Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 48, No. 1 (2026) pp. 33–39.

NEUTROSOPHIC BLOCK TOPOLOGICAL SPACES

INDUCED BY NEUTROSOPHIC GRAPHS

Mohanapriya Rajagopal¹, and Durgadevi Shanmugasundaram²

Abstract. This work introduces neutrosophic block topological spaces generated from

block decompositions of neutrosophic graphs, establishing formal topologies on vertex sets

where blocks represent maximal connected subgraphs without cut-vertices. Continuity

and homeomorphism theories prove structural preservation including cut-vertex corre-

spondence and block count invariance under topological equivalence.

Keywords: Neutrosophic Graph, Neutrosophic Block Topology, Cut-Vertices, Continuity,

Homeomorphism.

2010 AMS Subject Classiﬁcation: Primary: 54A05, 54A10; Secondary: 94C15.

1. Introduction and Preliminaries

Modern networks often involve uncertainty that classical models cannot adequately rep-

resent. Neutrosophic set theory, proposed by Smarandache, independently models truth,

indeterminacy, and falsity, allowing eﬀective representation of ambiguous information

[9, 10]. Extending this idea, neutrosophic graphs introduced by Broumi and Smaran-

dache [2, 3] and further developed by Akram and Shahzadi [1] assign neutrosophic values

to vertices and edges, capturing relational uncertainty. In classical graph theory, a block

is a maximal connected subgraph without a cut-vertex and plays a key role in struc-

tural analysis [4, 5]. Topological methods have further enriched graph analysis through

concepts such as continuity and connectedness [6, 8]. Recently, Macaso and Balingit

[7] introduced block topological spaces, linking graph decomposition with topology. Build-

ing on these studies, Mohanapriya Rajagopal and Durgadevi Shanmugasundaram

propose neutrosophic block topological spaces, integrating neutrosophic graphs with block-

based topology to formalize continuity and homeomorphism under uncertainty.

2. Neutrosophic Block Topology Induced by Neutrosophic Graph

+

Let N_Gbe a neutrosophic graph with connected components C₁, C₂, . . . , C_j, j ∈ Z .

Each component is ﬁrst examined for cut-vertices; components without cut-vertices are

considered neutrosophic blocks. If a component contains a cut-vertex (v, η(v), κ(v), λ(v)),

it is decomposed into branches at that vertex, where each branch consists of vertices

mutually reachable without passing through the cut-vertex. Each branch is then analyzed

recursively: branches without cut-vertices form blocks, while those containing cut-vertices

are further decomposed. This process continues until all resulting subgraphs contain no

cut-vertices, yielding the complete set of neutrosophic blocks of N_G.

33

34

Mohanapriya Rajagopal and Durgadevi Shanmugasudaram

Block Collection: The complete set of neutrosophic blocks consists of all components,

branches, and sub-branches that contain no cut-vertices.

Consider the neutrosophic

graph N_Gshown in Figure 1.

Figure 1. Neutrosophic Graph N_G

The components of N_Gshown in Figure 2 :

Figure 2. The Components of N_G: C₁, C₂

The distinct blocks of N_Gshown in Figure 3:

Figure 3. The Blocks of N_G

Deﬁnition 2.1. Let N_Gbe a neutrosophic graph possessing blocks B₁, B₂, . . . , B_kwith

+

k ∈ Z . The family of vertex sets Σ_B(N_G) = {V (B₁), V (B₂), . . . , V (B_k)}, generates a

topology on V (N_G) through the following construction. Deﬁne Γ_B(N_G) as the collection

of all ﬁnite intersections of sets from Σ_B(N_G), which serves as a topological base. The

topology τ_B(N_G) consists of all arbitrary unions of sets in Γ_B(N_G), together with the

empty set and V (N_G). We call τ_B(N_G) the neutrosophic block topology on N_G. The pair

(V (N_G), τ_B(N_G)) forms the neutrosophic block topological space associated with N_G. Sets

belonging to τ_B(N_G) are said to be τ_B(N_G)-open, while sets whose complements belong to

τ_B(N_G) are τ_B(N_G)-closed.

Example 2.2. Consider the graph H_Nshown in Figure 4

NEUTROSOPHIC BLOCK TOPOLOGICAL SPACES INDUCED BY NEUTROSOPHIC GRAPHS

35

Figure 4. H_N

The blocks of H_Nare shown in Figure 5 and Figure 6

Figure 5. The Blocks of H_N

Figure 6. The Blocks of H_N

ꢀ

ꢄ

{(v₁, η(v₁), κ(v₁), λ(v₁)), (v₂, η(v₂), κ(v₂), λ(v₂)), (v₃, η(v₃), κ(v₃), λ(v₃))},

ꢁ

ꢂ

ꢁ

ꢅ

{(v₃, η(v₃), κ(v₃), λ(v₃)), (v₄, η(v₄), κ(v₄), λ(v₄))},

Σ_B(H_N) =

.

ꢁ

ꢃ

ꢆ

{(v₃, η(v₃), κ(v₃), λ(v₃)), (v₅, η(v₅), κ(v₅), λ(v₅))}

Taking the ﬁnite intersections of sets in Σ_B(N_G), we obtain

ꢀ

ꢄ

ꢁ

{(v , η(v ), κ(v ), λ(v )), (v , η(v ), κ(v ), λ(v )), (v , η(v ), κ(v ), λ(v ))},

ꢁ

1

2

3

ꢁ

ꢂ

ꢁ

ꢅ

{(v₃, η(v₃), κ(v₃), λ(v₃)), (v₄, η(v₄), κ(v₄), λ(v₄))},

Γ_B(H_N) =

.

ꢁ

{(v , η(v ), κ(v ), λ(v )), (v , η(v ), κ(v ), λ(v ))},

ꢁ

3

5

ꢁ

ꢃ

ꢁ

ꢆ

{(v₃, η(v₃), κ(v₃), λ(v₃))}

36

Mohanapriya Rajagopal and Durgadevi Shanmugasudaram

Hence, the neutrosophic block topology is

ꢀ

ꢄ

∅,

ꢁ

{(v₃, η(v₃), κ(v₃), λ(v₃))},

ꢁ

{(v₁, η(v₁), κ(v₁), λ(v₁)), (v₂, η(v₂), κ(v₂), λ(v₂)), (v₃, η(v₃), κ(v₃), λ(v₃))},

{(v₃, η(v₃), κ(v₃), λ(v₃)), (v₄, η(v₄), κ(v₄), λ(v₄)), (v₅, η(v₅), κ(v₅), λ(v₅))},

{(v₁, η(v₁), κ(v₁), λ(v₁)), (v₂, η(v₂), κ(v₂), λ(v₂)), (v₃, η(v₃), κ(v₃), λ(v₃)), (v₄, η(v₄), κ(v₄), λ(v₄))},

{(v₁, η(v₁), κ(v₁), λ(v₁)), (v₂, η(v₂), κ(v₂), λ(v₂)), (v₃, η(v₃), κ(v₃), λ(v₃)), (v₅, η(v₅), κ(v₅), λ(v₅))},

V (H_G)

ꢁ

ꢂ

ꢁ

ꢅ

τ_B(H_N) =

.

ꢁ

ꢃ

ꢁ

ꢆ

2.1. Continuity in Neutrosophic Block Topological Space.

Deﬁnition 2.3. A function f : (V (N_G), τ_B(N_G)) → (V (N_H), τ_B(N_H)) is said to be

continuous if and only if for every U ∈ τ_B(NH), we have f⁻¹(U) ∈ τ_B(NG), where

f

⁻¹(U) = { (v, η(v), κ(v), λ(v)) ∈ V (NG) : f(v, η(v), κ(v), λ(v)) ∈ U }.

Theorem 2.4. Let N_Gand N_Hbe neutrosophic graphs. A function f : (V (N_G), τ_B(N_G)) →

(V (N_H), τ_B(N_H)) is continuous if and only if for every neutrosophic block B of N_H, the

set f⁻¹(V (B)) belongs to τ_B(N_G).

Proof. Assume f is continuous. Let B be any neutrosophic block of NH. Since

V (B) ∈ Σ_B(NH), it is τ_B(NH)-open. By continuity, f⁻¹(V (B)) ∈ τ_B(N_G).

Conversely, suppose that for every neutrosophic block B of N_H, f⁻¹(V (B)) ∈ τ_B(N_G).

Let U be an arbitrary τ_B(N_H)-open set. By deﬁnition of neutrosophic block topology,

ꢈ

ꢊ

ꢇ ꢉ

U =

V (B_j^α) ,

α

j

where each B^αis a neutrosophic block of N_H. Taking preimages,

j

ꢈ

ꢊ

ꢈ

ꢊ

ꢇ

ꢉ

ꢇ ꢉ

f

⁻¹(U) =

f⁻¹

V (B_j^α)

=

f

⁻¹(V (B_j^α)) .

α

j

α

j

By hypothesis, each f⁻¹(V (B^α)) is τ_B(NG)-open. Since τ_B(NG) is closed under ﬁnite

j

intersections and arbitrary unions, we have f⁻¹(U) ∈ τ_B(N_G). Hence, f is continuous.

Theorem 2.5. Let N_Gand N_Hbe neutrosophic graphs. If (v, η(v), κ(v), λ(v)) is a neu-

trosophic cut-vertex of N_Gand f : (V (N_G), τ_B(N_G)) → (V (N_H), τ_B(N_H)) is continuous,

then f((v, η(v), κ(v), λ(v))) is either a neutrosophic cut-vertex of N_Hor an isolated vertex

of N_H.

Proof. Since (v, η(v), κ(v), λ(v)) is a cut-vertex of the neutrosophic graph N_G, and

since every cut-vertex induces a singleton which is τ_B(N_G)-open (see [7], Theorem 3),

it follows that {(v, η(v), κ(v), λ(v))} ∈ τ_B(N_G).

Assume f

is continuous.

If

f((v, η(v), κ(v), λ(v)))

is

neither

a cut-vertex nor an isolated vertex in N_H, then

{ f((v, η(v), κ(v), λ(v))) } ∈/ τ_B(NH).

However,

by continuity,

the preimage of

{ f((v, η(v), κ(v), λ(v))) } under f must be τ_B(N_G)-open, which contains (v, η(v), κ(v), λ(v)).

NEUTROSOPHIC BLOCK TOPOLOGICAL SPACES INDUCED BY NEUTROSOPHIC GRAPHS

37

This leads to a contradiction. Therefore, f((v, η(v), κ(v), λ(v))) must be either a cut-vertex

or an isolated vertex of N_H.

Theorem 2.6. Let N_Gbe a neutrosophic graph and N_Ha neutrosophic empty graph (only

isolated vertices) with |V (N_G)| = |V (N_H)|. Then every bijection f : (V (N_G), τ_B(N_G)) →

(V (N_H), τ_B(N_H)) is continuous.

Proof. Since N_His an empty graph, every vertex of N_His isolated. Hence, each sin-

gleton {(w, η(w), κ(w), λ(w))} forms a block of N_H, and therefore Σ_B(N_H) consists of

all singletons. Consequently, τ_B(N_H) is the discrete topology on V (N_H) (see [7], The-

orem 2). Thus every subset of V (N_H) is τ_B(N_H)-open. Let U ⊆ V (N_H), then U =

ꢋ

{{(w, η(w), κ(w), λ(w))} : (w, η(w), κ(w), λ(w)) ∈ U}. Taking preimages,f⁻¹(U) =

ꢋ

{f⁻¹({(w, η(w), κ(w), λ(w))}) : (w, η(w), κ(w), λ(w)) ∈ U}. Since f is bijective, each

preimage is a singleton in V (N_G); hence f⁻¹(U) is a union of singletons. As τ_B(N_G) is

closed under arbitrary unions, f⁻¹(U) ∈ τ_B(N_G), and therefore f is continuous.

□

2.2. Homeomorphism in Neutrosophic Block Topological Space.

Deﬁnition 2.7. Let N_Gand N_Hbe neutrosophic graphs with block topological spaces

(V (N_G), τ_B(N_G)) and (V (N_H), τ_B(N_H)) respectively. A function f : (V (N_G), τ_B(N_G)) →

(V (N_H), τ_B(N_H)) is called a neutrosophic block topological homeomorphism if:

(i) f is bijective (one-to-one and onto),

(ii) f is continuous from (V (N_G), τ_B(N_G)) to (V (N_H), τ_B(N_H)), and

(iii) f⁻¹: V (N_H) → V (N_G) is continuous from (V (N_H), τ_B(N_H)) to (V (N_G), τ_B(N_G)).

In this case, we say (V (N_G), τ_B(N_G)) and (V (N_H), τ_B(N_H)) are homeomorphism, written

∼

as (V (N ), τ (N ))

(V (N ), τ (N )).

=

G

B

G

H

B

H

Theorem 2.8. Let N_Gand N_Hbe neutrosophic graphs. If f : (V (N_G), τ_B(N_G)) →

(V (N_H), τ_B(N_H)) is a neutrosophic block topological homeomorphism, then a vertex

(v, η(v), κ(v), λ(v)) ∈ V (N_G)

is a neutrosophic cut-vertex if and only if f(v, η(v), κ(v), λ(v)) ∈ V (N_H) is a neutrosophic

cut-vertex. Moreover, the neutrosophic degree of (v, η(v), κ(v), λ(v)) is preserved under f.

Proof. Suppose (v, η(v), κ(v), λ(v)) ∈ V (N_G) is a neutrosophic cut-vertex. Then

{(v, η(v), κ(v), λ(v))} ∈ τ_B(N_G).

Since f is a homeomorphism, f({(v, η(v), κ(v), λ(v))}) = {f(v, η(v), κ(v), λ(v))} ∈ τ_B(N_H).

By the neutrosophic analogue of the block topology characterization, if

{f(v, η(v), κ(v), λ(v))}

is open in τ_B(N_H), then f(v, η(v), κ(v), λ(v)) must be either a cut-vertex or an isolated ver-

tex of N_H.If f(v, η(v), κ(v), λ(v)) were isolated, then f(v, η(v), κ(v), λ(v)) would have no

adjacent vertices in N_H. But since f is bijective and preserves adjacency through continu-

ity, this contradicts (v, η(v), κ(v), λ(v)) being a cut-vertex in N_G. Thus, f(v, η(v), κ(v), λ(v))

38

Mohanapriya Rajagopal and Durgadevi Shanmugasudaram

must be a cut-vertex. Since f is a homeomorphism, it preserves neutrosophic values as-

sociated with open sets. Therefore, the degree of f(v, η(v), κ(v), λ(v)) equals the degree

of (v, η(v), κ(v), λ(v)). The reverse follows symmetrically by applying the same reasoning

to f⁻¹: (V (N_H), τ_B(N_H)) → (V (N_G), τ_B(N_G)).

□

Theorem 2.9. Let N_Gand N_Hbe neutrosophic graphs. If f : (V (N_G), τ_B(N_G)) →

(V (N_H), τ_B(N_H)) is a neutrosophic block topological homeomorphism, then N_Gand N_H

have the same number of neutrosophic blocks. Moreover, for each block B of N_G, there

exists a unique block B^′of N_Hsuch that f(V (B)) = V (B^′).

Proof. Let B be a neutrosophic block of N_G. Then V (B) ∈ Σ_B(N_G), hence V (B) is

τ_B(N_G)-open. Since f is a homeomorphism, f(V (B)) ∈ τ_B(N_H). By the deﬁnition of

block topology, τ_B(N_H)-open sets corresponding to maximal connected subgraphs without

cut-vertices are precisely the vertex sets of blocks of N_H. Thus, f(V (B)) = V (B^′) for

some block B^′of N_H.

To show uniqueness: If f(V (B)) were equal to both V (B₁^′) and V (B₂^′) for distinct

blocks B^′, B^′of N_H, then these blocks would share more than one vertex, contradicting

1

2

the fact that blocks intersect in at most one cut-vertex. Since f is bijective, every block

of N_Harises as the image of exactly one block of N_G. Hence, N_Gand N_Hhave the same

number of blocks.

□

Theorem 2.10 (Composition of Homeomorphisms). Let N_G, N_G, and N_Gbe neutro-

1

2

3

sophic graphs. If

f : (V (N_G), τ_B(N_G)) → (V (N_G), τ_B(N_G)) and g : (V (N_G), τ_B(N_G)) → (V (N_G), τ_B(N_G))

1

2

3

are neutrosophic block topological homeomorphisms, then the composition

g ◦ f : (V (N_G), τ_B(N_G)) → (V (N_G), τ_B(N_G))

1

3

is also a neutrosophic block topological homeomorphism.

Proof. Since f and g are bijective, their composition g ◦ f is bijective. Continuity: Let

W ∈ τ_B(N_G) be open. Since g is continuous, g⁻¹(W) ∈ τ_B(N_G). Since f is continuous,

3

2

f⁻¹(g⁻¹(W)) = (g ◦ f)⁻¹(W) ∈ τ_B(N_G). Hence, g ◦ f is continuous.

1

Continuity of the inverse: We have (g ◦ f)⁻¹= f⁻¹◦ g⁻¹. Since f⁻¹and g⁻¹are

continuous, their composition is continuous. Therefore, g ◦ f is a homeomorphism.

□

∼

Corollary 2.11. The relation “is homeomorphic to” ( ) on neutrosophic block topological

=

spaces is an equivalence relation.

Proof. Reﬂexivity: For any N_G, the identity homeomorphism theorem shows that

∼

(V (N ), τ (N ))

(V (N ), τ (N )).

=

G

B

G

B

G

∼

Symmetry: If (V (N_G), τ_B(N_G))

f⁻¹is a homeomorphism by inverse homeomorphism theorem. so (V (N_H), τ_B(N_H))

(V (N_H), τ_B(N_H)) via a homeomorphism f, then

=

∼

=

(V (N_G), τ_B(N_G)).

NEUTROSOPHIC BLOCK TOPOLOGICAL SPACES INDUCED BY NEUTROSOPHIC GRAPHS

39

∼

Transitivity: If (V (N_G), τ_B(N_G))

(V (N_G), τ_B(N_G)) and (V (N_G), τ_B(N_G))

=

1

2

(V (N_G), τ_B(N_G)), then by Composition of homomorphism theorem, their composition

3

∼

gives (V (N ), τ (N ))

(V (N ), τ (N )).

G₃G₃

=

G₁

B

G₁

B

∼

Therefore,

is an equivalence relation.

□

=

3. Conclusion

This work develops a rigorous framework for analyzing networks under uncertainty by

integrating neutrosophic graph theory with topological concepts. It introduces neutro-

sophic block topological spaces to formally capture complex relational patterns, charac-

terized through cut-vertex-based open sets and the preservation of block structures under

continuous and homeomorphic mappings. The framework strengthens classical graph

topology by accommodating uncertainty in relational structures and oﬀers a solid mathe-

matical foundation for complex network analysis. Future research may extend this study

to dynamic and weighted neutrosophic networks, temporal block topologies, and evolving

multi-criteria optimization models.

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size, IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2444–2451, 2016.

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[5] R. Diestel, Graph Theory (5th ed.), Springer, 2017.

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[7] J.B.C. Macaso and C.M.R. Balingit, The block topological space and block topological graph induced by

undirected graphs, European Journal of Pure and Applied Mathematics, 17 (2), 663–675, 2024; DOI:

10.29020/nybg.ejpam.v17i2.5135.

[8] J.R. Munkres, Topology (2nd ed.), Prentice Hall, 2000.

[9] A.A. Salama and S.A. Alblowi, Neutrosophic set and neutrosophic topological spaces, IOSR Journal of Math-

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(Received, December 22, 2025)

(Revised, January 03, 2026)

¹Department of Mathematics,

Sri Krishna Adhithya College of Arts and Science,

Coimbatore, India - 641042

Email¹: wishesmona@gmail.com

²Sri Krishna Adhithya College of Arts and Science,

Coimbatore, India - 641042

Email²: durga.sitha@gmail.com