EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6372
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Planarity of Intuitionistic Pythagorean Fuzzy Graphs
and Its Application in Decision Making

Obbu Ramesh1, Nainaru Tarakaramu2,∗, G. YuvaRoopa Lakshmi3,
Sharief Basha S4, Sarvar Iskandarov5, Akbar Toyirov6, Jushkinbek Yuldoshev7,
Sardor Sabirov8, K. K. Prashanth9

1 Department of Mathematics, Presidency University, Bengaluru-560064, Karnataka, India
2 Department of Mathematics, School of Liberal Arts and Sciences, Mohan Babu University,
Sree Sainath Nagar, Tirupati-517102, Andhra Pradesh, India
3 Department of Mathematics, Govt. Degree College for Women, Wanaparthy (Nandi Hills),
509103, Telangana, India
4 Department of Analytics, School of Computer Science and Engineering, Vellore Institute
of Technology, Vellore 632014, Tamil Nadu, India
5 Department of Mathematical Analyze, Urgench State University, Urgench City, Uzbekistan
6 Department of Information Technology and Exact Sciences, Termez University of
Economics and Service, Termez, Uzbekistan
7 Department of Pedagogy and Primary Education Methodology, Urgench Innovation Uni-
versity, Urgench, Uzbekistan
8 Department of General Professional Sciences, Mamun University, Khiva, Uzbekistan
9 Department of Mathematics, New Horizon College of Engineering, Bangalore-560103,
Karnataka, India

Abstract. This paper introduces the concept of Intuitionistic Pythagorean Fuzzy Graphs (IPFGs)
and outlines their key characteristics as a means of addressing such uncertain scenarios by exam-
ining Pythagorean uncertain planarity values using weak, strong, and significant edges. An Intu-
itionistic Pythagorean Fuzzy Graph is a generalization of a traditional graph that can solve certain
problems beyond the capabilities of both classical graph theory and fuzzy graph theory. The ap-
proach based on Pythagorean fuzzy graphs offers greater flexibility in handling human judgment
data compared to other fuzzy models.
This paper defines the concept of planarity in an Intuitionistic Pythagorean Fuzzy Graph using the
notions of intersecting value and Intuitionistic Pythagorean Fuzzy Planarity Value. The upper and
lower bounds of this planarity value are then established through a series of interrelated theorems.
Finally, the weak and strong planarity of an Intuitionistic Pythagorean Fuzzy Graph is delineated,
and the results of the analysis are discussed. The paper poses a challenge that demonstrates the
application of the proposed concept.
2020 Mathematics Subject Classifications: 05C10, 03E72, 68T37
Key Words and Phrases: Intuitionistic Pythagorean fuzzy graph, Crossing point, Strong Pla-
narity and Weak Planarity

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6372

Email addresses: nainaru143@gmail.com (N. Tarakaramu)

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



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 2 of 17

Nomenclature

PFPGs - Pythagorean Fuzzy Planar Graphs

IPFGs - Intuitionistic Pythagorean Fuzzy Graphs

IFGs - Intuitionistic Fuzzy Graphs

IFSs - Intuitionistic Fuzzy Sets

IFPR - Intuitionistic Fuzzy Preference Relations

PFDGs - Pythagorean Fuzzy Dual Graphs

ζrs - The strength of the Pythagorean fuzzy edge rs

ζP - The crossing rate at the point

ξ - Intuitionistic Pythagorean uncertain planarity rate

1. Introduction

The concept of graphs is recognized as one of the branches of mathematics that has un-
dergone tremendous development in recent years due to its numerous applications. Because
of its applications in a wide range of fields, including physics, electrical engineering, biol-
ogy, and operations research, graph theory is rapidly becoming central to mathematics.[1]
Euler’s polyhedral formula, which relates the faces, edges, and vertices of a polyhedron,
serves as the foundation for the theory of planar graphs.[2] New concept of Planar graphs
are used to clearly layout and shape complicated radio electronic circuits, planetary gear-
boxes, chemical molecules and railway maps inside the modern-day technology. Subway
tunnels, railway lines, Pipelines, metro traces and electric powered transmission traces are
vital while modelling an urban town. Crossing is advantageous because it saves space and
is inexpensive, but it also has some disadvantages. Although crossing such lines poses a
significant threat to human life, it is possible with the right precautions. When compared
to crossing between two crowded routes, the crossing between the uncrowded route and
crowded route poses fewer dangers. The relations ”uncrowded route” and ”crowded route”
are mentioned to as ”weak edge” and ”strong edge” in fuzzy graphs, respectively. Fuzzy
planar graph theory is made possible by allowing for crossings of this kind [3].
However, the modern fuzzy set concept began with Zadeh’s 1965 definition of the fuzzy
subset of a crisp set as a generalization of the standard subset [4]. Allowing to the tradi-
tional idea, any component has whichever a membership degree of one if it is related to
the subset being studied or a membership degree of zero if it is not. However, the fuzzy
idea states that any element with association degree among zero and one is a member
of the studied fuzzy subset, which is nearer to human thought due to the fact that our
thoughts work with the estimated type and do not know everything exactly. This can, for
instance, recognize the gray gradients of the white and black colours. The indistinct idea



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 3 of 17

has involved a lot of attention due to its numerous presentations in industries, communica-
tions, and artificial intelligence, despite the fact that it is considered a new mathematical
field. Consequently, numerous scientists and researchers have focused on improving this
concept and implementing it across all math disciplines. One of these fields where the
fuzzy idea can be put into practice is graph theory.[5] Rosenfeld first proposed this theory
in 1975 when he studied fuzzy relations on fuzzy sets and established the structure of
fuzzy graphs by extending several graph-theoretical concepts through analogous conclu-
sions. He also discussed concepts such as cut vertices, trees, forests, paths, connectedness,
and bridges. Additionally, Mordeson introduced the concepts of fuzzy diagrams and fuzzy
hyper diagrams.[6] Rakesh Jaiswal explored the application of fuzzy graph coloring to the
traffic light problem in another study.[7] Lavanya also examined the concept of fuzzy total
coloring and its application in job scheduling. [8] Eslahchi discovered the vertex strength
of fuzzy graphs in a different study. Deep structure was added to fuzzy graph theory in re-
cent years that are a study on fuzzy labelling graphs [9], a graph associated to a poly group
with respect to an automorphism [10], Sustainable goals in combating human trafficking
[11], Fuzzy graph theory with applications to human trafficking [12], Fuzzy graph theory
[13] and Product of bipolar fuzzy graphs and their degree [14]. Then referred about intu-
itionistic fuzzy graphs, their operations [15], and their Signless laplacian energies [16].Naz
et al. [17] based on Akram and Davvaz’s IFGs [18] proposed PFGs and their applications.
In, some PFG-related findings were discussed [19]. Naz and Akram conducted research
on the Pythagorean fuzzy graph energy [20]. IFGs2k was dealt with by Dhavudh and
Srinivasan [21]. Verma and co. and Akram and others [22] suggested a few PFG opera-
tions. In recent times, Akram et al. [23] introduced a few graphs in an environment that
was Pythagorean fuzzy. Sahoo et al did their research on intuitionistic fuzzy competition
graphs [24], Product of intuitionistic fuzzy graphs, Colouring of Mixed Fuzzy Graph [25],
Certain Types of Edge Irregular Intuitionistic Fuzzy Graphs [26] and Colouring of Mixed
Fuzzy Graph and its Application in COVID19 [27]. Shaik Noorjahan and Sharief Basha
Shaik proposed using correlation coefficient measurements between each option and the
optimal choice to calculate the ranking order of the alternatives in the study [28]. Shaik
Noorjahan and Shaik Sharief Basha investigated the use of the intuitionistic fuzzy rough
preference relation method for processing the intuitionistic fuzzy rough weighted average
[29]. Naveen Kumar Akula and Shaik Sharief Basha proposed a novel method for calcu-
lating the comparative position loads of establishments by manipulating the undecided
corroboration of IFPR and the correlation coefficient between one IFPR and other items
[30]. Noorjahan S and Sharief Basha S proposed a novel method for calculating the relative
position loads of establishments by varying the correlation coefficient between the intu-
itionistic fuzzy rough preference relation of one entity and the other items, as well as the
undecided corroboration within the intuitionistic fuzzy rough preference relation [31]. [32]
Areej Almuhaimeed presents the concepts of a normal HX-subgroup and a Pythagorean
fuzzy HX-Subgroup. [33] Using the Pythagorean fuzzy soft interior operator, A.A. Azzam
and M. Aldawood present the idea of Pythagorean fuzzy soft somewhat open sets and
expand its use to Pythagorean fuzzy soft topological spaces. This paper extends graph
theoretical results within the intuitionistic Pythagorean fuzzy environment. The structure



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 4 of 17

and applicability of planar graphs are fascinating. For instance, when designing complex
radio electronic circuits, components can be arranged so that the conductors connecting
them do not cross, and the concept of planar graphs can be applied to address this issue.
The precise structuring of a street or communication network is made possible through the
concepts of intuitionistic Pythagorean fuzzy graphs (IPFGs), Pythagorean fuzzy planar
graphs (PFPGs), and Pythagorean fuzzy dual graphs (PFDGs), which are discussed in
this study. By utilizing these diagrams, some real-world problems can be analysed and
designed effectively. Planarity is a significant property that is central to this research.
Non-planar IPFGs are also critically analysed. Intuitionistic Pythagorean fuzzy planar
graphs and Pythagorean fuzzy dual graphs are found to be closely related. By introduc-
ing the concepts of crossing rate and intuitionistic Pythagorean fuzzy planarity rate, this
paper defines the terms intuitionistic Pythagorean fuzzy planar graphs and planarity of
intuitionistic Pythagorean fuzzy planar graphs, and examines related results. Applications
of IPFPGs are also discussed.

2. Preliminaries

This section provides a brief discussion of intuitionistic fuzzy sets, intuitionistic fuzzy
diagrams, Pythagorean fuzzy sets, and graph theory.

2.1. Intuitionistic fuzzy graph

Sets whose elements have varying degrees of membership and non-membership are
known as intuitionistic fuzzy sets. The concept of intuitionistic fuzzy sets was introduced
by Krassimir Atanassov in 1983 as an extension of Lotfi Zadeh’s fuzzy set theory, which
itself extends the classical notion of a set.

Definition 1 ([34]). Let us take X is a non-empty set. An IFS A is defined in X is
A = {x, µA(x), γA(x) : x ∈ X}, where µA : X → [0, 1] and γA : X → [0, 1] represents the
membership and non-membership functions of A and it satisfies the condition 0 ≤ µA(x)+
γA(x) ≤ 1,∀x ∈ X with the degree of indeterminacy given by πA(x) = 1− µA(x)− γA(x)

2.2. Intuitionistic Fuzzy graph

Intuitionistic fuzzy relations and intuitionistic fuzzy graphs (IFGs) were first intro-
duced by Atanassov. The field of mathematics and its applications has seen exponential
growth in research on the theory of intuitionistic fuzzy sets (IFSs). This includes devel-
opments in information sciences as well as conventional mathematics.

Definition 2 ([35]). An IFG is well-defined as G = (V,E, µ, γ),where V denotes the set
of vertices, E represents the set of edges, µ is a fuzzy relationship function demarcated on
V ×V , and γ is a fuzzy non-relationship function expressed as µ(vi, vj) by µij and γ(vi, vj)
by γij , such that

(i) 0 ≤ µij + γij ≤ 1, (i, j = 1, 2, 3, . . . , n)



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 5 of 17

(ii) 0 ≤ µij , γij , πij ≤ 1, here πij = 1− µij − γij , (i, j = 1, 2, 3, . . . , n)

2.3. Pythagorean Fuzzy Set

Definition 3 ([36]). A definition of the Pythagorean fuzzy set R in a finite discourse
universe S = (s1, s2, . . . , sn) is given by

R = (x, µR(s), γR(s)/s ∈ S),

Where µR(s) : S → [0, 1] and γR(s) :→ [0, 1] and 0 ≤ µ2
R(s) + γ2R(s) ≤ 1, ∀s ∈ S, where

the numbers µR(s) and γR(s) represent the ranking of membership and the ranking of
non-membership, respectively, of s ∈ S in R.

Additionally, the hesitation degree, also known as the Pythagorean index, can be defined

as for each Pythagorean fuzzy set R in S ηR(s) =
√
(1− µ2

R(s)− γ2R(s))

Definition 4 ([20]). A PFG on a non-empty set X is a pair G= (P, Q), where P is a
PFS on X and Q is a PFR on X such that µR(u, v) ≤ min(µR(u), µR(v)) and γR(u, v) ≤
max(γR(u), γR(v)), and 0 ≤ µ2

R(u, v) + γ2R(u, v) ≤ 1, ∀(u, v) ∈ X.
A Pythagorean fuzzy graph (PFG) specifies the degree of membership of an element

in a given set, where the degree of non-membership is simply one minus the degree of
membership. By contrast, an intuitionistic Pythagorean fuzzy graph (IPFG) allows for both
a degree of membership and a degree of non-membership, which are largely independent;
the only condition is that their sum does not exceed one.

Pythagorean fuzzy graphs, as a new extension of Euler’s graph theory, offer an im-
proved way to represent complex graphical problems. Unlike intuitionistic Pythagorean
fuzzy graphs, which require that the sum of the membership and non-membership degrees
for any edge or vertex does not exceed one, this approach provides additional advantages.
[37] R. R. Yager provided a comparison between the fuzzy set and the intuitionistic fuzzy
set through Figure 1.

Figure 1: Comparison between IFS and PFS



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 6 of 17

3. Planarity of Intuitionistic Pythagorean Fuzzy Graphs

In this segment, the concept of planarity of intuitionistic Pythagorean fuzzy diagrams
is outlined, based on several crucial interrelated classifications such as crossing rate and
Pythagorean fuzzy planarity rate. Let G = (M,N) be an intuitionistic Pythagorean fuzzy
graph and for an assured geometric representation and it takes dual edges((rs), µN ((rs))i, γN ((rs))i)
and ((uv), µN ((uv))j , γN ((uv))j) which intersect at a particular point P, wherever i,j are
permanent integers. By fuzzy logic, if one at smallest of µN (rs)i and µN (uv)j is nearby to
0 in that case, the crossing won’t matter much in the representation. But then if together
of µN (rs)i and µN (uv)j are closed to 1, in that case, the crossing is matter much in the
illustration.

Based on this concept, the crossover rate and the intuitionistic Pythagorean fuzzy
planarity rate are explained.

Definition 5. The strength of the Pythagorean fuzzy edge rs is defined as

ζrs = (µrs, γrs) = (
µN (rs)i

µM (r) ∧ µM (s)
,

γN (rs)i
γM (r) ∧ γM (s)

)

Definition 6. Let G= (M,N) be an intuitionistic Pythagorean fuzzy diagram through a
convinced symmetrical picture which takes the crossing argument ‘P’ among the intuitionis-
tic Pythagorean an edge rs of IPFG is known as Strong if µrs ≥ 0.5 and γrs ≤ 0.5 otherwise
known as weak.uncertain edges ((rs), µN (rs)i, γN (rs)i) and ((uv), µN (uv)j , γN (uv)j). The
crossing rate at the point is then defined as

ζp = (µp, γp) = (
µrs + µuv

2
,
γrs + γuv

2
)

If the total of crossing arguments in an intuitionistic Pythagorean uncertain graph upsurge,
then the planarity value of IPFG declines. Hence the rate of ζP upsurges, then the planarity
value declines. So that ζP is proportionate inversely to planarity of IPFG. What defines
an intuitionistic Pythagorean fuzzy graph is the knowledge of its planarity value.

Definition 7. Let G= (M, N) be an intuitionistic Pythagorean fuzzy graph with a sure
geometric picture which have the crossing points P1, P2, . . . , Pn and now express the intu-
itionistic Pythagorean uncertain planarity rate ξ of this symmetrical representation of ‘G’
by

ζ = (ζµ, ζγ) =

(
1

1 + µξ1 + µξ2 +−−+µξn

,
1

1 + γξ1 + γξ2 +−−+γξn

)
Every symmetrical representation of an intuitionistic Pythagorean uncertain graph is pla-
nar with a specific planarity rate, because it is obvious that 0 ≤ ξ ≤ 1.

Example 1. Let G= (M, N) be an intuitionistic Pythagorean fuzzy graph

In the above Figure 2, there two crossing point ’P1 and p2’.
The crossing value at the point P1 is

ζp1 = (
0.7 + 0.4

2
,
0.3 + 0.3

2
) = (0.55, 0.3)



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Figure 2: IPFG

and the crossing value at the point p2 is :

ζp2 = (
0.6 + 0.8

2
,
0.4 + 0.1

2
) = (0.7, 0.25)

so, the inutionistic Pythagorean uncertain planarity rate of this geometric representation
of G

ξ = (ξµ, ξγ) = (
1

1 + 0.55 + 0.7
,

1

1 + 0.3 + 0.25
)

ξ = (0.44, 0.65)

Theorem 1. For any intuitionistic Pythagorean fuzzy graph with geometric representation
which has ‘k’ and ‘l’ crossing points of membership and non-membership standards are:(

1

1 + k
≤ ξµ ≤ 1

1 + kµ
,

1

1 + l
≤ ξ ≤ 1

1 + lΩ

)
where η = min{µσ(r) : r ∈ V } and Ω = min{γσ(r) : r ∈ V }

Proof. by the definition 6, we have

ξ = (ξµ, ξγ) =

(
1

1 + µζ1 + µζ2 + · · ·+ µζk

,
1

1 + γζ1 + γζ2 + · · ·+ γζl

)
and definition 5, then

ζP = (µP , γP ) =

(
µrs + µuv

2
,
γrs + γuv

2

)
Since (µrs, γrs) ≤ (1, 1), for any intuitionistic Pythagorean fuzzy graph edge ((rs), µN (rs)i, γN (rs)i)
in G, then

ζP (i,j) ≤
(
1 + 1

2
,
1 + 1

2

)
= (1, 1)

For {i = 1, 2, . . . , k} and {j = 1, 2, . . . , 1}, given by

(1 + µζ1 + µζ2 + · · ·+ µζk , 1 + γζ1 + γζ2 + · · ·+ γζl) ≤ (k, l)



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 8 of 17

So,

ξ =

(
1

1 + µζ1 + µζ2 + · · ·+ µζk

,
1

1 + γζ1 + γζ2 + · · ·+ γζl

)
≥

(
1

1 + k
,

1

1 + l

)
On the otherhand, (µrg, γrs) ≥ σ(r) ∧ σ(s), for any intuitionistic Pythagorean fuzzy two
edges in G.

((rs), µN ((rs))i, γN ((rs))i) and ((uv), µN ((uv))j , γN ((uv))j)

Since η = min
{
µσ(r) : r ∈ V, γσ(r) : r ∈ V

}
, then (µrs, γrs) ≥ (η,Ω)

ζP (i,j) ≥
(
η + η

2
,
Ω+ Ω

2

)
= (η,Ω)

It has µζ1 + µζ2 + · · ·+ µζn ≥ kη and γζ1 + γζ2 + · · ·+ γζl ≥ lΩ Thus, for {i = 1, 2, . . . , k}
and {j = 1, 2, . . . , 1}, given by

ξ =

(
1

1 + µζ1 + µζ2 + · · ·+ µζk

,
1

1 + γζ1 + γζ2 + · · ·+ γζl

)
≤

(
1

1 + kη
,

1

1 + lΩ

)

Theorem 2. For any intuitionistic Pythagorean fuzzy graph G with symmetrical repre-
sentation which takes (k, l) crossing points and which are edges steady.

ξ =

(
1

1 + kα
,

1

1 + lβ

)
Where α = µrs and β = γrs, for any intuitionistic Pythagorean fuzzy two edges

((rs), µN (rs)i, γN (rs)i) and ((uv), µN (uv)j , γN (uv)j) in G.

Proof. According to definition 5, we have

ζP = (µP , γP ) =

(
µrs + µuv

2
,
γrs + γuv

2

)
Since G is edge stable, then µrs = µuv = α,γrs = γuv = β, for any intuitionistic
Pythagorean fuzzy two edges ((rs), µN (rs)i, γN (rs)i and ((uv), µN (uv)i, γN (uv)i in G

ζP (i,j) =

(
α+ α

2
,
β + β

2

)
= (α, β)

For {i = 1, 2, . . . , k} and j = {j = 0, 1, 2, . . . , 1}, given by

ξ =

(
1

1 + µζ1 + µζ2 + · · ·+ µζk

,
1

1 + γζ1 + γζ2 + · · ·+ γζl

)

ξ =

(
1

1 + kα
,

1

1 + lβ

)



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Theorem 3. For any intuitionistic Pythagorean fuzzy graph G with geometric represen-
tation which has (k, l) crossing points, all of them in between the Pythagorean fuzzy edges,
which are ineffective.

ξ >

(
1

1 + kα
,

1

1 + lβ

)
Where α = maxi{µσ(r) : r ∈ V } and also β = maxi{γσ(r) : r ∈ V }

Proof. According to the defintion 2, we have

ζP = (µP , γP ) =

(
µrs + µuv

2
,
γrs + γuv

2

)
On account that each n crossing arguments in G are among non-effective Pythagorean
uncertain edges, then

µrs < max{σ(r), σ(s)}, µuv < max{σ(u), σ(v)}

And γrs < max{σ(r), σ(s)}, γuv < max{σ(u), σ(v)}

MEanwhile α = maxi{µσ(r) : x ∈ V } and also β = maxi{γσ(r) : x ∈ V }

ζP = (µP , γP ) =

(
µrs + µuv

2
,
γrs + γuv

2

)

ζP <

(
α+ α

2
,
β + β

2

)
= (α, β)

For {i = 1, 2, . . . , k} and j = {j = 0, 1, 2, . . . , 1}, given by

ξ =

(
1

1 + µζ1 + µζ2 + · · ·+ µζk

,
1

1 + γζ1 + γζ2 + · · ·+ γζl

)

ξ >

(
1

1 + kα
,

1

1 + lβ

)

Definition 8. Let G = (M, N) be an intuitionistic Pythagorean fuzzy graph with a sure
geometric picture which have the crossing points P1, P2, . . . , Pn so that m of these crossing
points gratifies that ζP = (µP , γP ) ≥ 0.5, for i = (0, 1, 2, . . . , k) and j = (0, 1, 2, . . . , l) then
said geometric picture of G intuitionistic Pythagorean fuzzy diagram if

ξ = (ξµ, ξγ) > (0.5, 0.5) & m <
n

2

This indicates that an intuitionistic Pythagorean fuzzy graph is the appropriate geometric
representation of G if the crossing points that satisfy zetaP ≥ 0.5 are fewer than half of
the entire number of crossing arguments n and the intuitionistic Pythagorean fuzzy graph
planarity rate is greater than 0.5. In this case, this symmetrical representation of G is
referred to as an intuitionistic Pythagorean uncertain weak planar diagram.



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Example 2. Let G = (M, N) be an intuitionistic Pythagorean fuzzy diagram denoted in
a symmetrical representation presented in Figure 3 below

Figure 3: IPFG

In the figure 3, there two crossing point ’P1 and p2’and Their crossing values are given
below :

ζp1 = (
0.4 + 0.3

2
,
0.4 + 0.1

2
) = (0.35, 0.25)

and ζp2 = (
0.8 + 0.7

2
,
0.2 + 0.2

2
) = (0.75, 0.2)

so, the inutionistic Pythagorean uncertain planarity rate of this geometric representation
of G

ξ = (ξµ, ξγ) = (
1

1 + 0.35 + 0.75
,

1

1 + 0.25 + 0.2
)

ξ = (0.48, 0.69)

Therefore, the weak planar from the geometric representation GI based on membership
planarity value and non-membership planarity value. Already, if ξγ(= 0.69) > 0.5, so the
geometric illustration of G is weak planar since the formula m < n

2 is not satisfied.

Theorem 4. If G be an intuitionistic Pythagorean fuzzy graph with geometric illustration
which have n crossing points satisfy that ζPi < 0.5, for all values of i = 1, 2, . . . , n, then

ξ = (ξµ, ξγ) >

(
2

2 + n
,

2

2 + n

)
Proof. Meanwhile ζpi < 0.5 for i = 1, 2, . . . , n

It is given by 1 + µζ1 + µζ2 + · · ·+ µζn < (0.5)n and 1 + γζ1 + γζ2 + · · ·+ γζn < (0.5)n

ξ =

(
1

1 + µζ1 + µζ2 + · · ·+ µζn

,
1

1 + γζ1 + γζ2 + · · ·+ γζn

)

ξ = (ξµ, ξγ) >

(
1

1 + 0.5n
,

1

1 + 0.5n

)



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Multiplying and dividing by ’2’ on the right hand side, we get

ξ = (ξµ, ξγ) >

(
1

1 + n
2

,
1

1 + n
2

)

Corollary 1. If G be an intuitionistic Pythagorean fuzzy graph with geometric illustration
which have 1 or 2 crossing points gives that ζPi < 0.5 for all crossing points, then this
geometric illustration of G is an intuitionistic Pythagorean fuzzy strong planar graph.

Proof. Redering to Theorem 4, then

ξ = (ξµ, ξγ) >

(
2

2 + n
,

2

2 + n

)
For n=1,

ξ = (ξµ, ξγ) >

(
2

2 + 1
,

2

2 + 1

)
ξ = (ξµ, ξγ) >

(
2

3
,
2

3

)
ξ = (ξµ, ξγ) = (0.66, 0.66)

For n=2,

ξ = (ξµ, ξγ) >

(
2

2 + 2
,

2

2 + 2

)
ξ = (ξµ, ξγ) >

(
2

4
,
2

4

)
ξ = (ξµ, ξγ) = (0.5, 0.5)

So, this geometric representation of G is an intuitionistic Pythagorean fuzzy strong planar
graph.

4. Application

Water from the river reaches people’s homes through sturdy pipelines that twist, turn,
and cross various regions within cities or village streets, carrying water under high pres-
sure. To reduce this high-pressure water force, a device known as a water tank is used.
Dispensing water from the tank is straightfor-ward, as it connects to different houses
through smaller pipes. However, when connecting houses in different areas, crossings be-
tween these small pipes may occur. Sometimes, such crossings are benefi-cial because
they save space and reduce costs. However, pipe crossings can also cause problems, as
they may lead to pipe breakage and water leakage, resulting in water loss and failure to
deliver water to its intended destination. To overcome this issue, crossings between pipes



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 12 of 17

should be minimized, and high-quality pipes should be used for installation. The real-
world methodology of intuitionistic Pythag-orean fuzzy graphs can be applied to model
and manage such situations, helping to reduce the risk of damage and associated costs.

Take into account a water tank wherein unique regions are related as proven in figure.
Every region v1, v2,. . . , v8 is signified by using a vertex and every pipe connection be-
tween exclusive areas through small pipe is signified by edges. The membership score of
a vertex characterizes the likelihood of a breakdown, whereas the non-membership score
suggests the probability of no breakdown occurring in the connections between various
areas. Similarly, the mem-bership score of an edge indicates the degree of risk associated
with a pipe connection between areas, while the non-membership score denotes the absence
of such a connection risk.

Figure 4: IPFG

Because the quantity of crossings increases, the charge of destruction will increase.
Therefore, the size of the planarity cost is vital. In the figure 4, there are eight crossings
points’ P1, P2, P3, P4, P5, P6, P7 and P8 between pair of pipe connections (v1v4, v2v3), (v1v4, v2v5), (v1v4, v2v6), (v2v5, v3v4), (v2v6, v3v4), (v2v6, v4v5), (v3v6, v4v5)
and (v2v5, v3v6) espectively.
The Strength of pipes are

v1v4 =

(
0.6

0.3
,
0.3

0.6

)
= (2, 0.5); v2v3 =

(
0.4

0.4
,
0.6

0.2

)
= (1, 2)

v2v5 =

(
0.8

0.1
,
0.2

0.6

)
= (8, 0.33); v2v6 =

(
0.2

0.7
,
0.6

0.2

)
= (0.28, 3)

v3v4 =

(
0.1

0.4
,
0.3

0.2

)
= (0.25, 1.5); v4v5 =

(
0.4

0.1
,
0.5

0.6

)
= (4, 0.83)

v3v6 =

(
0.7

0.4
,
0.2

0.2

)
= (1.75, 1)

Thus, the intuitionistic Pythagorean uncertain planarity rate is ξ = (0.055, 0.098)
Because the planarity rate is at a lowest, it designates the opportunity of excessive damage,
to reduce the planarity, it can trade the graphical illustration as proven in the under
determine.



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 13 of 17

Figure 5: IPFG

Hence that the number of crossover factors is inversely proportional to planarity. Mean-
while the number of crossing factors reduction. In figure 5, there is only one crossings point
P1 between pair of pipe connections (v3v6, v4v5) respectively.
The Strength of pipes are

v4v5 =

(
0.4

0.1
,
0.5

0.6

)
= (4, 0.83)

v3v6 =

(
0.7

0.4
,
0.2

0.2

)
= (1.75, 1)

Thus, the Pythagorean uncertain planarity rate is ξ = (0.15, 0.35) increase and score of
ruin reductions. In addition, the illustration in the preceding figure 5 reveals that P1 is the
only crossing left that cannot be decreased; however, the likelihood of a pipe connection
and the rate of destruction caused by it can be reduced by using high-quality pipes between
v1 and v2, v4 and v6. For that reason, the crossing could be less dangerous. As a result,
finish that Pythagorean uncertain linking model may be used for tracing and identifying
the rate of damage. Through probing and taking greater unique security procedures,
the share of damage may be decreased and plenty of human problems can be minimized.

Methodology:Reducing crossings and maintaining planarity between the connections of
the wate tank pipes.

INPUT: A discrete set of units U = {U1, U2, U3, . . . , Un}, a set of water tank connections
E = {E1, E2, E3, . . . , Em} between the pipes and a set of point of intersections C =
{C1, C2, C3, . . . , Cn}.
OUTPUT: Minimized crossing and increased planarity value.

1. Begin

2. Compute the strength of the edge Ei, where i = 1, 2, . . . , n by using the formula

ζrs = (µrs, γrs) =

(
µN (rs)j

µM (r) ∧ µM (s)
,

γN (rs)j
γM (r) ∧ γM (s)

)



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 14 of 17

3. 3. Determine the intersecting point between edges by using the formula

ζP = (µP , γP ) =

(
µrs + µuv

2
,
γrs + γuv

2

)
4. 4. Find the intuitionistic Pythagorean fuzzy planarity value defined as

ξ = (ξµ, ξγ) =

(
1

1 + µζ1 + µζ2 + · · ·+ µζn

,
1

1 + γζ1 + γζ2 + · · ·+ γζn

)
5. Keep the graphical representation of the edges Ejand Ek, if ξµ > 0.5 and ζγ < 0.86

otherwise change the graphical representation.

6. If there are no new crossings in the graphical representation of the edges Ej and Ek,
then modify it,if not, maintain the prior depiction.

7. The crossing and planarity values will be reduced and raised, respectively, by altering
the graphical depiction of the edges Ej and Ek.

8. end

5. Comparative Analysis

Pythagorean fuzzy graphs, a widely used extension of fuzzy and intuitionistic fuzzy
graphs, effectively represent structural relationships among multiple objects when the
connections between them are uncertain.

Therefore, the method developed in this article is more comprehensive than existing
approaches based on fuzzy and intuitionistic fuzzy graphs. The results clearly demonstrate
that the proposed technique, which utilizes the intersection points between edges and the
intuitionistic Pythagorean fuzzy planarity value, provides a more refined mechanism for
handling higher levels of uncertainty and hesitation.

Pythagorean Fuzzy Graphs (PFGs) represent a more generic model capable of man-
aging a wider range of uncertainty due to the Pythagorean property. Although both
Intuitionistic Fuzzy Graphs (IFGs) and PFGs are employed to model uncertainty in graph
topologies, PFGs are better suited to depict scenarios in which the sum of the degrees
of membership and non-membership exceeds 1. In contrast, IFGs cannot effectively rep-
resent such cases, even if the squares of these degrees satisfy the required constraints.
Compared to fuzzy sets and intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets and
PFGs offer enhanced flexibility for managing human judgment data. PFGs are viewed as
more powerful and ap-plicable than fuzzy graphs and IFGs when it comes to managing
uncertainty and incomplete data.



Obbu Ramesh et.al / Eur. J. Pure Appl. Math, 18 (3) (2025), 6372 15 of 17

6. Conclusion

A fuzzy graph can effectively represent the uncertainty present in various types of net-
works. Com-pared to classical, fuzzy, and intuitionistic fuzzy models, Pythagorean fuzzy
models offer greater pre-cision, flexibility, and compatibility for such systems. In this pa-
per, we introduce the concept of intui-tionistic Pythagorean fuzzy graphs (IPFGs). We
have developed a set of operational laws for IPFGs and examined their related theorems
in detail. We also determine the planarity of IPFGs formed through certain operations.
The intersecting value and the Pythagorean fuzzy planarity value are then used to define
the planarity of a Pythagorean fuzzy graph. In addition, we demonstrate that certain
theorems establish bounds for both general and specific planarity values of a Pythagorean
fuzzy graph. Subsequently, related theorems and applications of strong and weak pla-
narity are presented and de-fined. Finally, the applicability of the proposed generaliza-
tion of fuzzy graph theory is demonstrated through an application based on intuitionistic
Pythagorean fuzzy preference graphs (IPFGs). Our future research will focus on: (1)
interval-valued intuitionistic Pythagorean fuzzy graphs, (2) simplified inter-val-valued in-
tuitionistic Pythagorean fuzzy graphs, and (3) hesitant intuitionistic Pythagorean fuzzy
graphs. Further studies should also explore planarity, expressions, and dual structures in
Pythagorean fuzzy graphs.

Author Declarations

Conflict of interest: The authors have no conflicts to disclose. Author Con-
tributions:Obbu Ramesh: Mathematical Analysis, Investigation (equal). Nainaru
Tarakaramu: Editing review, Super vision (equal). YuvaRoopa Lakshmi: Devel-
oped the theory and performed the computations (equal). Sharief Basha S: Super vi-
sion (equal). Sarvar Iskandarov: Investigation (equal). Akbar Toyirov: Verified the
Methodology (equal). Jushkinbek Yuldoshev: Formulations, Verified the Methodology
(equal). Sardor. Sabirov: Conceived of the presented idea (equal).K. K. Prashanth:
Review editing, Developed the theory and performed the computations (equal).

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