Equitable Domination in Turiyam Graphs with Network Applications

Abdata Guluma Erana; V N SrinavasaRao Repalle; Fekadu Tesgara Agama

doi:10.12688/f1000research.181882.1

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Research Article

Equitable Domination in Turiyam Graphs with Network Applications

[version 1; peer review: awaiting peer review]

Abdata Guluma Erana¹, V N SrinavasaRao Repalle ², Fekadu Tesgara Agama³

PUBLISHED 01 Jun 2026

Author details Author details

¹ Mathematics, Wollega University, Nekemte, Oromia, 395, Ethiopia
² Mathematics, Wollega University, Nekemte, Oromia, Ethiopia
³ Mathematics, Wollega University, Nekemte, Oromia, 395, Ethiopia

Abdata Guluma Erana
Roles: Conceptualization, Methodology, Resources, Validation, Writing – Original Draft Preparation

V N SrinavasaRao Repalle
Roles: Formal Analysis, Methodology, Supervision, Writing – Review & Editing

Fekadu Tesgara Agama
Roles: Formal Analysis, Investigation, Supervision, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

Abstract

Background

Turiyam graphs provide a mathematical framework for representing systems with uncertainty, inconsistency, incomplete information, and refusal conditions. This study introduces the concept of equitable domination in Turiyam graphs to achieve balanced dominance among neighboring vertices and reduce excessive dependence on individual nodes in uncertain network structures.

Methods

The study defines equitable dominating sets in Turiyam graphs and investigates their theoretical properties. Relationships between equitable domination and classical domination concepts are examined through mathematical analysis. Fundamental properties, bounds, and characterizations of equitable domination are established within the Turiyam graph framework.

Results

The analysis shows that equitable domination supports balanced control and efficient resource allocation in uncertain networks. Equitable dominating sets generally require a slightly larger number of dominating vertices than minimum dominating sets, but they provide improved balance, fault tolerance, and network stability. The study also identifies potential applications in communication systems, wireless sensor networks, cloud computing, and social interaction models where reliable connectivity and fair workload distribution are important.

Conclusions

Equitable domination in Turiyam graphs provides an effective approach for analyzing and designing complex systems operating under uncertain conditions. The proposed framework extends domination theory in generalized graph structures and offers a foundation for future studies on optimization, decision-making, and network management in uncertain environments.

Keywords

Domination theory, Fairness in networks, Wireless sensor networks, Turiyam graph, equitable domination

Corresponding author: V N SrinavasaRao Repalle

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 Erana AG et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Erana AG, Repalle VNS and Agama FT. Equitable Domination in Turiyam Graphs with Network Applications [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:849 (https://doi.org/10.12688/f1000research.181882.1) First published: 01 Jun 2026, 15:849 (https://doi.org/10.12688/f1000research.181882.1) Latest published: 01 Jun 2026, 15:849 (https://doi.org/10.12688/f1000research.181882.1)

1. Introduction

The idea of uncertainty in real-world systems was first addressed by L.A. Zadeh in 1965 through the introduction of fuzzy set theory.¹ In graph theory, the representation of fuzzy graphs and several fuzzy analogues of connectivity were later introduced by Rosenfeld.² The study of dominating sets in graphs began with the works of Berge and Ore,³ while paired domination was initiated by Teresa et al.⁴ V.R. Kulli further expanded the theory by developing the concept of domination in graphs.⁵ Cockayne⁶ was the first to define the independent domination number in graphs, and Swaminathan and Dharmalingam⁷ introduced the concept of equitable domination.

A. Meenakshi contributed significantly to the field by developing and analyzing paired equitable domination⁸ as well as inflated graphs and their counterparts.^9,10 Strong and weak domination in fuzzy graphs were defined and studied by Nagoor Gani and M. Basher Ahmed.¹¹ K.T. Atanassov introduced intuitionistic fuzzy relations and intuitionistic fuzzy graphs (IFGs),¹² while Shannon and Atanassov,¹³ along with M.G. Karunambigai et al., ¹⁴ identified IFGs as a specific class of fuzzy graphs. Additionally, strong and weak domination in fuzzy graphs were investigated by Jayalakshmi and Harinarayanan.²² The terms “order,” “degree,” and “magnitude” in IFGs were established by Shajitha Begum and A. Nagoor Gani.¹⁵ Domination in split intuitionistic fuzzy graphs was introduced by S. Anu Priya and A. Nagoor Gani,¹⁶ and Dombi fuzzy graphs were later examined by the same researcher.¹⁷ Equitable domination in neutrosophic graphs was developed by A. Meenakshi and J. Senbagamalar.¹⁸ Further advancements in Turiyam theory including Turiyam relations, Turiyam graphs, and their applications were proposed by Ganati G.A.^19–21 Domination and paired domination in Turiyam graphs were subsequently developed by Erana A.G., Srinivasa Rao Repalle V.N., and Agama F.T.²³

In this study, we construct and analyze the equitable domination parameter in Turiyam graphs using strong arcs. The motivation for our work arises from the existing research on domination and equitable domination in Turiyam graphs, as well as applications involving strong arcs. The split domination work in intuitionistic fuzzy graphs found in¹⁶ primarily focused on vertex and edge cardinality; in contrast, our study extends the investigation to equitable domination in Turiyam graphs, aiming to determine the optimal degree, size, and order of their vertices. Inspired by the concepts of weak and strong domination in fuzzy graphs,¹³ we further explore both strong and weak equitable domination in Turiyam graphs using the score function.

2. Preliminaries

Definition 2.1

¹¹ If $A_{IF}$ is finite vertex set of $G_{IF} = (A_{IF}, B_{IF})$ such that;

(i) $μ_{1} : A_{IF} \to [0, 1]; γ_{1}$ : $A_{IF} \to [0, 1]$ , indicate the values of membership for the degrees of truth and falsehood respectively, with $0 \leq μ_{1} (v_{s}) + γ_{1} (v_{s}) \leq 1$ for each $v_{s} \in V$ and,
(ii) $B_{IF} \subseteq A_{IF} \times A_{IF}$ Where $μ_{2} : A_{IF} \times A_{IF} \to [0, 1]$ ; $γ_{2} : A_{IF} \times A_{IF} \to [0, 1]$ are such that $μ_{2} {(k_{i}, k_{j})} \leq min {μ_{1} (k_{i}), μ_{1} (k_{j})}$ ; $γ_{2} {(k_{i}, k_{j})} \geq max {γ_{1} (k_{i}), γ_{1} (k_{j})$ } and where $0 \leq μ_{2} {(k_{i}, k_{j})} + μ_{2} {(k_{i}, k_{j})} \leq 1 \forall (k_{i}, k_{j}) \in$ $B_{IF}$ is IFG.

Definition 2.2

¹¹If $μ_{2} (k_{i}, k_{j}) = min {μ_{1} (k_{1}), μ_{1} (k_{2})}$ and $γ_{2} {(k_{i}, k_{j})} = max {γ_{1} (k_{1}), γ_{1} (k_{2})$ }. Then, an arc ( $u_{d}, v_{d}$ ) is a strong arc.

Definition 2.3

¹¹ The degree of a node $u_{d}$ in an IFG, $G_{IF} = (A_{IF}, B_{IF}),$ is the sum of the weights of strength arcs incident at $u_{d}$ and Neighborhood of $u_{d}$ . Additionally, $δ (G_{IF}) = min {d G_{IF} (u_{d}) / u_{d} \in A_{IF}}$ is the smallest degree of $G_{IF}$ , and $∆ (G_{IF}) = max {d G_{IF} (u_{d}) / u_{d} \in A_{IF}}$ is the greatest degree of $G_{IF}$ .

Definition 2.4

¹¹ Node $u_{d} \in A_{IF}$ in an IFG, $G_{IF} = (A_{IF}, B_{IF})$ is an isolated node if $μ_{2} (k_{i}, k_{j}) = 0$ and $γ_{2} (k_{i}, k_{j}) = 0 .$

Definition 2.5

¹¹ Let IFG be $G_{IF} = (A_{IF}, B_{IF})$ . Then 𝐺’s cardinality is defined as

| G | = | \sum_{a_{i} \in A_{NS}} \frac{1 - T_{A_{NS} - F_{A_{NS}}}}{2} + \sum_{a_{i} a_{j} \in B_{NS}} \frac{1 - T_{B_{NS} - F_{B_{NS}}}}{2} |

Definition 2.6

¹¹ If there is a strength of arc between $u_{if}$ and $v_{if} \in A_{IF}$ , we say that $u_{if}$ dominates $v_{if}$ in $G_{IF}$ where $G_{IF} = (A_{IF}, B_{IF})$ . A subset $D_{d} \subseteq A_{IF}$ is a dominance set in $G_{IF}$ if for each $v_{if} \in A_{IF} - D_{d}$ , there is $u_{if} \in D_{d}$ dominates $v_{if}$ .

Definition 2.7

⁴ Let $X_{sp}$ be a generic element and a point space. One neutrosophic collection with values $A_{NS}$ (SVNS) is specified by $T_{A_{NS}} (X_{sp})$ , $I_{A_{NS}} (X_{sp}),$ and $F_{A_{NS}} (X_{sp})$ .

For each point $x^{'}$ in $X_{sp}; T_{A_{NS}} (X_{sp})$ , $I_{A_{NS}} (X_{sp})$ , and $F_{A_{NS}} (X_{sp}) \in [0, 1] .$ A SVNS,

$A_{NS} = {x^{'}$ : $T_{A_{NS}} (x^{'})$ , $I_{A_{NS}} (x^{'})$ , $F_{A_{NS}} (x^{'}) \geq x^{'} \in X_{sp}}$ .

Definition 2.8

⁴ Consider the sets $A_{NS} = (T_{A_{NS}}, I_{A_{NS}}, F_{A_{NS}})$ and $B_{NS} = (T_{B_{NS}}, I_{B_{NS}}, F_{B_{NS}})$ on a set $X_{sp}$ be single valued neutrosophic set. If $A_{NS} = (T_{A_{NS}}, I_{A_{NS}}, F_{A_{NS}})$ is a SVN relation on $B_{NS} = (T_{B_{NS}}, I_{B_{NS}}, F_{B_{NS}})$ then, $T_{B_{NS}} (x^{'}, y^{'}) \leq min {T_{A_{NS}} (x^{'}), T_{A_{NS}} (y^{'})$ }.

$I_{B_{NS}} (x^{'}, y^{'}) \leq min {I_{A_{NS}} (x^{'}), I_{A_{NS}} (y^{'})}$ , and $F_{B_{NS}} (x^{'}, y^{'}) \geq max {T_{A_{NS}} (x^{'}), T_{A_{NS}} (y^{'})}$ for all $x^{'}, y^{'}$ in $X_{sp}$ .

Definition 2.9

⁴ If $T_{E_{s}} {(a_{i}, a_{j})} = min {T_{V_{s}} (a_{i}), T_{V_{s}} (a_{j})}$ , $I_{E_{s}} {(a_{i}, a_{j})} = min {I_{V_{s}} (a_{i}), I_{V_{s}} (a_{j})}$ , $F_{E_{s}} {(a_{i}, a_{j})} = max {F_{V_{s}} (a_{i}), F_{V_{s}} (a_{j})}$ of $G_{NS} = (A_{NS}, B_{NS})$ of the neutrosophic graph then, an arc $(a_{i}, a_{j})$ of $G_{NS}$ is strong where $(a_{i}, a_{j}) \in E_{s}$ and $a_{i} & a_{j} \in V_{s}$ .

Definition 2.10

^4,24 Let u be a node in a NSG, $G_{NS} = (A_{NS}, B_{NS})$ . The degree of a node $u_{d}$ is represented by deg ( $u_{d}$ ), which is the total weight of the strong arcs that incident at that node. $N (u_{d}) = {v_{d} \in A_{NS} / (u_{d}, v_{d})$ is a strong arc} represents the neighborhood of $u_{d}$ .

$δ (G_{NS}) = min {d G_{NS} (u_{d}) / u_{d} \in A_{NS}}$ is the smallest degree of $G_{NS}$ and $∆ (G_{NS}) = max {d G_{NS} (u_{d}) / u_{d} \in A_{NS}}$ is the maximum degree of $G_{NS}$ .

Definition 2.11

²⁵ The order of $G_{NS}$ , $G_{NS} = (A_{NS}, B_{NS})$ is represented by

$| A_{NS} | = | \sum_{a_{i} \in A_{NS}} \frac{2 + T_{A_{NS} - (0.5) I_{A_{NS}} - F_{A_{NS}}}}{3} |$ , and the size of $G_{NS}$ , $G_{NS} = (A_{NS}, B_{NS})$ is represented by $| B_{NS} | = | \sum_{a_{i} \in A_{NS}} \frac{2 + T_{B_{NS} - (0.5) I_{B_{NS}} - F_{B_{NS}}}}{3} |$ .

Definition 2.12

²⁵ If there is at least one $u_{d} \in D_{NS}$ that dominates $v_{d}$ for each $v_{d} \in A_{IF} - D_{NS}$ , then the subset $D_{NS} \subseteq A_{NS}$ is considered a dominance set. The symbol $γ_{NG} (G_{NS})$ represents the dominance number, which is the minimum dominating set’s cardinality.

Definition 2.13

²⁵ If no appropriate subset of $D_{NS}$ is a dominance set of $G_{NS}$ , then the dominance set $D_{NS}$ of $A_{NS}$ is a minimum.

3. Result

3.0 Domination of Turiyam Graph Using Strength of Arc

Definition 3.0.1

Assume u is a node in $T_{G}$ . The degree of a node $u_{d}$ is explained as the total weight of the powerful arcs that occurred at $u_{d}$ and is represented by deg ( $u_{d}$ ). The neighbourhood $u_{d}$ , $N (u_{d}) = {v_{d} \in A_{TS} / (u_{d}, v_{d})}$ is a strong arc, and $δ (G_{NS}) = min {d G_{NS} (u_{d}) / u_{d} \in A_{NS}}$ is the smallest degree of $G_{NS}$ and $∆ (G_{NS}) = max {d G_{NS} (u_{d}) / u_{d} \in A_{NS}}$ is the greatest degree of $G_{NS}$ .

Definition 3.0.2

The order of $G_{TS}$ , $G_{TS} = (A_{TS}, B_{TS})$ is;

$| A_{TS} | = | \sum_{a_{i} \in A_{TS}} \frac{3 + T_{A_{TS} - (0.5) I_{A_{TS}} - F_{A_{TS}}}}{4} |$ , and the size of $G_{TS}$ , $G_{TS} = (A_{TS}, B_{TS})$ in a Turiyam graph is;

| B_{TS} | = | \sum_{a_{i} \in A_{TS}} \frac{3 + T_{B_{TS} - (0.5) I_{B_{TS}} - F_{B_{TS}}}}{4} |

Definition 3.0.3

The subset $D_{TS} \subseteq A_{TS}$ of $G_{TS}$ is an overwhelming set which is symbolized as $γ_{TG} (G_{TS})$ , if there is at least one $u_{d} \in D_{TS}$ that dominates $v_{d}$ for each $v_{d} \in A_{IF} - D_{TS}$ .

Definition 3.0.4

The dominance set $D_{TS}$ of $G_{TS}$ is minimum, if no proper subset of $D_{TS}$ is an overwhelming of $G_{TS}$ .

Example 3.1:

Let $G_{TS} = (A_{TS}, B_{TS})$ be $T_{G}$ shown in Figure 1.

From Figure 1, the arcs $a_{2} a_{4}$ and $a_{4} a_{7}$ are not strong arcs. The degrees of the vertices are given as follows:

deg (a_{1}) = (0.5,0.5,0.6,0.7),

deg (a_{2}) = (0.8,0.8,0.95,1.05),

deg (a_{3}) = (0.25,0.3,0.35,0.35),

deg (a_{4}) = (0.25,0.25,0.3,0.35),

deg (a_{5}) = (1.4,0.95,0.85,1.15),

deg (a_{6}) = (0.6,0.3,0.2,0.4),

deg (a_{7}) = (0.7,0.65,0.7,0.8), and

deg (a_{8}) = (0.4,0.3,0.35,0.45) .

Hence, the smallest truth, indeterminacy, falsity, and liberal membership values of $G_{TS}$ are:

δ (G_{TS}) = min {T (u_{d}) : u_{d} \in A_{TS}} = 0.25,

δ (G_{IS}) = min {I (u_{d}) : u_{d} \in A_{TS}} = 0.25,

δ (G_{FS}) = min {F (u_{d}) : u_{d} \in A_{TS}} = 0.2,

δ (G_{LS}) = min {L (u_{d}) : u_{d} \in A_{TS}} = 0.35 .

Therefore, the minimum degree of $G_{TS}$ is

δ (G_{TS}) = min {d_{G_{TS}} (u_{d}) : u_{d} \in A_{TS}} = (0.25,0.25,0.2,0.35) .

Similarly, the greatest truth, indeterminacy, falsity, and liberal membership values of $G_{TS}$ are:

Δ (G_{TS}) = max {d_{G_{TS}} (u_{d}) : u_{d} \in A_{TS}} = 1.4,

Δ (G_{IS}) = max {d_{G_{IS}} (u_{d}) : u_{d} \in A_{TS}} = 0.95,

Δ (G_{FS}) = max {d_{G_{FS}} (u_{d}) : u_{d} \in A_{TS}} = 0.95,

Δ (G_{LS}) = max {d_{G_{LS}} (u_{d}) : u_{d} \in A_{TS}} = 1.15 .

Thus, the maximum degree of $G_{TS}$ is

Δ (G_{TS}) = max {d_{G_{TS}} (u_{d}) : u_{d} \in A_{TS}} = (1.4,0.95,0.95,1.15) .

Theorem 3.0.1:

$G_{TS} = (A_{TS}, B_{TS})$ is minimal. If and only if one of the following criteria is met for each $v_{d} ϵ D_{TS}$ , where $D_{TS}$ is Turiyam dominance set.

i. There is a node $u_{d} ϵ A_{TS} - D_{TS}$ such that $N (u_{d}) \cap D_{TS} = {v_{d}}$
ii. $v_{d}$ is an isolate in $≺ D_{TS} ≻$

Proof: Assume that a vertex $v_{d}$ of $D_{TS}$ does not satisfy any of the previously mentioned requirements. As a result, $N (u_{d}) \cap D_{TS} \neq {v_{d}}$ for each node $u_{d} ϵ A_{TS} - D_{TS}$ . Additionally, $D_{TS} - v_{d}$ will be a minimal and overpowering of $G_{TS}$ , which is contrary to the assumption, if $v_{d}$ is not an isolate in $≺ D_{TS} ≻$ , according to condition (ii).

Theorem 3.0.2:

A dominance set is a subset $D_{TS} \subseteq A_{TS}$ of $G_{TS}$ , where $G_{TS} = (A_{TS}, B_{TS})$ . Then, there are two nodes $u_{d}, v_{d} ϵ A_{TS} - D_{TS}$ such that at least one vertex of $D_{TS}$ is present in every $u_{d} - v_{d}$ path.

Proof:

Presume that $D_{TS}$ is a dominant set of G. Since each vertex in $A_{TS} - D_{TS}$ is dominated by at least one vertex of $D_{TS}$ , there is a $u_{d} - v_{d}$ path that comprises at least one vertex of $D_{TS}$ .

Theorem 3.0.3:

For any Turiyam graph $T_{G}$ , with order $p$

γ {TG}^{(G_{TS})} \leq p - ∆_{T} (G_{TS})

γ {TG}^{(G_{TS})} \leq p - ∆_{I} (G_{TS})

γ {TG}^{(G_{TS})} \leq p - ∆_{F} (G_{TS})

γ {TG}^{(G_{TS})} \leq p - ∆_{L} (G_{TS})

Proof:

The dominant set has fewer vertices than $n$ since $\sum_{i = 1}^{n} d (v_{i}) \leq p$ , where $v_{i}$ is the number of nodes in the strong arc. This is demonstrated by taking $G_{TS} = (A_{TS}, B_{TS})$ to be a $T_{G}$ of order $p$ . Additionally, by the concept of $∆ (G_{TS})$ , the maximum truth values membership of $v_{i}$ among all nodes of $G_{TS}$ and $γ {TG}^{(G_{TS})} \leq \sum_{i = 1}^{n} d (v_{i}) \leq p - ∆_{T} (G_{TS})$ , Furthermore, we verify the others for falsity membership, indeterminacy membership, and liberal membership as follows:

γ {TG}^{(G_{TS})} \leq \sum_{i = 1}^{n} d (v_{i}) \leq p - ∆_{I} (G_{TS}), γ {TG}^{(G_{TS})} \leq \sum_{i = 1}^{n} d (v_{i}) \leq p - ∆_{F} (G_{TS}), and γ {TG}^{(G_{TS})} \leq \sum_{i = 1}^{n} d (v_{i}) \leq p - ∆_{L} (G_{TS}) .

Figure 1. Example of domination of Turiyam Graph using strong arc.

3.1 Equitable dominance of a Turiyam graph (TG)

Definition 3.1.1

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph. A dominating set $D_{TS} \subseteq A_{TS}$ is called an equitable dominating set if for every vertex $u \in A_{TS} \ D_{TS}$ , there exists a vertex $v \in D_{TS}$ such that

| deg (u) - deg (v) | \leq 1 .

The equitable domination number of $G_{TS}$ , denoted by $γ_{ef - TG} (G_{TS}),$ is the minimum cardinality of an equitable dominating set in $G_{TS}$ .

Definition 3.1.2

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph. An equitable dominating set $D_{TS} \subseteq A_{TS}$ is said to be minimum if no proper subset of $D_{TS}$ is an equitable dominating set of $G_{TS}$ .

Example 3.1.1

Let be $T_{G}$ with $G_{TS} = (A_{TS}, B_{TS})$ and let we have arcs $a_{1} a_{2}$ , $a_{2} a_{4}$ , $a_{4} a_{7}$ which are weak arcs and their degrees can be given as, deg $(a_{1}) = (0.25, 0.25, 0.3, 0.35)$ ,

deg (a_{2}) = (0.75, 0.65, 0.55, 0.75),

deg (a_{3}) = (0.25, 0.3, 0.35, 0.35),

deg (a_{4}) = (0.25, 0.25, 0.3, 0.35),

deg (a_{5}) = (1.4, 1.05, 0.75, 1.2),

deg (a_{6}) = (0.6, 0.3, 0.2, 0.4),

deg (a_{7}) = (0.7, 0.65, 0.7, 0.8) and,

deg (a_{8}) = (0.4, 0.3, 0.35, 0.45) .

Let $D_{ed - TS}^{(1)} = {a_{2}, a_{4}, a_{5}, a_{7}} .$ Now, consider the set $A_{TS} ∖ D_{ed - TS}^{(1)} = {a_{1}, a_{3}, a_{6}, a_{8}}$ .

For each vertex $a_{i} \in A_{TS} \ D_{ed - TS}^{(1)}$ , there exists a vertex $a_{j} \in D_{ed - TS}^{(1)}$ such that:

1. $a_{i} a_{j} \in B_{TS}$ (i.e., $a_{i}$ is adjacent to $a_{j}$ ), and
2. The degree condition holds component-wise:
$| {deg}_{k^{(ai)}} - {deg}_{k^{(aj)}} | \leq 1, for all components k .$

Indeed:

• $a_{1}$ is adjacent to $a_{2}$ or $a_{4}$ ,
• $a_{3}$ is adjacent to $a_{2}$ ,
• $a_{6}$ is adjacent to $a_{5}$ ,
• $a_{8}$ is adjacent to $a_{7}$ , and in each case, the component-wise difference of degrees is less than or equal to 1. Hence, $D_{ed - TS}^{(1)} = {a_{2}, a_{4}, a_{5}, a_{7}}$ is an equitable dominating set of $G_{TS}$ .

Theorem 3.1.1

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph and let $D_{TS} \subseteq A_{TS}$ be a balanced equitable dominating set. Then $D_{TS}$ is a minimum equitable dominating set if and only if every vertex $v \in D_{TS}$ satisfies at least one of the following conditions:

1. There exists a vertex $u \in A_{TS} \ D_{TS}$ such that $v$ is the only vertex in $D_{TS}$ adjacent to $u$ , that is, $N (u) \cap D_{TS} = {v};$
2. The vertex $v$ is isolated within the induced subgraph on $D_{TS}$ , i.e., $N (v) \cap D_{TS} = \emptyset .$
(⇒) Assume that $D_{TS}$ is a minimum equitable dominating set.

Suppose to the contrary, that there exists a vertex $v \in D_{TS}$ which does not satisfy either of the stated conditions. Then:

$i . v$ is not the unique neighbor of any vertex in $A_{TS} \ D_{TS}$ , meaning every such vertex dominated by $v$ is also dominated by another vertex in $D_{TS}$ ; and

$ii . v$ is not isolated in the induced subgraph $⟨ D_{TS} ⟩$ , so it has at least one neighbor within $D_{TS}$ .

These observations imply that removing $v$ from $D_{TS}$ does not affect the domination of vertices outside the set and does not disrupt the internal structure required for equitable domination. Hence, the set $D_{TS}^{'} = D_{TS} \ {v}$ would still be an equitable dominating set. This contradicts the minimality of $D_{TS}$ . Therefore, every vertex in $D_{TS}$ must satisfy at least one of the given conditions.

(⇐) Conversely, assume that every vertex $v \in D_{TS}$ satisfies at least one of the conditions.

Suppose that $D_{TS}$ is not minimum. Then there exists a proper subset $D_{TS}^{'} \subset D_{TS}$ that is also an equitable dominating set. This means there exists a vertex $v \in D_{TS}$ such that $v \notin D_{TS}^{'}$ .

We consider two cases:

i. If $v$ uniquely dominates some vertex $u \in A_{TS} \ D_{TS}$ , then removing $v$ leaves $u$ undominated, contradicting the assumption that $D_{TS}^{'}$ is a dominating set.
ii. If $v$ is isolated in $⟨ D_{TS} ⟩$ , then it does not share its role with other vertices in $D_{TS}$ . Removing it would violate the conditions required for equitable domination.
In both cases, a contradiction arises. Therefore, no proper subset of $D_{TS}$ can be an equitable dominating set. Hence, $D_{TS}$ is a minimum equitable dominating set if and only if every vertex in $D_{TS}$ satisfies at least one of the stated conditions.

Theorem 3.1.2

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph. Then the domination number of $G_{TS}$ is less than or equal to its equitable domination number, that is, $γ_{TG} (G_{TS}) \leq γ_{ed - TG} (G_{TS}) .$

Proof:

Let $D_{ed}$ be a minimum equitable dominating set of $G_{TS}$ . Then, $| D_{ed} | = γ_{ed - TG} (G_{TS}) .$

Since every equitable dominating set is also a dominating set, it satisfies all domination requirements along with additional equitable conditions. Therefore, $D_{ed}$ is also a dominating set of $G_{TS}$ . However, it may not be the smallest dominating set, and let $D$ be a minimum dominating set of $G_{TS}$ , so that $| D | = γ_{TG} (G_{TS}) .$ Since $D$ is the smallest among all dominating sets, we must have $| D | \leq | D_{ed} | .$ Hence, $γ_{TG} (G_{TS}) \leq γ_{ed - TG} (G_{TS}) .$

Example 3.1.2:

Consider the diagram below and let $G_{TS} = (A_{TS}, B_{TS})$ be the $T_{G}$ shown in the below Figure 2.

Figure 2. Example of equitable domination of Turiyam graph.

As we have seen from Figure 2, above all arcs are strength arcs and the only possible balanced dominance set is $D_{ed} - Ts = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}}$ , since $| deg (a_{i}) - deg (aj) | \geq 1$ for each $j$ = 2, 3, 4 & 5.

Theorem 3.1.3

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph and let $D_{ed - TS} \subseteq A_{TS}$ be an equitable dominating set. Then there exist two vertices $u, v \in A_{TS} \ D_{ed - TS}$ such that every path between $u$ and $v$ contains at least one vertex from $D_{ed - TS}$ .

Proof:

Assume that $D_{ed - TS}$ is an equitable dominating set of $G_{TS}$ . Suppose, for contradiction, that for every pair of vertices $u, v \in A_{TS} \ D_{ed - TS}$ , there exists at least one path between $u$ and $v$ that does not contain any vertex from $D_{ed - TS}$ . This means that the subgraph induced by $A_{TS} \ D_{ed - TS}$ is path-connected, i.e., all vertices outside the dominating set are connected without involving any vertex from $D_{ed - TS}$ . Now consider any vertex $x \in A_{TS} \ D_{ed - TS}$ . Since $D_{ed - TS}$ is a dominating set, there exists a vertex $y \in D_{ed - TS}$ such that $x$ is adjacent to $y$ .

However, due to the assumed connectivity of $A_{TS} \ D_{ed - TS}$ , the vertices outside $D_{ed - TS}$ are sufficiently interconnected. This implies that the domination provided by some vertices in $D_{ed - TS}$ may become redundant, and it may be possible to remove at least one vertex from $D_{ed - TS}$ while still maintaining domination. This contradicts the defining property of an equitable dominating set, which requires a structured and balanced domination across the graph. Therefore, our assumption is false.

Hence, there must exist at least one pair of vertices $u, v \in A_{TS} \ D_{ed - TS}$ such that every path between them contains at least one vertex of $D_{ed - TS}$ .

Theorem 3.1.4

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph of order $p = | A_{TS} |$ . Then the equitable domination number of $G_{TS}$ satisfies the following bounds:

γ_{ef - TG} (G_{TS}) \leq p - Δ_{T} (G_{TS}),

γ_{ef - TG} (G_{TS}) \leq p - Δ_{I} (G_{TS}),

γ_{ef - TG} (G_{TS}) \leq p - Δ_{F} (G_{TS}),

γ_{ef - TG} (G_{TS}) \leq p - Δ_{L} (G_{TS}),

where

Δ_{T}, Δ_{I}, Δ_{F},

and

Δ_{L}

denote the maximum values of the respective degree components in

G_{TS}

.

Proof:

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph with $p = | A_{TS} |$ , and let $v \in A_{TS}$ be a vertex such that its degree attains the maximum in one of the components, say $Δ_{T} (G_{TS})$ .

By definition, $v$ is adjacent to at least $Δ_{T} (G_{TS})$ vertices (with respect to the $T$ -component). Consider the set $D = A_{TS} \ N (v),$ where $N (v)$ denotes the neighborhood of $v$ .Then:

i. All vertices in $N (v)$ are directly dominated by $v$ ,
ii. The remaining vertices belong to $D$ .

Thus, the set $D \cup {v}$ forms a dominating set of $G_{TS}$ . Now observe that: $| D \cup {v} | = p - Δ_{T} (G_{TS}) .$ Since the equitable domination number is defined as the minimum cardinality among all equitable dominating sets, we obtain $γ_{ef - TG} (G_{TS}) \leq p - Δ_{T} (G_{TS}) .$ Using the same argument for the other degree components $Δ_{I} (G_{TS})$ , $Δ_{F} (G_{TS})$ , and $Δ_{L} (G_{TS})$ , we similarly obtain: $γ_{ef - TG} (G_{TS}) \leq p - Δ_{I} (G_{TS}), γ_{ef - TG} (G_{TS}) \leq p - Δ_{F} (G_{TS}), γ_{ef - TG} (G_{TS}) \leq p - Δ_{L} (G_{TS}) .$

3.2 Strong and weak Equitable Domination in Turiyam Graph

It is more challenging to handle the concepts of strong and weak domination in Turiyam graphs because the degree of edge membership values is determined by the membership values of the incident vertices. To address this issue, which is analogous to the concept of strong and weak equitable domination, we employ an edge cardinality score function for each edge and a vertex cardinality score function for every vertex. In a Turiyam graph, the existence of both strong and weak equitable domination is determined by the degree of the Turiyam graph.

Definition 3.2.1

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph. The node score function and edge score function of $G_{TS}$ are defined as follows:

1. The node score function $v_{sf} : A_{TS} \to ℝ$ is given by
$v_{sf} (a) = \frac{3 + T_{A_{TS}} (a) - 0.5 I_{A_{TS}} (a) - F_{A_{TS}} (a) - L_{A_{TS}} (a)}{4}, \forall a \in A_{TS} .$
2. The edge score function $e_{sf} : B_{TS} \to ℝ$ is given by

e_{sf} (ab) = \frac{3 + T_{B_{TS}} (ab) - 0.5 I_{B_{TS}} (ab) - F_{B_{TS}} (ab) - L_{B_{TS}} (ab)}{4}, \forall ab \in B_{TS} .

Definition 3.2.2

The degree of the vertex $u$ , denoted by $deg (u)$ , is defined as the sum of the scores of all strong arcs incident to $u$ . That is, $deg (u) = \sum_{\begin{matrix} uv \in B_{TS} \\ uv is a strong arc \end{matrix}} e_{sf} (uv),$ where $e_{sf} (uv)$ denotes the edge score function of the arc $uv$ . The neighborhood of $u$ , denoted by $N (u)$ , is defined as $N (u) = {v \in A_{TS} | uv \in B_{TS} and uv is a strong arc} .$

The minimum degree and maximum degree of $G_{TS}$ are defined, respectively, as

δ (G_{TS}) = min {deg (u) | u \in A_{TS}},

Δ (G_{TS}) = max {deg (u) | u \in A_{TS}} .

Definition 3.2.3

The order of $G_{TS} = (A_{TS}, B_{TS})$ is the sum of the score function of node cardinality of each vertex and is represented by $G_{TS}$ . The size of $G_{TS} = (A_{TS}, B_{TS})$ is the total of the score of the function of edge cardinality of each edge.

Definition 3.2.4

Consider the Turiyam graph $G_{TS} = (A_{TS}, B_{TS})$ . If $deg (u_{d}) \geq deg (v_{d})$ and $u_{d}$ is a member of the equitable dominance set, we say that $u_{d}$ is extremely equitable and predominates $v_{d}$ for any $u_{d}, v_{d} \in A_{TS}$ .

Definition 3.2.5

Take the Turiyam graph $G_{TS} = (A_{TS}, B_{TS})$ into consideration. For each $u_{d}, v_{d} \in A_{TS}$ , we say that $u_{d}$ weakly equitable dominates $v_{d}$ if deg $(u_{d}) \leq$ deg $(v_{d})$ is a member of the equitable dominance set.

Definition 3.2.6

If there is at least one $u_{d} \in D_{ed - {Ts}^{S}}$ such that $u_{d}$ is extremely equitable and predominates $v_{d}$ for each $v_{d} \in A_{TS} - D_{ed - {TS}^{S}}$ , then the dominance set $D_{ed - {TS}^{S}}$ of $A_{TS}$ of a Turiyam graph $G_{TS} = (A_{TS}, B_{TS})$ is a strong equitable dominance set. As the minimum cardinality of a strong equitable dominating set, a strong equitable domination number is denoted as $γ_{ed - {TG}^{S} (G_{Ts})}$ .

Definition 3.2.7

If there is at least one $u_{d} \in D_{ed - {Ts}^{W}}$ such that $u_{d}$ weakly equitable dominates $v_{d}$ for each $v_{d} \in A_{TS} - D_{ed - {TS}^{W}}$ , then a dominance set $D_{ed - {TS}^{W}}$ of $A_{TS}$ of a Turiyam graph $G_{TS} = (A_{TS}, B_{TS})$ is a weak equitable dominance set. A weak equitable dominance number, denoted as $γ_{ed - {TG}^{W} (G_{Ts})}$ , is the minimal cardinality of a weak equitable dominance set.

Theorem 3.2.1

For every Turiyam graph $TG$ , $G_{TS} = (A_{TS}, B_{TS})$ , $γ_{ed - {TG}^{W} (G_{Ts})} \leq γ_{ed - {TG}^{S} (G_{Ts})}$ or $γ_{ed - {TG}^{W} (G_{Ts})} \geq γ_{ed - {TG}^{S} (G_{Ts})}$

Proof:

Let: $D_{S}$ be a minimum strong edge-dominating TG set of $G_{TS}$ , so $| D_{S} | = γ_{ed - TG}^{S} (G_{TS})$ .

Step 1: Relationship between strong and weak conditions

By definition: Every strong edge-dominating TG set satisfies stricter conditions than a weak edge-dominating TG set and hence, any set that is strong edge-dominating is also weak edge-dominating.

Therefore, $D_{S} is also a weak edge - dominating TG set .$

Step 2: Minimality argument

Since $D_{S}$ is a weak edge-dominating TG set, but not necessarily minimum for the weak case, we compare it with the minimum weak set.

Let: $D_{W}$ be a minimum weak edge-dominating TG set, so $| D_{W} | = γ_{ed - TG}^{W} (G_{TS})$ . By minimality of $D_{W}$ , $| D_{W} | \leq | D_{S} |$

Step 3: Substitute parameters $γ_{ed - TG}^{W} (G_{TS}) \leq γ_{ed - TG}^{S} (G_{TS})$ . Thus, for every Turiyam graph $G_{TS}$ , $γ_{ed - TG}^{W} (G_{TS}) \leq γ_{ed - TG}^{S} (G_{TS})$

Theorem 3.2.2:

For any Turiyam graph $TG$ , $G_{TS} = (A_{TS}, B_{TS})$

i. $γ ed - {TG}^{s} (G_{Ts}) \leq o (G_{Ts}) - ∆ (G_{Ts})$
ii. $γ ed - {TG}^{s} (G_{Ts}) \leq o (G_{Ts}) - δ (G_{Ts})$

Proof.

Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph. Since $\sum_{i = 1}^{n} d (v_{i}) \leq p$ , where nodes in a strong arc are represented by $v_{i}$ , but an equitable dominant set has fewer vertices than $n$ . Additionally, according to the concept of minimal strong equitable dominance set and $∆ (G_{Ts}),$ the maximum truth membership values of $v_{i}$ among all of the nodes of $(G_{Ts})$ , we have $γ ed - T_{G} ({G_{Ts}}^{s}) \leq \sum_{i = 1}^{n} d (v_{i}) \leq p - ∆ (G_{Ts})$ . Similarly, we prove $γ ed - T_{G} ({G_{Ts}}^{s}) \leq p - δ (G_{Ts})$ .

Theorem 3.2.3:

For any Turiyam graph $TG$ , $G_{TS} = (A_{TS}, B_{TS})$

i. $γ_{ed - {TG}^{W} (G_{Ts})} \leq O (G_{Ts}) - ∆ (G_{Ts})$
ii. $γ_{ed - {TG}^{W} (G_{Ts})} \leq O (G_{Ts}) - δ (G_{Ts})$

Proof:

Follows from the above theorem

Theorem 3.2.4:

For any Turiyam graph $TG$ , $G_{TS} = (A_{TS}, B_{TS})$

i. $γ_{ed - {TG}^{s} (G_{Ts})} \geq O (G_{Ts}) / (∆ (G_{Ts}) + 1)$
ii. $γ_{ed - {TG}^{W} (G_{Ts})} \geq O (G_{Ts}) / (∆ (G_{Ts}) + 1)$

Proof: Let $G_{TS} = (A_{TS}, B_{TS})$ be a Turiyam graph and its strong equitable dominance number be $γ_{ed - {TG}^{s} (G_{Ts})}$ .

| O (G_{Ts}) - γ_{ed} - {TG}^{s} (G_{Ts}) | \leq \sum_{i = 1}^{n} d (v_{i}) \leq γ_{ed - {TG}^{s} (G_{Ts})} ∆ (G_{Ts}) \leq γ_{ed - {TG}^{s} (G_{Ts})} (∆ (G_{Ts}) + 1)

Hence, $γ_{ed - {TG}^{s} (G_{Ts})} \geq O (G_{Ts}) / (∆ (G_{Ts}) + 1)$ . Similarly, $γ_{ed - {TG}^{W} (G_{Ts})} \geq O (G_{Ts}) / (∆ (G_{Ts}) + 1)$ .

3.3 Applications of Equitable Dominance of Turiyam graphs in Networks

In networks (such as communication, computer, social, or sensor networks), domination in Turiyam graphs is used to identify a minimal set of key nodes (dominators) that can efficiently monitor, control, or provide service to the rest of the network. Turiyam domination accounts for uncertain, indeterminate, and contradictory relations in networks (extending neutrosophic graphs). Equitable Turiyam domination adds a fairness criterion: all dominator nodes must cover nearly equal numbers of neighbors, balancing workload or service responsibility.

3.3.1 Communication Networks (Wi-Fi/LAN)

Turiyam domination: Selects the minimal number of access points (APs) so every device is within range of at least one AP, even if some connections are uncertain or fluctuating. While Equitable Turiyam domination: Ensures that no AP is overloaded; each AP covers approximately the same number of devices, preventing bottlenecks.

Example 1:

At a university campus, APs are installed in three buildings.

➢ Turiyam domination identifies two APs sufficient to cover the whole campus.
➢ Equitable domination redistributes load so each AP serves ~20–25 students instead of one AP covering 40 while another covers 5.

Key Insight: From the Table 1 and example $1$ , we understand that Turiyam domination is about efficiency (minimize resources) and Equitable Turiyam domination is about efficiency + fairness (minimize resources while balancing service). Both are critical: first ensures coverage, second ensures sustainability. Now by considering the below Figure 3, as a network connection between different buildings we can apply equitable dominating Turiyam graph as follows:

Figure 3 shows that a campus with three buildings and five network nodes A, B, C, D and E. Each node represents a device (Ap, hub or logical controller). Edges are labeled with weights between 0 and 1(these are the link “truth”/reliability values). The two blue nodes A and D are highlighted as the chosen equitable dominating set; they jointly cover the network while keeping the load roughly balanced between them. As represented in diagram above;

Building (icons): Spatial anchors (library, class and cafeteria buildings) useful for mapping nodes to physical region.

Blue Nodes (A, D): The dominators in the diagram (selected controllers).

Yellow Nodes (B, C, E): Regular/client nodes that need coverage.

Edge labels (numbers like 0.8, 0.6, 0.4, 0.7, 0.3 …), link reliability/truth membership $t_{uv}$ between node $u$ and $v$ closer 1 which is more reliable link.

Turiyam nodes value (not drawn in the picture) and each node also have four Turiyam components (t, i, f and l). To get domination strength we need to combine node and edge values.

A node $u$ dominates a node $v$ if the combined reliability of $u$ (its truth) and the link $u - v$ is high enough. In Turiyam graphs we account for uncertainty and refusal. So, we typically compute scholar domination strength $D (u \to v)$ and compare it to a threshold $θ$ . If $D (u \to v) \geq θ$ , then $u$ effectively covers $v$ . Equitable domination requires that the numbers of nodes assigned to each dominator be nearly equal (most formal definition: difference $\leq 1$ ). This avoids overloading a single controller and spreading responsibility more fairly. In diagram A and D were chosen because their coverage responsibilities are roughly balanced given the link strengths shown.

To compute domination strengths;

D (u \to v) = t_{u} . t_{uv} . (1 - f_{u}) . (1 - l_{u}) Were,

$t_{u} :$ Node $u$ ’s truth/reliability, $t_{uv} :$ link reliability between $u$ and $v$ (edge label on diagram)

$f_{u}, l_{u}$ : Node falsity and liberal/refusal (reduce effective domination)

If the diagram show edge $A - B$ labeled $0.4$ and if node $A$ has $t_{A} = 0.95, f_{A} = 0.02, l_{A} = 0.1$ then, $D (A \to B) = 0.95 \times 0.4 \times (1 - 0.02) \times (1 - 0.01) \approx 0.369$ . If we use a threshold $θ = 0.35$ that value $\approx 0.369$ means $A$ dominates $B$ . If $θ = 0.4$ it would not. Also, we can use the same computation for every ordered pair $u \to v$ that has a non-zero link to form the domination matrix.

Now to verify $A, D$ as equitable dominating set we use the following algorithms:

Step 1: Pick a threshold $θ$

Decide how conservative we want coverage to be (example $θ = 0.35$ is moderate, $θ = 0.2$ is permissive).

Step 2: Assemble Turiyam inputs

If we already have node (t, i, f, l) values from measurement) and we read link $t_{uv}$ from the diagram’s edge/label’s

Step 3: Compute $D (u \to v)$ for each pair $u, v$ use the formula above that yields a numeric matrix (rows dominators, column targets).

Step 4: Build coverage sets $N_{θ} (u) = {(u : D (u \to v) \geq θ}$ . For each candidate dominator $u,$ list which nodes it reliably covers at $θ$ .

Step 5: Check domination (Coverage union)

A proposed dominating set $S$ (here $= {A, D})$ is valid if $U_{u \in S} N_{θ} (u) = V$

Step 6: Check equitability

Compute coverage size $C_{u} = | N_{θ} (u) \cap (node assigned to u) |$ . For a simple check assign each covered node to the dominator with the highest.

Step 7: Refine if needed

If {A, D} covers everything but is not equitable, try adding equitable a third dominator (e.g C or B) or lowering/raising $θ$ depending on your priorities (fairness versus fewer dominators).

Table 1. Comparison between Turiyam domination and Equitable domination.

Aspect	Turiyam domination	Equitable Turiyam domination
Objective	Minimal dominators	Minimal + balanced dominators
Load distribution	May be unbalanced	Balanced across dominators
Network lifetime (sensors)	Shorter if hubs overloaded	Longer due to fairness
Communication quality	Coverage assured	Coverage + fairness assured
Application	Minimal resource use	Resource efficiency + fairness

Figure 3. Application of equitable domination in Turiyam graph.

Summary of the diagram

Node $A$ is located near the library building and has medium-strength links to nodes $B$ and $D$ . It likely covers ( $A, B$ , and possibly $C / D$ ) at a moderate $θ$ . Node $D$ is located near the cafeteria building and connects to nodes $A$ , $C$ , and $E$ . It likely covers ( $D$ , and possibly $C$ and $E$ ) at a moderate $θ$ . Together, nodes $A$ and $D$ are positioned so that the union of their high-confidence coverage reaches all five nodes. Because each serve approximately half of the clients $A$ covering the left/central clients and $D$ covering the right/central clients-the workload is balanced. This balanced distribution reflects the equitable property demonstrated in the picture.

3.3.2 Equitable Domination in Social Networks

A Turiyam Graph $(TG)$ model’s social networks more realistically than classical or even neutrosophic graphs, because it represents truth $(T),$ indeterminacy ( $I)$ , falsity ( $F),$ and latent $(L)$ components of relationships. Each connection (friendship, influence, or communication) between two individuals $(u, v)$ is defined as: $T (u, v) = (t_{uv}, i_{uv}, f_{uv}, l_{uv})$ , Where:

$t_{uv}$ : Strength or reliability of connection (e.g., consistent interaction).

$i_{uv}$ : Uncertainty in the relationship (e.g., inconsistent communication).

$f_{uv}$ : Contradiction (e.g., misinformation, conflict).

$l_{uv}$ : Latent potential (e.g., shared background or unseen similarity).

In a social network, a dominating set ( $D$ ) is a group of leaders or influencers who can reach or influence all members of the network. A Turiyam dominating set accounts for uncertainty in connections, selecting minimal influencers under varying levels of trust or indeterminacy. An equitable dominating set (EDS) ensures that influence responsibility is fairly distributed, and no single leader dominates excessively or becomes overburdened.

Formally: $| N_{T} (u) | - | N_{T} (v) | \leq 1$ for all $, v \in$ $D$ . Meaning, each influencer reaches approximately the same number of followers with similar influence strength.

Example 2:

Consider the Scenario -Online Student Community

Setting: A university has a student social media platform connecting students across departments. There are five main representatives or leaders:

A (Admin), B (Tech Club Head), C (Sports Leader), D (Cultural Club Head), and E (General Member). Their influence strengths vary due to student interests, department ties, and posting frequency. Consider the diagram below to illustrate the example.

From Figure 4, we have the weights between each edge vary from one another and it indicates the relationship between everyone. These relationships can also be represented as Table 2.

Figure 4. Turiyam connection diagram.

Table 2. Turiyam connection values.

Edge	$(t, i, f, l)$	Description
$A - B$	$(0.8, 0.10, 0.05, 0.05)$	Strong admin-tech coordination
$A - C$	$(0.7, 0.15, 0.10, 0.05)$	Moderate link via events
$A - D$	$(0.9, 0.05, 0.03, 0.02)$	Very strong link
$B - E$	$(0.6, 0.25, 0.10, 0.05)$	Medium student engagement
$C - E$	$(0.5, 0.30, 0.15, 0.05)$	Weak indirect link
$D - E$	$(0.7, 0.20, 0.05, 0.05)$	Cultural connection strong

From Table 2, we observe that A and D have very strong links, and we can see the domination and its equitable domination analysis step by step:

Step 1: Domination $(θ = 0.5)$ . Nodes dominate others if $(t_{uv} \geq 0.5)$ and $(f_{uv} \leq 0.1) .$

$A$ dominates ${B, C, D}$ , $B$ dominates ${A, E}$ , $D$ dominates ${A, E}$ , $E$ is dominated by ${B, D}$ , and $C$ only dominated by ${A}$ . Hence, dominating set = {A, D} (all nodes covered).

Step 2: Check Equity

Being dominator $A$ has influenced members ${B, C, D}$ which count 3, and dominator $D$ has influenced members ${A, E}$ which count 2. Therefore, the difference $3 - 2 = 1$ → Equitable domination achieved. In this community network; Admin (A) and Cultural Leader (D) form an equitable dominating pair. Both cover complementary subgroups: technical, sports, and cultural clusters. Their influence load differs only by one node which means fair, sustainable community leadership. The benefits of this domination include, preventing one leader (Admin) from overloading communication tasks, each subgroup (sports, tech, and culture) is equally represented, and balanced influence promotes fair participation and diversity. Additionally, Turiyam graph is superior here because of; It models uncertain, fuzzy, and contradictory social ties, Latent factors (L) represent potential friendships or shared goals not yet active and Equitable domination ensures information diffusion fairness even under uncertainty.

The mathematical intuition of Example $(2)$ can be explained by the domination strength from $A$ to $B$ . $D_{T} (A, B) = t_{A} \times t_{AB} \times (1 - f_{A}) \times (1 - l_{A})$

Equitable domination optimizes the set $(D)$ such that; Max $\sum_{v} D_{T} (u, v) -$ min $\sum_{v} D_{T} (u, v) \leq 1$ , ensuring balanced influence across all dominators.

Real-world applications of equitable domination in Turiyam graphs are simplified and presented in Table 3. From Figure 4 and Table 3, the model of Turiyam graph captures truth, uncertainty, falsity, latent potential and its goal is balanced social influence via equitable domination to reduced overload, increased fairness and better engagement. The network shows student leaders forming balanced influencer coverage and the result impact on sustainable and inclusive community communication.

Table 3. Equitable domination impact of Example 2.

Context	Description	Equitable domination impact
University Platforms	Student group leaders managing posts and forums	Avoid single-user burnout, equal content reach
Online Learning Communities	Mentors guiding study groups	Fair mentor workload
NGO Volunteer Networks	Coordinators leading campaigns	Equal engagement levels
Corporate Teams (Slack/Teams)	Managers or moderators	Fair communication load and reduced bias

4. Conclusion

The concept of equitable dominance in Turiyam graphs offers a powerful framework for analyzing and designing real-world networks under conditions of uncertainty, contradiction, and partial truth. Unlike classical domination, which focuses solely on minimizing the number of dominators, equitable domination ensures that the responsibility of domination is fairly balanced across the selected nodes. Overall, equitable domination in Turiyam graphs represents a next-generation approach to network optimization-ensuring coverage, fairness, and sustainability simultaneously. It bridges theoretical graph models with practical engineering needs, making it particularly valuable for modern complex systems where both efficiency and equity are essential.

Data availability statement

This study is theoretical and mathematical in nature, no experimental or observational datasets were generated or analyzed during the study.

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Author details Author details

¹ Mathematics, Wollega University, Nekemte, Oromia, 395, Ethiopia
² Mathematics, Wollega University, Nekemte, Oromia, Ethiopia
³ Mathematics, Wollega University, Nekemte, Oromia, 395, Ethiopia

Abdata Guluma Erana
Roles: Conceptualization, Methodology, Resources, Validation, Writing – Original Draft Preparation

V N SrinavasaRao Repalle
Roles: Formal Analysis, Methodology, Supervision, Writing – Review & Editing

Fekadu Tesgara Agama
Roles: Formal Analysis, Investigation, Supervision, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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https://doi.org/10.12688/f1000research.181882.1

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Erana AG, Repalle VNS and Agama FT. Equitable Domination in Turiyam Graphs with Network Applications [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:849 (https://doi.org/10.12688/f1000research.181882.1)

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