in any open subgraph are pairwise topologically independent vertices, hence every open subgraph in (G, τ _M ) is weak stable open subgraph. Conversely, if every open subgraph in (G, τ _M ) is weak stable open subgraph, then by Definition 5.21, we have the vertices in any open subgraph are pairwise topologically independent vertices, then by Definition 5.21, there exists an open subgraph in (G, τ _M ) containing u but not containing v or conversely, hence (G, τ _M ) is T₀-space. □

Theorem 5.26. Every open subgraph in (G, τ _M ) is weak stable open subgraph if G is acyclic graph.

Proof. If G is acyclic graph, then by Theorem 4.4, (G, τ _M ) is T₀-space and by Theorem 5.25, hence every open subgraph in (G, τ _M ) is weak stable open subgraph. □

The converse of the above theorem need not be true in general. See Example 2.2, every open subgraph in (G, τ _M ) is weak stable open subgraph, but G is not acyclic graph.

Theorem 5.27. Every open subgraph in (G, τ _M ) is weak stable open subgraph if and only if each cycle block C _p in G has p or p - 1 cut vertices.

Proof. If each cycle block C _p in G has p or p - 1 cut vertices, then by Theorem 4.8, (G, τ _M ) is T₀-space and by Theorem 5.25, hence every open subgraph in (G, τ _M ) is weak stable open subgraph. Conversely, let H be an open subgraph in (G, τ _M ) is weak stable open subgraph, then be Definition 5.21, any two vertices u and v in H are pairwise weak topologically independent vertices containing one of them but not the other or conversely, that is one of u and v is in different blocks, so u is cut vertex or v is cut vertex or u and v are cut vertices in the some block, that is, each cycle block C _p in G has cut vertices of p or p - 1. □

Theorem 5.28. A maximal block topological space (G, τ _M ) is weak topologically independent graph space if and only if (G, τ _M ) is T₀-space.

Proof. Since a maximal block topological graph space (G, τ _M ) is topologically independent space, then there exists an open subgraph for any two distinct vertices u and v containing u but not containing v, this is equivalent to the definition of T₀- space, hence (G, τ _M ) is T₀-space. Conversely, if (G, τ _M ) is T₀-space, then there exists an open subgraph for any two distinct vertices u and v containing u but not containing v, hence (G, τ _M ) is topologically independent graph space. □

Corollary 5.29. A maximal block topological space (G, τ _M ) is weak topologically independent graph space if G is acyclic graph.

Proof. Follows from Theorems 4.4 and 5.28. □ The converse of the above theorem need not be true in general.

See Example 2.2, a maximal block topological space (G, τ _M ) is weak topologically independent space, but G is not acyclic graph.

Corollary 5.30. A maximal block topological space (G, τ _M ) is weak topologically independent graph space if and only if each cycle block C _p in G has p or p - 1 cut vertices.

Proof. Follows from Theorems 4.8 and 5.28. □

6 Conclusion

In this paper, we generated a topological space via simple graph G by blocks and cut vertices in G called maximal block topological space, some properties and results of this type of topology was presented. Some topological concepts namely, derived, closure and interior subgraphs were defined. Also, we proved that in a maximal block topological space both T₀-space and T_1/2-space are equivalent. Finally, irreducibility and topologically independent of maximal block topological space were introduced.