EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5675
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Determining Electrical Vehicle Charging Stations Sites
Using Dominance in Neutrosophic Fuzzy Directed

Graphs

Mohammed Alqahtani

Department of Basic Sciences, College of Science and Theoretical Studies,
Saudi Electronic University, P.O. Box 93499, Riyadh 11673, Saudi Arabia

Abstract. In this paper, we begin by describing several forms of effective arcs in neutrosophic
fuzzy-directed graphs, based on the fundamental principles introduced in neutrosophic fuzzy-
directed graphs. We then investigate several forms of dominance in neutrosophic fuzzy directed
graphs, looking at their significance and uses in decision-making situations. Our study fills two
important gaps in the literature: it extends the concept of domination from fuzzy directed graphs
to neutrosophic fuzzy directed graphs, and it provides a detailed characterization of dominations
within this advanced framework. The concept of dominations in fuzzy graphs, fuzzy directed
graphs, intuitionistic fuzzy graphs, neutrosophic fuzzy graphs, and picture fuzzy graphs is well-
documented in the literature. First, we characterize several kinds of effective arcs that are
particular to neutrosophic fuzzy directed graphs, such as semi-κ effective arc, semi-φ effective
arc, and semi-ϖ effective arc. Next, we provide the notions of dominations and domination
numbers connected to these arcs in different types of fuzzy-directed graphs that are neutrosophic.
In particular, our analysis of minimum dominating sets provides valuable characterizations of
dominations in these graphs. We also study the smallest dominating sets and dominance numbers
for neutrosophic fuzzy dipaths and dicycles, and we provide some interesting results. Finally,
we suggest an algorithm to handle decision-making problems, such figuring out the ideal places
to launch EV charging station in different metropolitan regions, by utilizing the ideas presented
in this research. In regard to the development of electric mobility and environmentally friendly
urban growth, this strategy offers a far superior technique of identifying EV charging stations.

2020 Mathematics Subject Classifications: 05C72, 05C69, 05C90, 68R10, 90B90

Key Words and Phrases: Neutrosophicgraph, fermatean neutrosophic graph, fermatean
neutrosophic digraphs, generalized fermatean neutrosophic digraphs

1. Introduction

Zadeh initiated the theory of FSs in 1965 [54] as the development of the classical
theory of sets. FSs have been used subsequently in different areas, such as computer

DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5675

Email address: m.alqahtani@seu.edu.sa (Mohammed Alqahtani)

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



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 2 of 27

science, management, artificial intelligence, and decision making because they have su-
perior capability in handling with the large amount of information in contrary to the
classic logic. Due to the flexibility for source selection, there have been a number of
extensions to FSs. Interval-valued fuzzy sets (IVFSs) are among the first generalizations
that Zadeh originally introduced in 1975 [55]. As opposed to FSs where membership
values are single numbers in the range [0,1], IVFSs permit the membership grade to be a
subinterval of [0,1]. However, non-membership was not explicitly explored in either FSs
or IVFSs. Atanassov [14] proposes the concept of non-membership and starts the study
of the IFSs, where the membership and non-membership are two independent factors
integrated considering the fact that the sum of membership and non-membership should
not be over 1. In the past few decades, Graph theory has emerged as one powerful tool
for analyzing real-world problems across divergent sciences. Various kinds of graphical
models have been used in various branches of studies including pattern management,
computer networking, and decision making. Fuzzy logic grows from uncertainty in real-
world problems and has developed into a useful and potent resolution. In recent years, a
new approach to model uncertainty is known as fuzzy graphs (FGs). First proposed by
Rosenfeld [42], FGs are considered more efficient and elastic in comparison with classical
graphs. Because of this flexibility, Their applications have been investigated in numerous
fields. Some new terms to the FG theory were introduced by Bhattacharya [15], while
advanced operation was defined in [32]. The complement of FGs was for the first time
in [50], whereas, the average connectivity in the FGs has been described by Poulik et al.
[37]. Most of the authors have generalized the classical graph concepts towards FGs that
has led to the use of this concept in areas like social studies and road network analysis.
Later improvements consist of the extension of aggregation operators in a complicated
fuzzy setting by Ubaid ur Rehman [51]; Jana et al. [23] proposed the Dombi operator
using Pythagorean fuzzy knowledge under multi-attribute decision-making. The edge
coloring of FGs has been investigated by Mahapatra et al. [28]. A new concept called
fuzzy-directed graphs (FDGs) was defined by Deepak Mordeson and developed in [33]
and [26]. Bipolar FDGs and their applications in decision-making are explored in [3].
The generalized forms of FGs have been defined as intuitionistic fuzzy graphs (IFGs)
in [46]. Similarly, more elaborate IFGs with networking applications were presented
in [53]. The dynamics of IFGs are explained by [45] through strong and effective arcs
only; categorical properties of IFGs are introduced by [41]. Competition graphs have
been defined by Akram et al. [7] under a complex intuitionistic fuzzy environment and
later extended in terms of IF-hypergraphs, Strong-IFGs, IF-cycles, and IF-trees [[6]-[4]].
Complex intuitionistic fuzzy power interaction aggregation operators by Ali et al. [13],
on decision-making techniques. The use of IFDGs for a decision support system was
considered in [5], and interval valued IFDG matrix for roadblock analysis in [47]. Bipo-
lar IFDGs were initialised using energy by Nithyanandham et al. [36] and applied to
decision-making.
Recent years witnessed extensive interest in neutrosophic graphs since they can deal with
imprecise, uncertain, and indeterminate data, which are so characteristic of the current
complex-world problems. Neutrosophic logic, which generalized the classical and fuzzy



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 3 of 27

logic by involving truth, uncertainty, and false parts, has given sufficient mathematical
support to various graph-oriented studies. An extended work of domination in neutro-
sophic graphs is being studied, which involves various applications in areas including
decision-making, communication network, and computational intelligence. Mullai and
Broumi examined the notion of dominating energy in neutrosophic graphs, which can
help to design efficient energy networks [35]. Fei used neutrosophic graphs to wireless
networks, proving the efficiency of application of NM to handle the uncertainties of
communication networks [20]. The recent study conducted by Khan et al. on covering
and paired domination under the neutrosophic context has extended the broad spec-
trum application of neutrosophic graphs in [25], Similarly, in single-valued neutrosophic
graphs, the authors analyzed complementary domination and edge domination in the
related field of graph-based analysis [[52],[29]]. A few recent works have been mainly
targeted at examining different operations of and using various kinds of neutrosophic
graphs, such as Dombi neutrosophic graphs, which have been discussed by Lakhwani
et al., [27]and domination in neutrosophic incidence graphs as concepts further devel-
oped by Mohamad et al., [31]. Secure dominance, integrity, and domination integrity
in neutrosophic graphs were also attempted, by Broumi and Jaikumar et al. (2023)
[[16],[22]]. Other extensions to the field of the present work include further studies
in fermatean neutrosophic Dombi fuzzy graphs, which have been studied by Sasikala
and Divya [43], and offered new insights into the aspect of neutrosophic graph theory.
Furthermore, by extension complex fermatean neutrosophic graphs have been used as
showcased by Broumi et al. [17] for decision-making purposes. Subsequent studies of
equitable domination in neutrosophic graphs focusing on strong arcs by Meenakshi and
Senbagamalar [30], as well as new definitions of domination in neutrosophic incidence
graphs by Smarandache et al. [48], have extended the theoretical framework of the
subject matter. The existence of domination of neutrosophic over graphs as studied by
[18], NarmadaDevi and the use of neutrosophic networks applying efficient domination
as described by Senbagamalar et al. [44] establishes the practical importance for neu-
trosophic graphs in handling decision-making dilemmas in conditions of uncertainty. In
continuation with their work, Kaviyarasu et al. [24]have attempted to forward circular
economy strategies for sustainable development concerning t-neutrosophic fuzzy graphs.
This study points to the applicability of neutrosophic fuzzy graph models improves the
problematic determination in the circular economy by managing all the large resources
and wastes effectively. The use of neutrosophic fuzzy logic here shows that these models
could be used to solve complex sustainability issues.

Furthermore, neutrosophic graphs are useful in infrastructure design. Alqahtani et
al. [21] used complex neutrosophic graphs to design the structure of the hospital by
incorporating multi-constrained and multi-uncertain factors into the planning model.
In the same vein, AL-Omeri and Kaviyarasu [9] discussed how the concept of neutro-
sophic graphs can be applied for the operational enhancement of earthquake response
centers, and how such mathematical models can better capture uncertain and variable
environmental conditions that affect disaster response planning and execution. Such
applications reveal the increasing importance of neutrosophic fuzzy graph theory in



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 4 of 27

modeling the current problems. For neutrosophic graphs, [12] also developed survey
results of planer and outer planar, suggesting a direction of further topological structure
investigation of this framework. That is, Sreelakshmi and Samundesvari [49]reported
the concept of total domination of USVNG in 2024 to expand the consideration of
dominance theory in the context of neutrosophic systems. In [39] This article explores
multi-objective fuzzy programming, which may relate to the decision-making framework
used in neutrosophic fuzzy graph theory. In [1] focus on parameter estimation techniques
can inform the modeling and decision-making processes in the EV station framework.
[2]The mathematical modeling approach used here could provide conceptual or tech-
nical inspiration for modeling domination in neutrosophic fuzzy directed graphs. [38]
The decision-making framework addressing under constraints and uncertainties could be
linked to EV charging locations under various constraints. [40] Directly relevant due to
its focus on neutrosophic methods, which align well with the methodology in your paper.
The set of works by Al-Masarwah and other authors gives important contributions to
fuzzy mathematics and its use in different fields [10]. The earlier studies provide an
initial understanding of fuzzy soft graph essentials, including some elements and opera-
tions concerning fuzzy uncertain graph models [11]. The subsequent works are devoted
to further development of the sophisticated classification of different types of fuzzy soft
graphs and the explorations of further mathematical properties of the fuzzy soft graphs,
which enhances the theory. The other extensions extend the above studies by defining
m-polar fuzzy points in BCK/BCI-algebras, which links fuzzy logic and algebraic sys-
tems [8]. This paper also introduces int-soft ideals in ordered semigroups, it expand
these ideas to ideal theory to provide further development for algebraic modeling [34].

In this study, we also define several new concepts of dominations in neutrosophic
fuzzy directed graphs(NFDGs) by using the concepts of the different types of effective
arcs. An algorithm created to address real-world issues, like figuring out the best places
for EV charging stations, demonstrates the work’s practical significance. This method
improves the efficacy and efficiency of decision-making procedures in transportation net-
works and urban planning by utilizing the special qualities of NFDGs. This study
illustrates the adaptability and usefulness of NFDGs in tackling contemporary issues by
bridging the gap between theoretical developments and real-world implementations. list
of Abbreviations given in Table 1.

1.1. Motivation

Different from the fuzzy directed graphs (FDGs) where only the degree of membership
associated with the directed edges is taken into consideration, in neutrosophic fuzzy
directed graphs (NFDGs) the degree of truth, indeterminacy and falsity attached to the
edges are taken into account, making NFDGs more general and flexible than FDGs.
Further, the domination concept has been introduced in fuzzy graphs (FGs), interval-
valued fuzzy graphs (IFGs), and Pythagorean fuzzy graphs (PFGs), and hence it inspired
us to define the domination concepts in NFDGs and explain their usefulness in decision-
making.



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Table 1: List of Abbreviations

Abbreviations Meaning

FDGs Fuzzy Directed Graphs
NFDGs Neutrosophic Fuzzy Directed Graphs
PFGs Pythagorean Fuzzy Graphs
FGs Fuzzy Graphs
IFGs Intuitionistic Fuzzy Graphs

NF-dipath Neutrosophic Fuzzy Dipath
NF-dicycle Neutrosophic Fuzzy Dicycle

MD Membership degrees
NMD Non Membership degrees
EV Electrical Vehicle
DS Dominating Set

MDS Minimal Dominating Set

1.2. Novelty

(i) In the case of NFDGs, we also discussed other types of effective arcs, including
semi-κ effective arcs, semi-φ effective arcs, and semi-ϖ effective arcs provided
adequate definitions for these terms.

(ii) Finally, a Dominating set, Minimal dominating set (MDS) and domination num-
ber were studied in relation to certain defined NFDGs.

(iii) Similarly, we defined neutrosophic fuzzy dipath (NF-dipath) and neutrosophic
fuzzy dicycle (NF-dicycle) in NFDGs and stated their domination-dependent char-
acteristics.

(iv) Last but not least, we described both theoretical and practical aspects of the
above-mentioned concepts in decision-making situations.

1.3. Structure of the article

This manuscript has six sections. Section 2 discusses key terminologies and concepts
associated with neutrosophic sets, fuzzy graphs, and their generalizations. Section 3
explains different kinds of useful arcs in NFDG and the domination concepts that depend
on them. In section 4 the author tackles a decision-making problem and shows how
domination works in NFDGs. Section 5 consists of a comparative analysis, and Section
6 contains the last comments including suggestions for future research.

2. Preliminaries

Definition 1. [54] If Z is a nonempty set and λ : Z → [0, 1] is the membership function,
then (λ, Z) is a fuzzy set.



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Definition 2. [31] An generalisation of a fuzzy set, a neutrosophic fuzzy set consists
of three independent membership functions for each element ρ in a universe U : falsity-
membership ϖ(ρ), truth-membership κ(ρ), and indeterminacy-membership φ(ρ). These
functions meet the requirement: κ(ρ), φ(ρ), ϖ(ρ) ∈ [0, 1] and κ(ρ) + φ(ρ) +ϖ(ρ) ≤ 3

Definition 3. The dominant set (DS) in G̃ is a subset X1 of X, provided that for any
w ∈ X1, there exists ρ ∈ U − X1 such that w dominates ρ. If X1 contains no suitable
DS of G̃, then a DS X1 of a fuzzy graph G̃ is a minimum dominating set (MDS). The
lowest (fuzzy) cardinality of a DS in G̃ is its dominant number (DN ).

Definition 4. [19] Consider the basic directed graph ĜD = (V,E), where V is a
nonempty finite set and E = {(ρ1, ρ2) | ρ1, ρ2 ∈ V, ρ1 ̸= ρ2}. The pair G̃D = (µV , µE) is
a fuzzy digraph (FDG), where µV : V → [0, 1] and µE : E → [0, 1] with µE((ρ1, ρ2)) ≤
µV (ρ1) ∧ µV (ρ2), for all ρ1, ρ2 ∈ V .

Definition 5. [19] Consider a hidden digraph ĜD = (V,E), in which V is the set
of vertices and E is the set of directed edges. Hence, G̃D = (µV , µE) is the fuzzy
directed graph of ĜD, where µV = {µV (ρ1) | ρ1 ∈ V } is the set of vertices (nodes),
and µE = {µE((ρ1, ρ2)) | ρ1, ρ2 ∈ V } represents the fuzzy directed edges from µV (ρ1) to
µV (ρ2) in a fuzzy digraph G̃D.

Definition 6. [19] A hidden digraph of an FDG G̃D = (µV , µE) is represented as
ĜD = (V,E). The arc (ρ1, ρ2) ∈ E is thus referred to be an effective arc of G̃D, if
µE(ρ1, ρ2) ≤ µV (ρ1) ∧ µV (ρ2), for all ρ1, ρ2 ∈ V .

3. Dominance via Effective Arcs in the NFDGs

First, we introduce several kinds of effective edges in NFDGs, such as semi-κ effective
arcs, semi-φ effective arcs, and semi-ϖ effective arcs etc . Using these arcs, we further
develop these ideas to present the idea of domination in NFDGs. Within this framework,
we analyze several significant domination-related NFDG features. We also provide some
interesting findings on Minimum MDS and DS in IFDGs.

In addition, we define the terms ”NF-dipath” and ”NF-dicycle” inside NFDGs and in-
vestigate the MDSs, Dominating Numbers (DNs), and related ideas that are connected
to them. We also provide some interesting findings about NF-dicycles and NF-dipaths
in NFDGs.

Throughout the discussion, GΓ = (Λ,Ω) will signify an NFDG of a hidden digraph
ĜΓ = (V,E), where Λ = (κΛ, φΛ, ϖΛ) and Ω = (κΩ, φΩ, ϖΩ). To begin, we define an
effective arc in an NFDG as follows:

Definition 7. An arc (x, y) in an NFDG GΓ is considered an effective arc if it meets the
requirements listed below: κΩ(x, y) = min{κΛ(x),κΛ(y)} φΩ(x, y) = min{φΛ(x), φΛ(y)}
and ϖΩ(x, y) = max{ϖΛ(x), ϖΛ(y)} where κΩ(x, y) + φΩ(x, y) + ϖΩ(x, y) ≤ 3 for all
(x, y) ∈ Ω. Next, we delineate the related categories of effective arcs in an NFDG,
namely the semi-κ effective arcs, semi-φ effective arcs, and semi-ϖ effective arcs



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Definition 8. A semi-κ effective arc of the NFDG GΓ is defined as,

κΩ(x, y) = min{κΛ(x),κΛ(y)}
φΩ(x, y) ̸= min{φΛ(x), φΛ(y)}

and ϖΩ(x, y) ̸= max{ϖΛ(x), ϖΛ(y)}

with κΩ(x, y) + φΩ(x, y) +ϖΩ(x, y) ≤ 3 for all (x, y) ∈ Ω.

Definition 9. A semi-φ effective arc of the NFDG GΓ is defined as,

κΩ(x, y) ̸= min{κΛ(x),κΛ(y)}
φΩ(x, y) = min{φΛ(x), φΛ(y)}

and ϖΩ(x, y) ̸= max{ϖΛ(x), ϖΛ(y)}

with κΩ(x, y) + φΩ(x, y) +ϖΩ(x, y) ≤ 3 for all (x, y) ∈ Ω.

Definition 10. A semi-ϖ effective arc of the NFDG GΓ is defined as,

κΩ(x, y) ̸= min{κΛ(x),κΛ(y)}
φΩ(x, y) ̸= min{φΛ(x), φΛ(y)}

and ϖΩ(x, y) = max{ϖΛ(x), ϖΛ(y)}

with κΩ(x, y) + φΩ(x, y) +ϖΩ(x, y) ≤ 3 for all (x, y) ∈ Ω.

Example 1. Concerning Fig. 1 NFDG, we have the following.

(i) Examine the arc (b, c), In this case κΩ(b, c) = min{κΛ(b),κΛ(c)} = min (0.4, 0.7) =
0.4, φΩ(b, c) = min{φΛ(b), φΛ(c)} = min (0.5, 0.9) = 0.5 ̸= 0.6, and ϖΩ(b, c) =
max{ϖΛ(b), ϖΛ(c)} = max (0.5, 0.7) = 0.7 ̸= 0.4. As a result, an arc (b, c) is
non-effective, whereas an arc (b, c) is semi-κ effective.

(ii) Examine the arc (c, d), In this case κΩ(c, d) = min{κΛ(c),κΛ(d)} = min (0.7, 0.8) =
0.7 ̸= 0.4, φΩ(c, d) = min{φΛ(c), φΛ(d)} = min (0.5, 0.6) = 0.5, and ϖΩ(c, d) =
max{ϖΛ(c), ϖΛ(d)} = max (0.5, 0.5) = 0.5 ̸= 0.3. As a result, an arc (c, d) is
non-effective, whereas an arc (c, d) is semi-φ effective.

(iii) Examine the arc (a, b), In this case κΩ(a, b) = min{κΛ(a),κΛ(b)} = min (0.5, 0.4) =
0.4 ̸= 0.3, φΩ(a, b) = min{φΛ(a), φΛ(b)} = min (0.6, 0.9) = 0.6 ̸= 0.5, and
ϖΩ(a, b) = max{ϖΛ(a), ϖΛ(b)} = max (0.8, 0.7) = 0.8. As a result, an arc (a,
b) is non-effective, whereas an arc (a, b) is semi-ϖ effective.

(iv) Examine the arc (d, a), In this case κΩ(d, a) = min{κΛ(d),κΛ(a)} = min (0.5, 0.8) =
0.5, φΩ(d, a) = min{φΛ(d), φΛ(a)} = min (0.6, 0.6) = 0.6, and ϖΩ(d, a) = max{ϖΛ(d), ϖΛ(a)} =
max (0.8, 0.5) = 0.8. As a result, an arc (d, a) is an effective arc.

Definition 11. Let GΓ = (Λ,Ω) will signify an NFDG of a hidden digraph ĜΓ = (V,E),
then



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Figure 1: NFDG .

(i) The effective neighborhood (EN) of a vertex x ∈ V is defined as

ℸE(x) = {y ∈ V | arc(x, y) is an effective arc},

while the closed effective neighborhood (CEN) of y is given by

ℸE [x] = ℸE(x) ∪ {x}.

(ii) The semi-κ EN of x ∈ V is defined as

ℸκE(x) = {x ∈ V | arc(x, y) is a semi-κ-effective arc},

and the closed semi-κ-EN of x is

ℸκE [x] = ℸκE(x) ∪ {x}.

(iii) The semi-φ EN of x ∈ V is defined as

ℸφE(x) = {x ∈ V | arc(x, y) is a semi-φ-effective arc},

and the closed semi-φ-EN of x is

ℸφE [x] = ℸφE(x) ∪ {x}.



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(iv) The semi-ϖ EN of x ∈ V is defined as

ℸϖE(x) = {x ∈ V | arc(x, y) is a semi-ϖ-effective arc},

and the closed semi-ϖ-EN of x is

ℸϖE [x] = ℸϖE(x) ∪ {x}.

(v) The minimum cardinality of effective neighborhoods in a graph GΓ is

δE(GΓ) = min{|ℸE(x)| | x ∈ V (GΓ)}.

(vi) The maximum cardinality of effective neighborhoods in a graph GΓ is

∆E(GΓ) = max{|ℸE(x)| | x ∈ V (GΓ)}.

Definition 12. Let GΓ be an NFDG with two vertices, x and y. The vertex x is said to
dominate the vertex y if the directed edge (x, y), is considered an effective edge.

Example 2. The directed edge (d, a) in the NFDG depicted in Fig. 1 is a effective edge,
however, the edges (a, b), (b, c) and (c, d) are not effective arcs. Because of the effective
edge, the vertex d dominates the vertex a. However, vertex a does not dominate b, vertex
b does not dominate c, and vertex c does not dominate d as there are no effective edges
between them.

Definition 13. In an NFDG, U is a subset of V and is considered a DS if for each
y ∈ V − U there exists x ∈ V such that x dominates y.

Definition 14. A dominating set U is called a MDS in an NFDG if it contains no other
dominating sets. The effective arc domination number (DN), represented by ζE(GΓ), is
the minimal fuzzy cardinality among all dominating sets in an NFDG. The minimal
effective edge dominating set’s total membership count is provided by n(ζE(GΓ)).

Example 3. From Fig. 2, it is easy to infer that the MDSs of an NFDG GΓ are
represented by {(0.9, 0.6, 0.2), (0.8, 0.5, 0.3)} and {(0.8, 0.5, 0.3), (0.6, 0.3, 0.5)}.

Definition 15. Let GΓ = (Λ,Ω) will signify an NFDG. The only DS of GΓ is (κΛ, φΛ, ϖΛ)
if

κΩ(x, y) ≤ min{κΛ(x),κΛ(y)}
φΩ(x, y) ≤ min{φΛ(x), φΛ(y)}

and ϖΩ(x, y) ≤ max{ϖΛ(x), ϖΛ(y)}

Example 4. It is straightforward to confirm that
{(0.5, 0.6, 0.8), (0.4, 0.9, 0.7), (0.9, 0.6, 0.2), (0.7, 0.5, 0.5), (0.8, 0.6, 0.5)} is the sole DS of
an NFDG GΓ shown in Fig. 3.



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Figure 2: MDS .

Definition 16. Let GΓ be an NFDG and let x, y be its two vertices.

(i) x (semi-κ-effective) dominates y, if the arc (x, y) is semi-κ-effective.

(ii) x (semi-φ-effective) dominates y, if the arc (x, y) is semi-φ-effective.

(iii) x (semi-ϖ-effective) dominates y, if the arc (x, y) is semi-ϖ-effective.

Example 5. It is clear from Fig.1 that the arc (a, b) is a semi-(ϖ) effective arc, in-
dicating that a semi-(ϖ)-effectively dominates b. In the same way, the arc (b, c) is a
semi-(κ)-effective arc; as a result, b semi-(κ)-effectively dominates c, and the arc (c, d)
is a semi-(φ)-effective arc; as a result, c semi-(φ)-effectively dominates d.

Theorem 1. Let GΓ = (Λ,Ω) be an NFDG, and let ĜΓ = (V,E) be its hidden digraph
with ℵ ⊆ V . A DS κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ) of GΓ is minimal if and only if for each
(κΛ (e) , φΛ (e) , ϖΛ (e) ∈ κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)), either

(ℸGΓ
) (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) ∩ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) = ∅

or

(ℸGΓ
) (κΛ (f) , φΛ (f) , ϖΛ (f)) ∩ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) = {κΛ (e) , φΛ (e) , ϖΛ (e)},

for some (κΛ (f) , φΛ (f) , ϖΛ (f)) ∈ (κΛ, φΛ, ϖΛ) \ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)).



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Figure 3: Dominating set in NFDG .

Proof. Let (κΛ (e) , φΛ (e) , ϖΛ (e)) ∈ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) such that (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ))
is an MDS of GΓ, then

(κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) \ (κΛ (e) , φΛ (e) , ϖΛ (e))

. Hence

(κΛ (f) , φΛ (f) , ϖΛ (f)) /∈ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) \ (κΛ (e) , φΛ (e) , ϖΛ (e))

Consider that (κΛ (f) , φΛ (f) , ϖΛ (f)) is not dominated by any of the members of
(κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) \ (κΛ (e) , φΛ (e) , ϖΛ (e)).
Now, we have the following cases:
Case 1. Let (κΛ (f) , φΛ (f) , ϖΛ (f)) = (κΛ (e) , φΛ (e) , ϖΛ (e)).
Then (κΛ (e) , φΛ (e) , ϖΛ (e)) is not dominated by any of the members of
(κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) \ (κΛ (e) , φΛ (e) , ϖΛ (e)), i.e.,
(ℸGΓ

)((κΛ (e) , φΛ (e) , ϖΛ (e))) ∩ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) = ∅.
Case2.Let(κΛ (f) , φΛ (f) , ϖΛ (f)) ̸= (κΛ (e) , φΛ (e) , ϖΛ (e)). Then (κΛ (f) , φΛ (f) , ϖΛ (f)) /∈
(κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)). Since (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) is an MDS of GΓ, it implies
that (κΛ (f) , φΛ (f) , ϖΛ (f)) is dominated by (κΛ (e) , φΛ (e) , ϖΛ (e)) ∈ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)).
Thus,

(ℸGΓ
)(κΛ (f) , φΛ (f) , ϖΛ (f)) ∩ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)) = {(κΛ (e) , φΛ (e) , ϖΛ (e))},

forsome(κΛ (f) , φΛ (f) , ϖΛ (f)) ∈ (κΛ, φΛ, ϖΛ) \ (κΛ (ℵ) , φΛ (ℵ) , ϖΛ (ℵ)). Likewise,
its opposite may also be demonstrated.



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 12 of 27

Definition 17. NF-dipaths Dp are effective arcs where the starting node of the next edge
is the same as its ending node.

Consider Dp = (Λ,Ω) as an NF-dipath of a hidden dipath Dn = (V,U), where n
denotes an integer with l ≥ 2. Then:
1.(κΛ, φΛ, ϖΛ) = {(κΛ(ek), φΛ(ek), ϖΛ(ek)) : ek ∈ Λ, for all k ∈ {1, 2, . . . , l}}.
2.(κΩ, φΩ, ϖΩ) = {(κΩ(ek, ek+1), φΩ(ek, ek+1), ϖΩ(ek, ek+1) : (ek, ek+1) ∈ Λ, for all k ∈
{1, 2, . . . , l − 1}}.
3.(κΩ(ek, ek+1), φΩ(ek, ek+1), ϖΩ(ek, ek+1) = {(κΛ(ek), φΛ(ek), ϖΛ(ek)), (κΛ(ek+1), φΛ(ek+1), ϖΛ(ek+1))},
for all k ∈ {1, 2, . . . , l − 1}.
4. The start and end vertices of a nontrivial NF-dipath are
(κΛ(e1), φΛ(e1), ϖΛ(e1)) and (κΛ(el), φΛ(el), ϖΛ(el)), respectively.

Theorem 2. An NF-dipath is represented by Dp. Following that, one of the prerequisites
needs to be met:
1. ζ(Dp) =

∑l/2
r=1(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1))

2. ζ(Dp) = minE, where

E =
{

l+1
2 −

∑(l+1)/2−r
r=1 (κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) + . . .∑(l+1)/2

r=(l+3)/2−k(κΛ(e2r−2), φΛ(e2r−2), ϖΛ(e2r−2)) : k ∈ {0, 1, 2, . . . , (l − 1)/2}
}
.

Proof. From the Remark 3
(κΛ, φΛ, ϖΛ) = {(κΛ(ek), φΛ(ek), ϖΛ(ek)) : ek ∈ Λ, for all k ∈ {1, 2, . . . , l}}.
(κΩ, φΩ, ϖΩ) = {(κΩ(ek, ek+1), φΩ(ek, ek+1), ϖΩ(ek, ek+1) : (ek, ek+1) ∈ Λ, for all k ∈
{1, 2, . . . , l−1}}. Because (κΩ(ek, ek+1), φΩ(ek, ek+1), ϖΩ(ek, ek+1) is an effective edge for
all k ∈ {1, 2, . . . , (l−1)}, it implies (κΛ(ek), φΛ(ek), ϖΛ(ek)) dominate (κΛ(ek+1), φΛ(ek+1), ϖΛ(ek+1))
for all k ∈ {1, 2, . . . , (l − 1)}.

Case 1: If l = 2r i.e., even, for positive integer r, then
{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))} is
clearly the MDS of Dp. Note that
{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))}
=

∑l/2
r=1(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)).

Therefore, ζ(Dp) =
∑l/2

r=1(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)). This proves (1).
Case 2: When l = 2r- 1, when l is odd, for any integer r. Thus,

{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el), φΛ(el), ϖΛ(el))};

{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−2), φΛ(el−2), ϖΛ(el−2)), (κΛ(el−1),
φΛ(el−1), ϖΛ(el−1))};

{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−4), φΛ(el−4), ϖΛ(el−4)),
(κΛ(el−3), φΛ(el−3), ϖΛ(el−3)), (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))};

...
{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . .



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 13 of 27

, (κΛ(el−5), φΛ(el−5), ϖΛ(el−5)), (κΛ(el−3), φΛ(el−3), ϖΛ(el−3)), (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))}.

MDSs for Dp. Take note of that,

(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , . . . , (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))

=

l+1
2∑

r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) .

Generally, we have

(κΛ(e1), φΛ(e1), ϖΛ(e1)) + (κΛ(e3), φΛ(e3), ϖΛ(e3)) + · · ·+ (κΛ(eg), φΛ(eg), ϖΛ(eg))

+ (κΛ(eg+1), φΛ(eg+1), ϖΛ(eg+1)) + · · ·+ (κΛ(el−3), φΛ(el−3), ϖΛ(el−3)) + (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))

=

g+1
2∑

r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) +

l+1
2∑

r= g+3
2

(κΛ(e2r−2), φΛ(e2r−2), ϖΛ(e2r−2)) .

Let
k + 1

2
=
m+ 1

2
for r ∈

{
0, 1, 2, . . . ,

l − 1

2

}
.

(κΛ(e1), φΛ(e1), ϖΛ(e1)) + (κΛ(e3), φΛ(e3), ϖΛ(e3)) + · · ·+ (κΛ(eg), φΛ(eg), ϖΛ(eg))

+ (κΛ(eg+1), φΛ(eg+1), ϖΛ(eg+1)) + · · ·+ (κΛ(el−3), φΛ(el−3), ϖΛ(el−3)) + (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))

=

l+1
2

−k∑
r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) +

l+1
2∑

r= l+3
2

−k

(κΛ(e2r−2), φΛ(e2r−2), ϖΛ(e2r−2)) .

Consequently, the minimum of

E =
{

l+1
2 −

∑(l+1)/2−r
r=1 (κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) +

∑(l+1)/2
r=(l+3)/2−k(κΛ(e2r−2), φΛ(e2r−2), ϖΛ(e2r−2)) :

k ∈ {0, 1, 2, . . . , (l − 1)/2}} represents a DN of an NF-dipath Dp.
It follows that ζ(Dp) = minE.

Example 6. A hidden directed route P6 = (V,C) has an NF-dipath Dp = (Λ,Ω). Next,
the MDS of Dp may be obtained from Fig.4.

{(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , (κΛ(e5), φΛ(e5), ϖΛ(e5))}

and the DN is, ζ(Dp) =
∑3

r=1(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)).

Example 7. From Fig. 5, let Dp = (Λ,Ω) be a NF-dipath of a hidden directed path
P5 = (V,C). Let:

A = {(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , (κΛ(e5), φΛ(e5), ϖΛ(e5))} ,

B = {(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , (κΛ(e5), φΛ(e5), ϖΛ(e5))} ,



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 14 of 27

Figure 4: NFDP .

C = {(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , (κΛ(e5), φΛ(e5), ϖΛ(e5))} .

Then, the MDS of Dp is {A,B,C}, and the DN is:

ζ(Dp) = min
{∑

(κΛ(e),φΛ(e),ϖΛ(e))∈A (κΛ(e), φΛ(e), ϖΛ(e)) ,
∑

(κΛ(f),φΛ(f),ϖΛ(f))∈B (κΛ(f), φΛ(f), ϖΛ(f)) ,∑
(κΛ(h),φΛ(h),ϖΛ(h))∈R (κΛ(h), φΛ(h), ϖΛ(h))

}
.

Figure 5: NFDP .

Theorem 3. Let Dp = (Λ,Ω) be a NF-dipath of a hidden directed path PN = (V,C). If
(κΛ(e), φΛ(e), ϖΛ(e)) = (κΛ(f), φLambda(f), ϖLambda(f)), then

ζ(Dp) =
l

2
(κΛ(e), φΛ(e), ϖΛ(e)) , ∀ e, f ∈ V.

Proof. 1. If n is even: By Theorem 2, we have:

ζ(Dp) =

n/2∑
r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) .

Since (κΛ(e), φΛ(e), ϖΛ(e)) = (κΛ(f), φLambda(f), ϖLambda(f)) for all e, f ∈ V , it im-
plies that:
(κΛ(e1), φΛ(e1), ϖΛ(e1)) = (κΛ(e3), φΛ(e3), ϖΛ(e3)) = · · · = (κΛ(el−1), φΛ(el−1), ϖΛ(el−1)) =

(κΛ(e), φΛ(e), ϖΛ(e)) .
Thus,

ζ(Dp) =

l/2∑
k=1

(κΛ(e), φΛ(e), ϖΛ(e)) =
n

2
(κΛ(e), φΛ(e), ϖΛ(e)) .



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 15 of 27

2. If m is odd: By Theorem 2, we have:

ζ(Dp) =

(m+1)/2−k∑
r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1))+

(l+1)/2∑
r=(l+3)/2−k

(κΛ(e2r−2), φΛ(e2r−2), ϖΛ(e2r−2)) ,

for all k ∈ {0, 1, 2, . . . , l}. Simplifying, we have:

ζ(Dp) =

(
l + 1

2
− j

)
(κΛ(e), φΛ(e), ϖΛ(e))+

[
l + 1

2
−
(
l + 3

2
− k + 1

)]
(κΛ(e), φΛ(e), ϖΛ(e)) .

This further simplifies to:

ζ(Dp) =
l + 1

2
(κΛ(e), φΛ(e), ϖΛ(e)) .

Hence, ζ(Dp) is either
l
2 (κΛ(e), φΛ(e), ϖΛ(e)) if l is even, or

l+1
2 (κΛ(e), φΛ(e), ϖΛ(e)) if

l is odd. Consequently,

ζ(Dp) =
l

2
(κΛ(e), φΛ(e), ϖΛ(e)) .

Definition 18. A dipath where the beginning and ending vertices are the same is called
an NF-dicycle.

Remark 1. Consider Dc = (Λ,Ω) to be an NF-dicycle of a hidden directed route Pl =
(V,C), where l denotes an integer with l ≥ 3. Then:

(i) (κΛ, φΛ, ϖΛ) = {(κΛ(ek), φΛ(ek), ϖΛ(ek)) : ek ∈ Λ, ∀k ∈ {1, 2, . . . , l}}.

(ii) (κΩ, φΩ, ϖΩ) = {(κΩ(ek, ek+1), φΩ(ek, ek+1), ϖΩ(ek, ek+1)), (κΩ(el, e1), φΩ(el, e1), ϖΩ(el, e1)) :
(ek, ek+1), (em, e1) ∈ Λ,∀k ∈ {1, 2, . . . , l − 1}}.

(iii) (κΩ(ek, ek+1), φΩ(ek, ek+1), ϖΩ(ek, ek+1)) = (κΛ(ek), φΛ(ek), ϖΛ(ek)), (κΛ(ek+1), φΛ(ek+1), ϖΛ(ek+1))
for all k ∈ {1, 2, . . . , l − 1}.

Theorem 4. Consider DC = (Λ,Ω) as an NFD-cycle for a hidden dipath Dl = (V,C),
where l ≥ 3. Then, one of the following requirements must exist:

(i)

ζ(DC) = min


l/2∑
r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)),

m/2∑
r=1

(κΛ(e2r), φΛ(e2r), ϖΛ(e2r))

 ;

(ii)
ζ(DC) = min(E ∪ F ),



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 16 of 27

where

X =

{
l+1
2 −

∑k
r=1(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) +

∑ l+1
2

r=n+3
2

−k
(κΛ(e2r−2), φΛ(e2r−2), ϖΛ(e2r−2)) :

k ∈
{
0, 1, 2, . . . , l−1

2

}}
,

and

F =

{
l+1
2 −

∑k
r=1(κΛ(e2r), φΛ(e2r), ϖΛ(e2r)) +

∑ l+1
2

r= l+3
2

−k
(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) :

k ∈
{
0, 1, 2, . . . , l−1

2

}}
.

Proof. Assume that R = {1, 2, . . . , l}. The arcs (κΩ(ek, ek+1), φΩ(ek, ek+1), ϖΩ(ek, ek+1))
as well as (κΩ(em, e1), φΩ(em, e1), ϖΩ(em, e1)) are effective for all k ∈ I. This im-
plies that (κΛ(ek), φΛ(ek), ϖΛ(ek)) dominates (κΛ(ek+1), φΛ(ek+1), ϖΛ(ek+1)) for all k ∈
{1, 2, . . . , l − 1} and (κΛ(el), φΛ(el), ϖΛ(el)) dominates (κΛ(e1), φΛ(e1), ϖΛ(e1)). Case
1: If, for any number r, l is even, that is, l = 2r. Following that, the specified collection

{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))}

and

{(κΛ(e2), φΛ(e2), ϖΛ(e2)), (κΛ(e4), φΛ(e4), ϖΛ(e4)), . . . , (κΛ(el), φΛ(el), ϖΛ(el))}

are both the DS and MDSs of Dc. Take note of that
{(κΛ(e1), φΛ(e1), ϖΛ(e1)) + (κΛ(e3), φΛ(e3), ϖΛ(e3)) + . . .+ (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))} =∑m/2

r=1 (κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) and {(κΛ(e2), φΛ(e2), ϖΛ(e2)) + (κΛ(e4), φΛ(e4), ϖΛ(e4)) + . . .+

(κΛ(el), φΛ(el), ϖΛ(el))} =
∑l/2

r=1(κΛ(e2r), φΛ(e2r), ϖΛ(e2r)).
Thus,

ζ(DC) = min


l/2∑
r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)),

l/2∑
r=1

(κΛ(e2r), φΛ(e2r), ϖΛ(e2r))

 .

This proves (1).
Case 2: For every integer r, if l = 2r − 1, that is, l, is odd. Next up are the sets
{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el), φΛ(el), ϖΛ(el))} ;
{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−2), φΛ(el−2), ϖΛ(el−2)), (κΛ(el−1),
φΛ(el−1), ϖΛ(el−1))} ;
{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−4), φΛ(el−4), ϖΛ(el−4)), (κΛ(el−3), φΛ(el−3),
ϖΛ(el−3)), (κΛ(el−1), ϖΛ(el−1))} ;

. . . ;

{(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), . . . , (κΛ(el−5), φΛ(el−5), ϖΛ(el−5)), (κΛ(el−3), φΛ(el−3),
ϖΛ(el−3)), (κΛ(el−1), φΛ(el−1), ϖΛ(el−1))}
are a few of the MDS and DS of Dc. Take note of that
(κΛ(e1), φΛ(e1), ϖΛ(e1)) + (κΛ(e3), φΛ(e3), ϖΛ(e3)) + . . .+ (κΛ(el), φΛ(el), ϖΛ(el)) =



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 17 of 27

∑ l+1
2

r=1(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)).
In general,
(κΛ(e1), φΛ(e1), ϖΛ(e1))+(κΛ(e3), φΛ(e3), ϖΛ(e3))+. . .+(κΛ(el−5), φΛ(el−5), ϖΛ(el−5))+
(κΛ(el−3), φΛ(el−3), ϖΛ(el−3))+(κΛ(el−1), φΛ(el−1), ϖΛ(el−1))+(κΛ(el), φΛ(el), ϖΛ(el)) =∑ l+1

2
r=1(κΛ(e2r), φΛ(e2r), ϖΛ(e2r)) +

∑ l+1
2

r= k+3
2

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)).

Let k+1
2 + k = l+1

2 .Then,

l + 1

2
−

k∑
r=1

(κΛ(e2r), φΛ(e2r), ϖΛ(e2r))+

l+1
2∑

r=m+3
2

−k

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)), ∀j ∈ {0, 1, 2, . . . , l − 1

2
}.

Thus,
ζ(Dc) = min(E ∪ F ).

This proves demonstrates (2).

Figure 6: NF Directed cycle with six verticees .

Example 8. The hidden dicycle D6 = (V,C) has an NF-dicycle Dc = (Λ,Ω). Determine
the sets

E = {(κΛ(e1), φΛ(e1), ϖΛ(e1)), (κΛ(e3), φΛ(e3), ϖΛ(e3)), (κΛ(e5), φΛ(e5), ϖΛ(e5))}

and

F = {(κΛ(e2), φΛ(e2), ϖΛ(e2)), (κΛ(e4), φΛ(e4), ϖΛ(e4)), (κΛ(e6), φΛ(e6), ϖΛ(e6))}

(see Fig. 6). The MDS of Dc is {E,F}, and the DN is given by

ζ(Dc) = min

{
3∑

r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) ,
3∑

r=1

(κΛ(e2r), φΛ(e2r), ϖΛ(e2r))

}
.



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 18 of 27

Example 9. Given a hidden dicycle D5 = (V,C), let Dc = (Λ,Ω) be an IF dicycle.
Consider

E1 = {(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , (κΛ(e5), φΛ(e5), ϖΛ(e5))} ,
E2 = {(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e2), φΛ(e2), ϖΛ(e2)) , (κΛ(e4), φΛ(e4), ϖΛ(e4))} ,
E3 = {(κΛ(e2), φΛ(e2), ϖΛ(e2)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , (κΛ(e5), φΛ(e5), ϖΛ(e5))} ,
E4 = {(κΛ(e1), φΛ(e1), ϖΛ(e1)) , (κΛ(e3), φΛ(e3), ϖΛ(e3)) , (κΛ(e4), φΛ(e4), ϖΛ(e4))} ,
E5 = {(κΛ(e2), φΛ(e2), ϖΛ(e2)) , (κΛ(e4), φΛ(e4), ϖΛ(e4)) , (κΛ(e5), φΛ(e5), ϖΛ(e5))} .

Thus, based on Fig. 7, Dc has an MDS of {Er : r = 1, 2, . . . , 5} and the DN is ζ(Dc) =

min
{∑5

k=1 (κΛ(ek), φΛ(ek), ϖΛ(ek)) ∈ Er

}
.

Figure 7: NF Directed cycle with five verticees .

Theorem 5. If Dc = (Λ,Ω) as a Nf- dicycle of a hidden dicycle Cp = (V,C),where
p ≥ 3. If (κΛ(e), φΛ(e), ϖΛ(e)) = (κΛ(f), φΛ(f), ϖΛ(f)) for all e, f ∈ V , then[

l

2

]
(κΛ(e), φΛ(e), ϖΛ(e)).

Proof. If n is even, by Theorem 4, we have:

l

2

∑
γ(Dc) = min


l
2∑

r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1)) ,

l
2∑

r=1

(κΛ(e2r), φΛ(e2r), ϖΛ(e2r))

 .



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 19 of 27

Because (κΛ(e), φΛ(e), ϖΛ(e)) = (κΛ(f), φΛ(f), ϖΛ(f)) for all e, f ∈ V , it implies

l

2

∑
γ(DC) =

(
l

2

)
(κΛ(e), φΛ(e), ϖΛ(e)),

using the same reasoning as in Theorem 3.26. Similarly, if m is odd, by Theorem 3.26:

l+1
2

−k∑
r=1

γ(Dc) =

l+1
2

−k∑
r=1

(κΛ(e2r−1), φΛ(e2r−1), ϖΛ(e2r−1))+

l+1
2∑

r= l+3
2

−k

(κΛ(e2r−2), φΛ(e2r−2), ϖΛ(e2r−2)) ,

for all k ∈ {0, 1, 2, . . . , l−1
2 }. Because (κΛ(e), φΛ(e), ϖΛ(e)) = (κΛ(f), φΛ(f), ϖΛ(f)) for

all e, f ∈ V , we obtain:

l + 1

2

∑
γ(Dc) =

(
l + 1

2

)
(κΛ(e), φΛ(e), ϖΛ(e)).

Hence, γ(Dc) is either
(
l
2

)
(κΛ(e), φΛ(e), ϖΛ(e)) if l is even, or

(
l+1
2

)
(κΛ(e), φΛ(e), ϖΛ(e))

if l is odd. Thus,

ζ(Dc) =

[
l

2

]
(κΛ(e), φΛ(e), ϖΛ(e)).

4. Selection of locations for EV charging stations using dominance in
neutrosophic fuzzy directed graphs

Many fuzzy graphs have been used to mimic real-world problems, such as sites for
infrastructure projects. Here, we provide a model based on dominance in Neutrosophic
Fuzzy Directed Graphs (NFDGs) to determine the best sites for public charging stations
for electric vehicles (EVs). For comparison, we first model the scenario using the princi-
ples of domination in Fuzzy Directed Graphs (FDGs), and then explore it through the
perspective of NFDG dominance. Adopting our suggested NFDG model produces more
consistent and trustworthy outcomes than the old FDG strategy.

4.1. Use FDG as a model

If a city planner is assigned the duty of installing public charging stations at different
points around the city. The planner is unable to construct charging stations at every
place because of financial restrictions. The objective is to determine the bare minimum
of charging stations necessary to coverage, guaranteeing that most EV users can readily
locate a station.
The prospective sites are depicted as nodes in this FDG model (Fig. 8), and their
membership degrees (MD) illustrate the proportion of nearby EV customers who would
utilize the charging stations. The key roads that connect these places are shown as
directed edges, and the MD of these roads indicates the probability that EV users will



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 20 of 27

Figure 8: FDG Model .

utilize them.
We ascertain that the set{L2, L6, L9} is a MDS by looking at the effective arcs, such as
(L1, L2), (L1, L4), (L9, L2), (L1, L4), (L7, L6), (L8, L9), (L7, L9), (L10, L9), (L8, L5), (L5, L6),
(L3, L6), (L3, L8), (L10, L7), (L4, L3), it shows that 85% of EV users near L2, 90% near
L6, and 95% nearL9 are likely to utilize the charging stations. These localities also
have significant MDs. These places are ideal for station placement since they are also
well-connected to other localities.

4.2. Use the NFDG model

We only took into account the likelihood of usage (liking) in the FDG-based model;
we neglected to account for any deterrents (disliking) that can influence EV consumers’
decisions. We use the NFDG approach to address this and offer a more in-depth ex-
amination. The NFDG is shown in Fig. 9, where the non-membership degrees (NMD)
represent the percentage of users who might avoid the charging stations because of things
like distance, traffic congestion, or safety concerns, and the MD of the nodes represents
the percentage of EV users who are likely to use them. Based on a resident poll, the
directed arcs show the likelihood that electric vehicle (EV) users will drive over the



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 21 of 27

roadways that link these places. We determine the dominating set and effective arcs by
using the concept of dominance in NFDGs. Using this technique, we discover once more
that the set{L2, L6, L9} forms an MDS, suggesting ideal charging station sites. Table
2 indicates that these locations are suitable as ideal sites since they have low Falsity-
membership (10% forL2, 5% for L6, and 3% for L9) and high Truth-membership (85%
for L2, 90% for L6, and 95% for L9).
A more thorough assessment of both positive and negative elements is made possible
by the usage of NFDGs, which promotes better decision-making. The city planner can
guarantee coverage and customer happiness by choosing these sites, which will promote
EV adoption and ensuring the charging network is implemented successfully.

Figure 9: NFDG Model .

Table 2: Membership Values of Locations

Location Truth-membership Falsity-membership Indeterminacy-membership

L1 0.75 0.2 0.05
L2 0.85 0.1 0.05
L3 0.7 0.25 0.05
L4 0.6 0.15 0.25
L5 0.5 0.2 0.3
L6 0.9 0.05 0.05
L7 0.7 0.25 0.05
L8 0.65 0.3 0.05
L9 0.95 0.03 0.02
L10 0.7 0.25 0.05



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 22 of 27

Pseudo-code for Neutrosophic Fuzzy System

Input: NFS of Locations Λ
Output: MDS

void FNFS_MDS () {

NFS Λ = getNFSOfLocations ();

int numOfLocations = count(Λ);
NFS Ω;
Set DS;

for (int e = 0; e < numOfLocations; e++) {

for (int f = 0; f < numOfLocations; f++) {

if (Λ(e, f)) {

κΩ(e, f) = min(κΛ(e),κΛ(f));
φΩ(e, f) = max(φΛ(e), φΛ(f));
ϖΩ(e, f) = min(ϖΛ(e), ϖΛ(f));

}

if (arc_is_effective(e, f, Ω)) {

DS.add(e);

}

}

}

Set MDS = find_MDS(DS);

}

bool arc_is_effective(int e, int f, NFS Ω) {

return (κΩ(e, f) > κthreshold &&

φΩ(e, f) < φthreshold &&

ϖΩ(e, f) < ϖthreshold);

}

struct NFS {

map <pair <int , int >, double > κ;
map <pair <int , int >, double > φ;
map <pair <int , int >, double > ϖ;

bool isAdjacent(int e, int f);

};

NFS getNFSOfLocations () {

NFS Λ;
return Λ;

}



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 23 of 27

int count(NFS Λ) {

return κΛ.size ();

}

5. Comparative analysis

The concept of domination inuzzy graphs (FGs) has proved very important in the
solving of real life problems that are characterized by uncertainties. Rosenfeld intro-
duced the concept of Fuzzy graph (FG) where vertices and arcs are associated with a
fuzzy membership degree under some conditions. After that, FGs have been used in
many area and have productive outcomes and motivated the extension such as fuzzy
directed graph (FDGs) where vertex and edge value are fuzzy set. It turns out that sim-
ple/certain cases are best addressed with FDFs while these very problems are not very
effective when dealing with complexity and uncertainty. This has led to more flexible
structures to enhance accuracy giving rise to the neutrosophic fuzzy graphs (NFGs).
Meanwhile, NFSs introduce the truth membership and falsity membership for both ver-
tices and edges of NFGs to overcome the drawback of the FGs and FDGs as the ef-
fectiveness and ineffectiveness are balanced. Moreover, this framework is developed by
Neutrosophic fuzzy directed graphs (NFDGs) which propose effective arcs for the better
accuracy of the right locations for EV charging station sites. For example, FDGs can
only show that it has an effectiveness of 95 percent for a location like L9 , where this sin-
gle membership value eliminates consideration of ineffectiveness it may introduce bias.
However, NFDG approaches use both truth and falsity values, leaded a more fitness
estimating for potential EV charging station sites by trade suitable-efficiency pairs, as
well as trade unsuitable-efficiency pairs.

In the example above, NFDGs enables one to evaluate various aspects which in turns
facilitate a better identification of the best sites to locate EV charging. Thus, NFDGs
offer more general and more flexible structure that would expand the ability to accurately
model and transform uncertain information in the given domain.

6. Conclusion

This study demonstrates the effectiveness of the NFDGs comparing to the traditional
FDGs. The utilization of effective arcs like semi-k, semi-∅, and semi-ψ arcs along with
the reconceptualization of domination types make the NFDGs a strong platform of
analyzing and solving uncertainty based situations. The flexibility of applying NFDGs
for modelling higher domination structures like DS or MDS contributes to making of
decisions and to real-world use. For example, the usability of NFDGs when solving the
problem of locating EV charging stations is clear, as the employment of such algorithms
provides better and more effective outcomes compared to straightforward FDGs. These
outcomes stress common usage of NFDGs, especially for considering real world problems,
where the interpretation of uncertainty is crucial.



M. Alqahtani / Eur. J. Pure Appl. Math, 18 (1) (2025), 5675 24 of 27

The extension of the proposed NFDG framework can also be applicable to various
domains such as disaster management, healthcare logistics, and smart transportation sys-
tems, where uncertainty prevails over decision-making. Further comparison with other
advanced graph models, including intuitionistic fuzzy graphs or bipolar fuzzy graphs
could further explicate the advantages and convenient of NFDGs. In addition to that, it
would be equally interesting to test the working of this framework in different conditions
such as the urban, rural, and the industrial area environments to establish its flexibility
and expansion.

Furthermore, identifying improved algorithms for calculation of domination sets and
other measures on NFDGs would increase the efficiency of their computation with respect
to large-size networks. Extension of the NFDG-based models with the state of the art
technologies like AI and machine learning could also pave way for decision making under
uncertainty, as and when they are more developed and applied across the implementation
arena.

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	Introduction
	Motivation
	Novelty
	Structure of the article

	 Preliminaries
	Dominance via Effective Arcs in the NFDGs
	Selection of locations for EV charging stations using dominance in neutrosophic fuzzy directed graphs
	Use FDG as a model
	Use the NFDG model

	Comparative analysis
	Conclusion