EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6611
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Folding on Topological Graphs and Their Fundamental
Group

Mohammed Abu-Saleem
1 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117,
Jordan

Abstract. We introduce the effect of folding on the fundamental groups of a connected topological
graph and its dual graph. We will deduce the limit of folding on the fundamental group of a
topological graph and its duality. We obtain the relations between the induced folding and the
induced retraction on the fundamental group. We apply retraction and folding operations to
reduce the size of the graph while preserving its essential structural properties, which can assist
us in developing mathematical software.

2020 Mathematics Subject Classifications: 05C10, 19D55, 55N10, 54-XX

Key Words and Phrases: Topological graph, Dual topological graph, Folding, Fundamental
group

1. Introduction

Graph theory serves as a mathematical representation that effectively investigates sev-
eral tangible, real-world problems. Many different concerns related to the fields of chem-
istry, physics, communication, computer science, the genetic code, social science, psychol-
ogy, and linguistics could potentially be articulated as problems related to graph theory.
Many different areas of mathematics, including matrix theory, group theory, and topologi-
cal structures, have strong relationships with graph theory [1]. Any graph G could indicate
a topological space such that every vertex corresponds to only one point and every edge
corresponds to a distinct arc, homeomorphic to the closed interval. The boundary points
of an arc denote the endpoints of the associated edge, the interiors of the arcs are mutu-
ally disjoint and do not intersect the points representing vertices, and this configuration
is referred to as a topological representation of G [2, 3].

A graph is defined as an ordered pair G = (V (G), E(G)), in which V (G) ̸= Φ is a set
and E(G) is a set that is disjoint from V (G), which are referred to as the vertices of G.
The elements of E(G) are referred to as the edges of G [1, 4, 5]. A graph H is called a
subgraph of a graph G, denoted H ⊆ G, if V (H) ⊆ V (G) and E(H) ⊆ E(G) [6]. A graph

DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6611

Email address: m abusaleem@bau.edu.jo (M. Abu-Saleem)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Abu-Saleem / Eur. J. Pure Appl. Math, 18 (3) (2025), 6611 2 of 7

G is planar if its edges meet only at their endpoints in the plane. We refer to a drawing of
this form as a planar embedding of the graph G [7]. Let G be a plane graph. To construct
a new graph d(G) of G, for all faces f of G, select a vertex f̂ , with respect to all edges e of
G, choose an edge ê. Then edge ê joins vertices f̂ and ĝ in d(G) iff edge e is common to the
boundaries of faces f and g in G, this graph is called the dual of G [1]. We refer to an edge
with coinciding vertices as a loop, while we refer to two vertices connected by more than
one edge as multiple edges. An infinite graph is defined as a graph where both the edge
set and the vertex set have infinite cardinality. A cycle graph is defined by the existence
of a singular cycle; we will denote the cycle graph with one vertex by C, and the cycle
graph with n- vertices is represented as Cn. The path graph is a graph that consists of a
single path; the path graph with n-vertices is represented as Pn [1, 4]. The cobblestone
path Jn is the graph formed by duplicating each edge of the n-vertex path Pn. A graph
with one vertex and n self-loops is called an n-bouquet Bn [2, 3]. Let Y be a space, and
let y0 be an element of Y . The set π1(Y, y0) consists of the homotopy classes of loops in
(Y, y0), and it is equipped with the product operation [u][v] = [u.v] referred to as the
fundamental group [8, 9]. The fundamental groups with particular spaces were looked at
[10, 11]. Let Y1 and Y2 be spaces such that y1 ∈ Y1 and y2 ∈ Y2, with Y1 ∩ Y2. The wedge
sum Y1 ⋎ Y2 is defined as the quotient of Y1 ∪ Y2 by the identification y1 ∼ y2. [8]. A
map Φ : V (G1) −→ V (G2) constitutes a homomorphism from G1 to G2 if it maintains edge
preservation; specifically, for each edge [u1, u2] in G1, [Φ(u1),Φ(u2)] must be an edge in G1

[3]. A retract of a graph G1 is a subgraph G2 of G1 for which there is a homomorphism
r : G1 → G2, denoted retraction, providing r(u) = u for every vertex u in G2 [8]. The
folding is a continuous map 𭟋 : G1 → G2 in which 𭟋(υ) ∈ V (G2) for every υ ∈ V (G1) and
𭟋(e) ∈ E(G2) for every e ∈ E(G1) [12, 13]. The topological and Banhatti indices for
various silicate and oxide networks, using fuzzy set extensions, were presented in [14, 15].
For further information about the folding and retraction on manifolds and graphs, see
[12, 16–19].

The central problem of this paper is to investigate the effects of folding and retraction
on various classes of graphs and to characterize how retraction and folding impact key
graph invariants. Transformations mainly consist of folding, limit folding, and rewrite
rules that help sort graphs, find unchanging features, and enhance complex networks while
keeping important information intact. Folding theory is the important transformation used
in structural topology. Its simplest form merges and eliminates edges and vertices. This
operation may remove multi-edges or loops by using folding and limit folding mapping in
a special family of graph theory. This study will improve our insight into graph theory by
looking at folding transformations, which will help us better understand graph structure
and classification.

2. The main result

Theorem 1. Given a connected graph G, then
(i) The folding 𭟋 : G −→ G, induces �̂� : π1(G) −→ π1(G) for which �̂�(π1(G)) = π1(𭟋(G)).
(ii) There exists a specific type of folding 𭟋 : G −→ G̃ that induces �̂� : π1(G) −→ π1(G̃),



M. Abu-Saleem / Eur. J. Pure Appl. Math, 18 (3) (2025), 6611 3 of 7

and for this type, rank (�̂�(π1(G))) ≥ rank (π1(G)).
Proof. (i) Let G be a connected graph. Then,

�̂�(π1(G)) = �̂�{[C] :where C is a cycle based at one vertex v0 ∈ G} = {[𭟋(C)] :where
𭟋(C) is a cycle based at 𭟋(v0) ∈ 𭟋(G)} = π1(𭟋(G)).
(ii) Let 𭟋 : G −→ G̃ be a folding map in such a way that an edge is folded into another
edge, then 𭟋(G) contains a new multiple edge that induces �̂� : π1(G) −→ π1(G̃ ) such
that �̂�(π1(G) = π1(𭟋(G)). Since rank (π1(𭟋(G))) ≥rank (π1(G)), it follows that rank
(�̂�(π1(G))) ≥rank (π1(G)).

Definition 1. Let {𭟋i : Gi−1 → Gi : i = 1, 2, . . .m} be a sequence of folding maps on a con-
nected graph, then we define the limit folding map as lim

m→∞
𭟋m(Gm−1) = lim

m→∞
𭟋m(𭟋m−1(. . . (𭟋1(G0) . . .).

Theorem 2. Let Jn be a cobblestone path, then
(i) There is folding 𭟋 : Jn → Jn which induces �̂� : π1(Jn) → π1(Jn) such that rank (�̂�(π1(Jn))) ≤
n− 1.
(ii) The folding 𭟋 : dual(Jn) → dual(Jn) induces �̂� : π1(dual(Jn)) → π1(dual(Jn)) for
which rank (�̂�(π1(Jn))) = 0.

Proof. (i) Consider the folding𭟋 : Jn → Jn such that𭟋(Jn) = Jm, m ≤ n−2, n ≥ 3, as
in Figure 1, for n = 5, which induces �̂� : π1(Jn) → π1(Jn) for which rank (�̂�(π1(Jn))) =
rank (π1(Jn−2)) = n− 1.

Figure 1:

(ii) Let 𭟋 : dual(Jn) → dual(Jn) be a folding such that 𭟋(dual(Jn)) = 𭟋( Pn+2) =
Pm and m ≤ n + 2 as in Figure 2, for n = 5, which induces �̂� : π1(dual(Jn)) →
π1(dual(Jn)) for which rank ( �̂�(π1(Jn))) = 0.

Theorem 3. Let J∞ be an infinite cobblestone path. Then, there are two types of foldings
𭟋 : J∞ → J∞ induce �̂� : π1(J∞ ) → π1(J∞) for which rank (�̂�(π1(J∞))) ≤ 1.



M. Abu-Saleem / Eur. J. Pure Appl. Math, 18 (3) (2025), 6611 4 of 7

Figure 2:

Proof. Let J∞ be an infinite cobblestone path, and consider 𭟋 : J∞ → J∞ to be a
folding such that 𭟋(J∞) = C2 as in Figure 3(a), which induces �̂� : π1(J∞ ) → π1(J∞) in
which rank (�̂�(π1(J∞))) = 1. Furthermore, let 𭟋 : J∞ → J∞ be a folding such that
𭟋(J∞) = I1 ⋎ p∞ ⋎ I2 , where I1 and I2 are arcs homeomorphic to the closed interval
[a, b] , as shown in Figure 3(b), which induces �̂� : π1(J∞ ) → π1(J∞) for which rank
(�̂�(π1(J∞))) = 0.

Figure 3:



M. Abu-Saleem / Eur. J. Pure Appl. Math, 18 (3) (2025), 6611 5 of 7

Theorem 4. Given an m-bouquet graph Bm, then the sequence of folding maps {𭟋i :
Bmi−1 → Bmi : i = 1, 2, . . . n} induces {�̂�i : π1

(
Bmi−1

)
→ π1(Bmi) : i = 1, 2, . . . n} such

that rank ( lim
n→∞

(�̂�n(π1 (Bmn−1)))) = 0.

Proof. Given an m-bouquet graph Bm .
Now, consider the following sequence of folding maps:
𭟋1 : Bm0 → Bm1

, 𭟋2 : Bm1
→ Bm2

, . . . , 𭟋n : Bmn−1 → Bmn, for which lim
n→∞

𭟋n(Bmn−1) =

v (one vertex) as in Figure 4, which induces
�̂�1 : π1 (Bm0) → π1 (Bm1

), �̂�2 : π1 (B1) → π1 (B2), . . . , �̂�n : π1

(
Bmn−1

)
→ π1(Bmn),

for which
lim
n→∞

(�̂�n(π1 (Bmn−1))) = π1( lim
n→∞

(𭟋n (Bmn−1))) = π1 (v).

Hence, rank ( lim
n→∞

(�̂�n(π1 (Bmn−1)))) = 0.

Figure 4:

Theorem 5. Given a connected graph Gk that is homeomorphic to k-bouquet graph Bk for k =

1, 2, . . . , n, then for any h, there is a folding 𭟋h :
n
⋎
k=1

Gk −→
n
⋎
k=1

Gk which induces a

folding �̂�h :
n∗

k=1
π1(Gk) → n∗

k=1
π1(Gk) such that �̂�h(

n∗
k=1

π1(Gk)) is a free group of rank

m− h, for h = 1, 2, . . . , n where, m = n(n+1)
2 .

Proof. Let 𭟋1 :
n
⋎
k=1

Gk −→
n
⋎
k=1

Gk be a folding such that 𭟋1(
n
⋎
k=1

Gk) = G1 ⋎ G2 ⋎

. . . ⋎ 𭟋1(Gr1) ⋎ . . . ⋎ Gn for r1 = 1, 2, . . . , n where 𭟋1(Gt) = 𭟋1(Bt) = Bt−1, folding

one loop into another, then we obtain the induced folding �̂�1 :
n∗

k=1
π1(Gk) −→

n∗
k=1

π1(Gk)

such that �̂�1(
n∗

k=1
π1(Gk)) = π1(𭟋1(

n
⋎
i=1

Gk)) and so �̂�1(
n∗

k=1
π1(Gi) ) ≈ π1(G1) ∗ π1(G2) ∗ · · · ∗

π1(𭟋1(Gr1
))∗· · ·∗π1(Gn). Since π1(𭟋1(Gr1

)) = π1(Br1−1), it follows that rank (π1(𭟋1(Gr1
))) =rank

(π1(Gr1
))− 1. Hence, rank (�̂�1(

n∗
i=1

π1(Gi) )) = n− 1. Moreover, let 𭟋2 :
n
⋎
k=1

Gi −→
n
⋎
k=1

Gi be

folding such that 𭟋2(
n
⋎
k=1

Gk) = G1⋎G2⋎ . . .⋎𭟋2(Gr1
)⋎ . . .⋎𭟋2(Gr2

)⋎ . . .⋎Gn for r1, r2 =

1, 2, . . . , n, r1 ≺ r2 , and 𭟋2(Gr1) = 𭟋2(Br1) = Br1−1, 𭟋2(Gr2) = 𭟋2(Br2) = Br2−1, then



M. Abu-Saleem / Eur. J. Pure Appl. Math, 18 (3) (2025), 6611 6 of 7

we get the induced folding �̂�2 :
n∗

k=1
π1(Gk) −→

n∗
k=1

π1(Gk) such that rank (�̂�2(
n∗

k=1
π1(Gk)) =

n− 2. We follow this approach, we have 𭟋n :
n
⋎
k=1

Gk −→
n
⋎
k=1

Gk such that

𭟋n(
n
⋎
k=1

Gk) =
n
⋎
k=1

𭟋n(Gk) and 𭟋n(Gk) = 𭟋n(Bk) = Bk−1, ∀k = 1, 2, . . . , n, which induces

a folding �̂�n :
n∗

k=1
π1(Gk) −→ n∗

k=1
π1(Gk) such that �̂�n(

n∗
k=1

π1(Gk)) is a free group of rank

m− n. Consequently, rank (�̂�h(
n∗

k=1
π1(Gk) )) = m− h, for h = 1, 2, . . . , n.

Theorem 6. Assume that G is a graph and H is a subgraph of it. For i = 1, 2, . . . n, let 𭟋i

and ri be chains of folding and retraction functions, respectively. Then, there is a chain
of a commutative diagram of a graph that induces a chain of a commutative diagram of
fundamental groups.

Proof. Given a commutative diagram:

G 𭟋1−−−→ G1
𭟋2−−−−−→ G2 −−−

lim
m→∞

𭟋m

−−−−−−−−→ only one vertex or one edge
↓ r1 ↓ r2 ↓ r3 ↓ lim

m→∞
rm

H 𭟋1−−−−−→ H 1
𭟋2−−−−→ H2 −−−

lim
m→∞

𭟋m

−−−−−−−−−→ only one vertex or one edge.
As the fundamental group constitutes a functor, we get the following chain of a commuta-
tive diagram of fundamental groups:

π1 (G) �̂�1−−−−−→ π1 (G1 )
�̂�1−−−−−−→ π1 (G2) −−−

lim
m→∞

�̂�m

−−−−−−−−→ {0}
↓ r̂1 ↓ r̂2 ↓ r̂3 ↓ lim

m→∞
r̂m

π1 (H)
�̂�1−−−−−→ π1 (H1)

�̂�2−−−−−−→ π1 (H2) −−−
lim

m→∞
�̂�m

−−−−−−−−→ {0}.

3. Conclusion

In topological graph theory, the effect of folding on special types of graphs and their
duals is introduced. Also, the folding induced by the fundamental groups is obtained. The
limits of folding on the induced fundamental group are presented. The relations between
the induced folding and the induced retraction on the fundamental group are deduced.

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