Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

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Intuitionistic Fuzzy Threshold Hypergraphs and Their Role in Chasing 

Fugitives with Multi-Bots 

 

Myithili.K.K1, Nandhini.C 2 

1Associate Professor & Head, Department of Mathematics(CA), Vellalar College for Women, Erode-638012, 

Tamilnadu, India, mathsmyth@gmail.com 

2Research Scholar, Department of Mathematics, Vellalar College for Women, Erode-638012, Tamilnadu, India, 

c.nandhini@vcw.ac.in 

 

Article History: 

Received: 15-04-2024 

Revised: 03-06-2024 

Accepted: 22-06-2024 

Abstract 

In this paper, Intuitionistic Fuzzy Threshold Hypergraph (IFTHG) is described with 

some definitions, such as adjacency level, strength, walk, hyperpath, score values, 

connected and disconnected IFTHGs. IFTHGs are essential for modeling complex 

relationships and uncertainties in emergency response scenarios within crowed areas. 

Furthermore, a novel method for capturing fugitives using IFTHG model is 

demonstrated. The proposed system initializes robots and implements a step-by-step 

algorithm upon detecting any intrusion, ultimately determining the nearest robot to 

capture the fugitives. 

Keywords: Intuitionistic fuzzy threshold hypergraph, multi robots, algorithm, fugitive 

chase. 

2000 Mathematics Subject Classification: 05C65 

 

1. INTRODUCTION  

To deal with the complexities of application base, the graph concept was expanded to 

provide a hypergraph, which is a set of all vertices including a collection of V subsets. The 

notions of hypergraph was introduced by Berge [5]. Chvatal and Hammer were the first to 

introduce threshold graphs(1973) in [6]. 

In set theory, Zadeh[13] created fuzzy sets as a technique of conveying ambiguity and 

vagueness. Fuzzy set theory has sparked attention to variety of fields. Atanassov[1, 3] came up 

with of Intuitionistic Fuzzy Sets(IFS) concept as a generalization of fuzzy sets & Atanassov 

added an additional module to fuzzy set(which specifies the degree of non-membership). The 

concept behind intuitionistic fuzzy relations and graphs were discussed in [2, 4]. 

Intuitionistic Fuzzy Graphs(IFGs), Intuitionistic Fuzzy Hypergraphs(IFHGs) and 

Intuitionistic Fuzzy Directed Hypergraphs(IFDHG) have been introduced in [9, 10, 11]. 

Some types of IFDHGs are discussed in [8]. A novel decision-making approach based on 

hypergraphs in IF environment has been discussed in [7]. Lanzhen Yang and Hua Mao [12] 

introduced intuitionistic fuzzy threshold graph and explained its applications. 

In this research paper, Section 2 elaborates on fundamental definitions, while Section 3 

includes mathematical definitions of IFTHG and its related extensions. Furthermore, Section 4 

outlines the implementation of IFTHG for fugitive determination, offering a discussion on 

simulation results and examples. 

mailto:mathsmyth@gmail.com
mailto:c.nandhini@vcw.ac.in


Communications on Applied Nonlinear Analysis 

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2. PRELIMINARIES 

In this section, we go through certain important concepts that related to our main concept. 

Definition 2.1. [3] Let a set 𝐸 be fixed. An Intuitionistic Fuzzy Set (IFS) 𝑈 in 𝐸 is an 

object of the form 𝑈 = {< 𝑢𝑖, 𝜇𝑖(𝑢𝑖), 𝜈𝑖(𝑢𝑖) > |𝑢𝑖 ∈ 𝐸}, where the function 𝜇𝑖: 𝐸 → [0,1] 

and 𝜈𝑖: 𝐸 → [0,1] determine the degree of membership & the degree of non-membership of the 

element 𝑢𝑖 ∈ 𝐸, respectively and for every 𝑢𝑖 ∈ 𝐸, 0 ≤ 𝜇𝑖(𝑢𝑖) + 𝜈𝑖(𝑢𝑖) ≤ 1. 

Definition 2.2. [3] Let 𝐸 be the fixed set and 𝑈 = {< 𝑢𝑖 , 𝜇𝑖(𝑢𝑖), 𝜈𝑖(𝑢𝑖) > |𝑢𝑖 ∈ 𝐸}, be 

an IFS. Six types of Cartesian products of 𝑛 subsets (crisp sets) 𝑈1, 𝑈2, … ,𝑈𝑛 of 𝑈 over 

𝐸 are defined as follows 

𝑈𝑖1 ×1 𝑈𝑖2 ×1 𝑈𝑖3 ×1 …×1 𝑈𝑖𝑛

= {〈(𝑢1, 𝑢2, … , 𝑢𝑛),∏𝜇𝑖 ,

𝑛

𝑖=1

∏𝜈𝑖

𝑛

𝑖=1

〉 |𝑢1 ∈ 𝑈1, 𝑢2 ∈ 𝑈2, … , 𝑢𝑛 ∈ 𝑈𝑛} 

𝑈𝑖1 ×2 𝑈𝑖2 ×2 𝑈𝑖3 ×2 …×2 𝑈𝑖𝑛

= {〈(𝑢1, 𝑢2, … , 𝑢𝑛),∑𝜇𝑖

𝑛

𝑖=1

−∑𝜇𝑖𝜇𝑗 + ∑ 𝜇𝑖𝜇𝑗𝜇𝑘
𝑖≠𝑗≠𝑘

−⋯+ (−1)𝑛−2

𝑖≠𝑗

∑ 𝜇𝑖𝜇𝑗𝜇𝑘…𝜇𝑛
𝑖≠𝑗≠𝑘…≠𝑛

+ (−1)𝑛−1∏𝜇𝑖 ,

𝑛

𝑖=1

∏𝜈𝑖

𝑛

𝑖=1

〉 |𝑢1 ∈ 𝑈1, 𝑢2 ∈ 𝑈2, … , 𝑢𝑛 ∈ 𝑈𝑛} 

𝑈𝑖1 ×3 𝑈𝑖2 ×3 𝑈𝑖3 ×3 …×3 𝑈𝑖𝑛

= {〈(𝑢1, 𝑢2, … , 𝑢𝑛),∏𝜇𝑖 ,

𝑛

𝑖=1

∑𝜈𝑖

𝑛

𝑖=1

−∑𝜈𝑖𝜈𝑗 + ∑ 𝜈𝑖𝜈𝑗𝜈𝑘
𝑖≠𝑗≠𝑘

−⋯+ (−1)𝑛−2

𝑖≠𝑗

∑ 𝜈𝑖𝜈𝑗𝜈𝑘…𝜈𝑛
𝑖≠𝑗≠𝑘…≠𝑛

+ (−1)𝑛−1∏𝜈𝑖

𝑛

𝑖=1

〉 |𝑢1 ∈ 𝑈1, 𝑢2 ∈ 𝑈2, … , 𝑢𝑛 ∈ 𝑈𝑛} 

𝑈𝑖1 ×4 𝑈𝑖2 ×4 𝑈𝑖3 ×4 …×4 𝑈𝑖𝑛
= {〈(𝑢1, 𝑢2, … , 𝑢𝑛),𝑚𝑖 𝑛(𝜇1, 𝜇2, … , 𝜇𝑛) ,𝑚𝑎𝑥⁡(𝜈1, 𝜈2, … , 𝜈𝑛)〉|𝑢1 ∈ 𝑈1, 𝑢2
∈ 𝑈2, … , 𝑢𝑛 ∈ 𝑈𝑛} 

𝑈𝑖1 ×5 𝑈𝑖2 ×5 𝑈𝑖3 ×5 …×5 𝑈𝑖𝑛
= {〈(𝑢1, 𝑢2, … , 𝑢𝑛),𝑚𝑎𝑥(𝜇1, 𝜇2, … , 𝜇𝑛) ,min(𝜈1, 𝜈2, … , 𝜈𝑛)〉|𝑢1 ∈ 𝑈1, 𝑢2
∈ 𝑈2, … , 𝑢𝑛 ∈ 𝑈𝑛} 

 



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𝑈𝑖1 ×6 𝑈𝑖2 ×6 𝑈𝑖3 ×6 …×6 𝑈𝑖𝑛

= {〈(𝑢1, 𝑢2, … , 𝑢𝑛),
∑ 𝜇𝑖
𝑛
𝑖=1

𝑛
,
∑ 𝜈𝑖
𝑛
𝑖=1

𝑛
〉 |𝑢1 ∈ 𝑈1, 𝑢2 ∈ 𝑈2, … , 𝑢𝑛 ∈ 𝑈𝑛} 

It must be noted that 𝑢𝑖 ×𝑠 𝑢𝑗 is an IFS, where 𝑠 = 1,2,3,4,5,6. 

Definition 2.3. [9] An Intuitionistic Fuzzy Graph (IFG) is of the form 𝐺 = (𝑈, 𝐸), 

where 

(i) 𝑈 = {𝑢1, 𝑢2, … , 𝑢𝑛} such that 𝜇𝑖: 𝑈 → [0,1] and 𝜈𝑖: 𝑈 → [0,1] denote the degrees of 

membership & non-membership of the element 𝑢𝑖 ∈ 𝑈 respectively and 0 ≤ 𝜇𝑖(𝑢𝑖) +

𝜈𝑖(𝑢𝑖) ≤ 1 for every 𝑢𝑖 ∈ 𝑈, 𝑖 = 1,2,… ,𝑛. 

(ii) 𝐸 ⊆ 𝑈×𝑈 where 𝜇𝑖𝑗: 𝑈 × 𝑈 → [0,1] and 𝜈𝑖𝑗: 𝑈 × 𝑈 → [0,1] are such that 𝜇𝑖𝑗 ≤

𝜇𝑖 ∧ 𝜇𝑗, 𝜈𝑖𝑗 ≤ 𝜈𝑖 ∨ 𝜈𝑗 and 0 ≤ 𝜇𝑖(𝑢𝑖) + 𝜈𝑖(𝑢𝑖) ≤ 1, where  𝜇𝑖𝑗 and 𝜈𝑖𝑗 are the 

membership & non-membership values of the edge (𝜈𝑖, 𝜈𝑗); the values of 𝜇𝑖 ∧ 𝜇𝑗 and 𝜈𝑖 ∨

𝜈𝑗  can be determined by any one of the cartesian products ×𝑠 where 𝑠 = 1,2,3,4,5,6, 

∀⁡𝑖⁡&⁡𝑗  given in Definition 2.2. 

Note: Throughout this paper, it is assumed that the fourth Cartesian product 

𝑈𝑖1 ×4 𝑈𝑖2 ×4 𝑈𝑖3 ×4 …×4 𝑈𝑖𝑛 = {〈(𝑢1, 𝑢2, … , 𝑢𝑛), 𝑚𝑖 𝑛(𝜇1, 𝜇2, … , 𝜇𝑛) ,

𝑚𝑎𝑥⁡(𝜈1, 𝜈2, … , 𝜈𝑛)〉|⁡𝑢1 ∈ 𝑈1, 𝑢2 ∈ 𝑈2, … , 𝑢𝑛 ∈ 𝑈𝑛}, is used to determine the edge membership 

𝜇𝑖𝑗  and the edge non-membership 𝜈𝑖𝑗. 

Definition 2.4. [10] An Intuitionistic Fuzzy Hypergraph(IFHG) is an ordered pair 𝐻 =

(𝑈,𝐸) where 

(i) 𝑈 = {𝑢1, 𝑢2, … , 𝑢𝑛,}, is a finite set of IF vertices, 

(ii) 𝐸 = {𝐸1, 𝐸2, … , 𝐸𝑚}  is a family of crisp subsets of 𝑈 , 

(iii) 𝐸𝑗 = {𝑢𝑖, 𝜇𝑗(𝑢𝑖), 𝜈𝑗(𝑢𝑖)|0 ≤ 𝜇𝑗(𝑢𝑖) + 𝜈𝑗(𝑢𝑖) ≤ 1}, 𝑗 = 1,2, … ,𝑚 

(iv) 𝐸𝑗 ≠ ∅, 𝑗 = 1,2, … ,𝑚 

(v)  ⋃ supp(𝐸𝑗) = 𝑈,𝑗 ⁡𝑗 = 1,2, … ,𝑚 

Here, the hyperedges 𝐸𝑗 are crisp sets of IF vertices, 𝜇𝑗(𝑢𝑖)⁡&⁡𝜈𝑗(𝑢𝑖)  denotes the degrees of 

membership & non-membership of vertex 𝑢𝑖 to hyperedge 𝐸𝑗. 

Notations - list[8] 

• < 𝜇(𝑢𝑖), 𝜈(𝑢𝑖) > or simply < 𝜇𝑖, 𝜈𝑖 > denote the degrees of membership & non-

membership of the vertex 𝑢𝑖 ∈ 𝑈 such that 0 ≤ 𝜇𝑖 + 𝜈𝑖 ≤ 1. 

• < 𝜇(𝑢𝑗), 𝜈(𝑢𝑗) > or simply < 𝜇𝑗 , 𝜈𝑗 > denote the degrees of membership & non-

membership of the hyperedge (𝑢𝑖 , 𝑢𝑗) ∈ 𝑈 ×𝑈, such that 0 ≤ 𝜇𝑗 + 𝜈𝑗 ≤ 1. 

• 𝜇𝑖𝑗 and 𝜈𝑖𝑗 are the membership & non-membership value of 𝑖𝑡ℎ vertex in 𝑗𝑡ℎ 

hyperedge. 

•  Support of an IFS 𝑈 in 𝐸 is denoted by 𝑠𝑢𝑝𝑝(𝐸𝑗) = {𝑢𝑖│𝜇𝑗(𝑢𝑖) >

0⁡&⁡𝜈𝑗(𝑢𝑖) > 0}. 



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3. Intuitionistic Fuzzy Threshold Hypergraph 

Definition 3.1. The Intuitionistic Fuzzy Threshold Hypergraph(IFTHG) is defined as 

ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) where, 

(i) 𝑈 = {𝑢1, 𝑢2, … , 𝑢𝑛,} is a finite set of IF vertices 

(ii) ℰ = {ℰ1, ℰ2, … , ℰ𝑚} is a family of crisp subsets of 𝑈  

(iii) ℰ𝑗 = {𝑢𝑖 , 𝜇𝑗(𝑢𝑖), 𝜈𝑗(𝑢𝑖)|0 ≤ 𝜇𝑗(𝑢𝑖) + 𝜈𝑗(𝑢𝑖) ≤ 1}, 𝑗 = 1,2, … ,𝑚 

(iv) ℰ𝑗 ≠ ∅, 𝑗 = 1,2, … ,𝑚 

(v) ⋃ supp(ℰ𝑗) = 𝑈,𝑗 ⁡𝑗 = 1,2, … ,𝑚 

(vi) an independent set 𝑉 ⊆ 𝑈 has a set of all distinct combinations of a non-adjacent 

vertices in ℍ𝔾 iff there exists a threshold values 𝑠1⁡&⁡𝑠2 > 0 such that 

∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 − 𝜈𝑗(𝑢𝑖)) ≤ 𝑠2𝑢𝑖∈𝑉

. 

 

Example 

Consider an IFTHG ℍ𝔾 = (𝑈, ℰ; 0.4,0.6) with 𝑈 = {𝑢1, 𝑢2, … , 𝑢7,}, ℰ = {ℰ1, ℰ2, ℰ3, ℰ4}. 

 

 
Fig 3.1: Intuitionistic Fuzzy Threshold Hypergraph HG 

Adjacency matrix of the above IFTHG is represented as follows: 

 
Note: Intuitionistic fuzzy hypergraphs is a special case of the intuitionistic fuzzy threshold 

hypergraphs. 



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Definition 3.2. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG. The adjacency level between two 

vertices 𝑢𝑖 and 𝑢𝑖+1, denoted by 𝛾(𝑢𝑖, 𝑢𝑖+1), is defined by 

⁡𝛾(𝑢𝑖, 𝑢𝑖+1) = max
𝑗
(min(𝜇𝑗(𝑢𝑖)) , (𝜇𝑗(𝑢𝑖+1))),min

𝑗
(max (𝜈𝑗(𝑢𝑖)) , (𝜈𝑗(𝑢𝑖+1))), 

where 𝑖 = 1,2,… ,𝑛⁡&⁡𝑗 = 1,2,… ,𝑚 for which ∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 − 𝜈𝑗(𝑢𝑖)) ≤𝑢𝑖∈𝑉

𝑠2. 

Definition 3.3. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG. The adjacency level between the 

hyperedges ℰ𝑗 and ℰ𝑘, denoted by 𝜎(ℰ𝑗, ℰ𝑘), is defined by 

𝜎(ℰ𝑗, ℰ𝑘) = max
𝑗
(min (𝜇𝑗(𝑢𝑖)) , 𝜇𝑘(𝑢𝑖)),min

𝑗
(max (𝜈𝑗(𝑢𝑖)) , 𝜈𝑘(𝑢𝑖)), 

where 𝑖 = 1,2,… ,𝑛⁡&⁡𝑗, 𝑘 = 1,2,… ,𝑚 for which ∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 −𝑢𝑖∈𝑉

𝜈𝑗(𝑢𝑖)) ≤ 𝑠2. 

Definition 3.4. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2)  be an IFTHG. The Strength 𝛿 of a hyperedge ℰ𝑗 is the 

minimum membership 𝜇𝑗(𝑢𝑖) and maximum non- membership 𝜈𝑗(𝑢𝑖) of vertices in the 

hyperedge ℰ𝑗. Then, 

𝛿(ℰ𝑗) = (min
𝑢𝑖

( 𝜇𝑗(𝑢𝑖)),max
𝑢𝑖

( 𝜈𝑗(𝑢𝑖)) 

for every 𝜇𝑗(𝑢𝑖) > 0, 𝜈𝑗(𝑢𝑖) > 0 for which ∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 − 𝜈𝑗(𝑢𝑖)) ≤𝑢𝑖∈𝑉

𝑠2. 

Definition 3.5. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG. Then the hyper walk is a sequence of 

vertices 𝑢1, 𝑢2, … , 𝑢𝑛, not necessarily distinct, if at least one of the 𝜇𝑗(𝑢𝑖 , 𝑢𝑖+1)⁡&⁡𝜈𝑗(𝑢𝑖 , 𝑢𝑖+1) are 

different from zero, for which ∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 − 𝜈𝑗(𝑢𝑖)) ≤ 𝑠2𝑢𝑖∈𝑉

. 

Definition 3.6. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG. Then the intuitionistic fuzzy threshold 

hyperpath 𝒫 of length 𝑘 in an IFTHG is defined as a sequence say, 𝑢1⁡, ℰ1⁡⁡, 𝑢2⁡⁡, ℰ2⁡, … , 𝑢𝑘, ℰ𝑘, 𝑢𝑘+1 

of distinct vertices 𝑢𝑖′s  and hyperedges ℰ𝑗′s such that 

(i) 𝜇𝑗(ℰ𝑗) > 0, for all 1 ≤ 𝑗 ≤ 𝑘 

(ii) 𝑢𝑖 , 𝑢𝑖+1 ∈ ℰ𝑗, for all 1 ≤ 𝑗 ≤ 𝑘 

for which ∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 − 𝜈𝑗(𝑢𝑖)) ≤ 𝑠2𝑢𝑖∈𝑉

. 

Definition 3.7. Consider  ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG on a non-empty set 𝑈 is 

connected if every two distinct vertices in ℍ𝔾 are linked by an intuitionistic fuzzy 

threshold hyperpath for which ∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 − 𝜈𝑗(𝑢𝑖)) ≤ 𝑠2𝑢𝑖∈𝑉

. Otherwise, ℍ𝔾 is 

disconnected. 

Definition 3.8. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG. Then the score value of a vertex 𝑢𝑖 

and hyperedge  ℰ𝑗 is denoted as 𝒮(𝑢𝑖) =
1−𝜈𝑖(𝑢𝑖)

2−𝜇𝑖(𝑢𝑖)−𝜈𝑖(𝑢𝑖)
 & 𝒮(ℰ𝑗) =

1−𝜈𝑗(ℰ𝑗)

2−𝜇𝑗(ℰ𝑗)−𝜈𝑗(ℰ𝑗)
  respectively, for 

which ∑ 𝜇𝑗(𝑢𝑖) ≤ 𝑠1⁡𝑢𝑖∈𝑉
 &  ∑ (1 − 𝜈𝑗(𝑢𝑖)) ≤ 𝑠2𝑢𝑖∈𝑉

. 



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Theorem 3.1. If 𝑑𝑢𝑣 ≤ 𝑠1 and (1 − 𝑑𝑢𝑣) ≤ 𝑠2, ∀𝑢, 𝑣 ∈ ℰ in the IFTHG, then the IFTHG is 

connected. 

Proof. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG, where 𝑈 represents the set of all vertices 

𝑢1, 𝑢2, … , 𝑢𝑛 and ℰ represents the set of all hyperedges ℰ1, ℰ2, … , ℰ𝑛. Let 𝑑𝑢𝑣 denote the distance 

between any two connected vertices u and v and let 𝑠1, 𝑠2 denotes the threshold values. 

Assume two arbitrary vertices u and v which are connected in the IFTHG.  ∃ a path 𝑃: 𝑢 =

𝑢1⁡ℰ1⁡𝑢2⁡ℰ2…𝑢𝑚⁡ℰ𝑚⁡𝑢𝑚+1 between any 2 vertices u & v in the IFTHG with the condition 

that the distance 𝑑𝑢𝑣 between u & v must satisfy the condition 𝑑𝑢𝑣 ≤ 𝑠1 and (1 − 𝑑𝑢𝑣) ≤ 𝑠2. 

Since all the hyperedges are connected with each other, it is proved that the IFTHG is connected.  

 

Theorem 3.2. Let ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be the IFTHG. Then ℍ𝔾 is disconnected iff every non-

empty subsets 𝑈1, 𝑈2 of 𝑈 such that 𝑈1 ∪ 𝑈2 = 𝑈,𝑈1 ∩𝑈2 = ∅ and there is no hyperedge 

ℰ𝑗 ∈ ℰ which has one vertex in 𝑈1 and another vertex in 𝑈2. 

Proof. Suppose that ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) is a disconnected IFTHG. Then ∃ a vertices 𝑢, 𝑣⁡𝜖⁡𝑈 

such that there is no path between u and v. 

Let 𝑈𝑢 = {𝑢 ∈ 𝑈|∃⁡a⁡path⁡between⁡𝑢⁡&⁡𝑣}. 

Then, clearly 𝑈𝑢 ≠ ∅.  

Let 𝑈𝑣 = 𝑈 −𝑈𝑢.  Since 𝑣 ∉ 𝑈𝑢 ⟹ 𝑣 ∈ 𝑈𝑣. 

Now, 𝑈𝑢 ≠ ∅,𝑈𝑣 ≠ ∅ and 𝑈𝑢 ∪ 𝑈𝑣 = 𝑈.(also 𝑈𝑢 ∩𝑈𝑣 = ∅). 

Suppose there exists a hyperedge ℰ𝑗 = {𝑢1, 𝑢2} ∈ ℰ such that 𝑢1 ∈ 𝑈𝑢, 𝑢2 ∈ 𝑈𝑣. 

Now 𝑢1 and 𝑢 are connected as ℰ𝑗 = {𝑢1, 𝑢2} ⟹ 𝑢⁡and⁡𝑢2⁡ are also 

disconnected. 

⇒ 𝑢2 ∈ 𝑈𝑢 

which is a contradiction. Hence the result holds good. 

Conversely, assume that ∃ a partition of 𝑈 such that 𝑈1 ∪𝑈2 = 𝑈,𝑈1 ∩ 𝑈2 = ∅, ⁡𝑈1 ≠

∅,𝑈2 ≠ ∅⁡and there exists no hyperedge ℰ𝑗 ∈ ℰ having one vertex in 𝑈1 and another 

vertex in 𝑈2. 

To prove: ℍ𝔾 is disconnected.  

Suppose ℍ𝔾 is connected. 

Take 𝑢𝜖𝑈1 ⊆ 𝑈,𝑣𝜖𝑈2 ⊆ 𝑈. 

⟹ 𝑢,𝑣 ∈ 𝑈1 ∪𝑈2 = 𝑈 [Since ℍ𝔾 is connected].  

Therefore, ∃ a path 𝑃: 𝑢 = 𝑢1ℰ1𝑢2ℰ2…𝑢𝑚ℰ𝑚𝑢𝑚+1 = 𝑣 connecting u and v. 

Since 𝑢𝜖𝑈1, 𝑣 ∈ 𝑈2 and 𝑈1 ∩ 𝑈2 = ∅. 

⟹ there exists an i such that 𝑢𝑖 ∈ 𝑈1, 𝑢𝑖+1 ∈ 𝑈2. Now ℰ = {𝑢𝑖 , 𝑢𝑖+1} ∈ ℰ such that one 

vertex is in 𝑈1 and another vertex is in 𝑈2. which is contradiction. 

⟹ ℍ𝔾 is disconnected IFTHG. 



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Theorem 3.3. Let ℍ𝔾 be an IFTHG, with vertices u, v ∈ U. There is a 𝑢 − 𝑣 hyper walk in 

ℍ𝔾 iff there exists a 𝑢 − 𝑣 hyperpath. 

Proof. Assume ℍ𝔾 = (𝑈, ℰ; 𝑠1, 𝑠2) be an IFTHG and W is a hyper walk in ℍ𝔾. The 

theorem is proved by using mathematical induction on length of W . If W be the length 1 or 2, 

then it is obvious that W is a hyperpath in ℍ𝔾. 

 Now, assume the result is possible for every hyperwalks of length less than k, and consider W 

has length k, which implies W is, 𝑢 = 𝑢0, 𝑢1, 𝑢2, … , 𝑢𝑘−1, 𝑢𝑘 = 𝑣 where the vertices are not 

necessarily distinct. If the vertices are distinct then W itself be a desired 𝑢 − 𝑣 hyperpath. If not, 

then assume j is a smallest integer such that 𝑢𝑗 = 𝑢𝑟 for some 𝑟 > 𝑗. Assume W1 is a 

hyperwalk in ℍ𝔾, is 

𝑢 = 𝑢0, 𝑢1, 𝑢2, … , 𝑢𝑗, 𝑢𝑟+1, … , 𝑢𝑘 = 𝑣. 

This hyperwalk has length strictly less than k, and then the induction hypothesis gives that W1 has a 

𝑢 − 𝑣 hyperpath in ℍ𝔾. This means that W contains a 𝑢 − 𝑣 hyperpath and proof is complete. 

The converse part is obviously holds. 

 

4. Utilizing the IFTHG Model for Robot-Based Security Applications 

The IFTHG model proposes employing robots as security guards in order to reduce risks and 

ensure public safety in situations when the robots are able to catch fugitives and keep control 

until the police arrives. Security bots would be alerted and transmit a message to the control room 

in the event of an unusual occurrence such as robbery, heist, gunshot or accident. 

Consider a large mall comprising two significant compartments with 11 and 10 robots 

respectively, each equipped with a primary control room. 

• In the IFTHG model, individual robots are represented as vertices, de- noted by  𝑈 =

{𝑢1, 𝑢2, . . . , 𝑢21, 𝐾, 𝐿}, while hyperedges could represent complex relationships or 

interactions between multiple robots in each block of mall which are denoted by 

ℰ = {ℰ1, ℰ2, . . . , ℰ10}. 

• Every robot is attached to two main processing systems, 𝐾 and 𝐿, which direct 

the robots on how to act. 

• There are two intuitionistic fuzzy threshold subhypergraphs(IFTsHGs) based on the 

major control systems 𝐾 and 𝐿. The first IFTsHG has 11 robots {𝑢1, 𝑢2, . . . , 𝑢11} 

linked with main control system 𝐾 with hyperedges {ℰ1, ℰ2, . . . , ℰ5}, and the second 

IFTsHG has 10 robots{𝑢12, . . . , 𝑢21} connected to 𝐿 with hyperedges {ℰ6, . . . ℰ9}⁡. 

• Additionally, the hyperedge S10 is connected to the major control systems 𝐾& 𝐿. 

 

To regulate the maximum and minimum values for bots, threshold values are essential. 

Vertices of IFTHGs indicate where the robots are situated and how accessible they are possible 

to escape routes. Almost every exit gate and crowded area has at least one robot. Robots have 

been spread out around the area and are ready to go on tasks before any alarm case. This 

algorithm is used to find in which gate the fugitives are using in the situation that the security 

department suddenly develops an alarm case concerning any suspicious persons. After this, the 

security sections goal is to pursue any suspicious fugitives until the police forces get involved in 



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this pursuit. 

 
Fig 4.1: Large mall with security robots 

Some notions of membership and non-membership in this scenario are denoted as follows: 

• 𝜇𝑗(𝑢𝑛) represents where the robots are located and how accessible they are to 

possible escape routes. 

• 𝑣𝑗(𝑢𝑛) reflect how far a robot is from these critical areas. 

• 𝜇𝑖𝑗(ℰ𝑚) represents the overall significance of a hyperedge in enhancing the system’s 

efficiency and effectiveness within addressing block. 

• 𝑣𝑖𝑗(ℰ𝑚) reflects the degree of non-belongingness or lack of significance of a 

hyperedge within addressing block. 

The following IFTHG depicts the pictorial representation of fig. 4.1 

 

 

Fig 4.2: IFTHG linked with multi robot system 



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The values of vertices are tabulated below: 

Table 1: Score values 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

S.No 𝑢𝑖/ℰj (𝜇, 𝜈) Score 

1 u1 (0.6,0.002) 0.7139 

2 u2 (0.5,0.0015) 0.6663 

3 u3 (0.55,0.002) 0.6892 

4 u4 (0.5,0.001) 0.6894 

5 u5 (0.4,0.001) 0.6248 

6 u6 (0.5,0.001) 0.6664 

7 u7 (0.6,0.0015) 0.7140 

8 u8 (0.3,0.004) 0.5875 

9 u9 (0.5,0.005) 0.6656 

10 u10 (0.3,0.05) 0.5758 

11 u11 (0.2,0.045) 0.5442 

12 u12 (0.5,0.001) 0.6664 

13 u13 (0.3,0.002) 0.5878 

14 u14 (0.4,0.0015) 0.6246 

15 u15 (0.6,0.003) 0.7137 

16 u16 (0.2,0.004) 0.5547 

17 u17 (0.5,0.001) 0.6664 

18 u18 (0.7,0.0015) 0.7688 

19 u19 (0.6,0.001) 0.7141 

20 u20 (0.7,0.002) 0.7689 

21 u21 (0.65,0.0015) 0.7404 

22 K (0.96,0.02) 0.9608 

23 L (0.92,0.04) 0.9423 

24 ℰ1 (0.5,0.002) 0.6662 

25 ℰ2 (0.4,0.01) 0.6226 

26 ℰ3 (0.5,0.015) 0.6633 

27 ℰ4 (0.3,0.005) 0.5870 

28 ℰ5 (0.2,0.05) 0.5429 

29 ℰ6 (0.3,0.002) 0.5878 

30 ℰ7 (0.2,0.004) 0.5546 

31 ℰ8 (0.5,0.0015) 0.6663 

32 ℰ9 (0.6,0.002) 0.7139 

33 ℰ10 (0.6,0.006) 0.7131 



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An overview of the proposed simulation is represented as follows: 

 

 

Fig 4.3: Flow chart of simulated IFTHG model 

 

This situation describes a scenario where there’s a main control system, referred 

to as K, which is responsible for overseeing and managing a certain environment or 

system. When the main control system detects an intrusion or a threat, it triggers an 

algorithm to initiate a series of calculations. 



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Fig:4.4 IFTsHG: Robots linked to K. 

The following tables 2 and 3 represents the adjacency matrix and score values respectively 

of fig. 4.4

 
If a thread is detected under the major control system 𝐾, then 𝐾 will promptly initiate 

the following algorithm, proceeding step by step. 



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Algorithm: 

Step 1: Initialize system and robots class 

Robot: 

def init (self, id, location): 

self.id = id self.location = location 

Step 2: Find any intrusion and determine its location intrusion detected 

= True 

Step 3: Find location of all robots def find 

robot locations(robots): 

return robot.id: robot.location for robot in robots robot locations = 

find robot locations(robots) 

Step 4: Find score value of robots 

def calculate score(1-non membership value/2 -membership value- non 

membership value): 

score values = 

for robot id, location in robot locations.items(): 

return score values 

Step 5: Calculate eigenvalues and eigenvectors A = 

np.array([]) 

eigenvalues, eigenvectors = np.linalg.eig(A) 

Step 6: Assign index values to robots based on eigenvalues sorted 

indices = np.argsort(eigenvalues) 

index values = robot: index for index, robot in enumerate(sorted indices) 

Step 7: Assign task to nearest robot 

task assigned = f”Task assigned to Robot nearest robot id” 

Step 8: Monitor the robots and situation situation 

under control = True 

Step 9: Check if situation is under control if situation 

under control: 

print(“Situation is under control. Continuing monitoring.”) else: 

print(“Situation is not under control. Activating alarm and executing emergency response 

plan. Notifying authorities.”) 

 

The score values of the robots indicate their accessibility. This plot illustrates the relationship 

between the score values of robots in two compartments. From this plotted graph, robots connected 

to the main control system 𝐿 demonstrate higher accessibility compared to those connected to 

𝐾. 

 



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Fig 4.5: Score values of the robots are linked to 𝐾⁡&⁡𝐿 

 

4.1 Simulation Results 

• Score values are employed to precisely locate security robots, utilizing the IFTHG 

model to calculate distances between them. These robots establish communication 

channels using radio frequency technologies, facilitating smooth information exchange 

and coordination in surveillance operations. 

• In order to obtain communication and distance information, the adjacency matrix, 

its eigenvalues, and related index values have been used. 

• The adjacency matrix illustrates the distances between robots that establish 

communication links with each other. 

• The highest eigenvalue represents the principal eigenvector of the system, from which 

the main control system directs the nearest robot seize the fugitives. 

 

5. CONCLUSION 

This paper explains about the basic concepts of IFTHG and some of its properties. IFTHGs provide a 

more flexible approach to real-time decision making by representing uncertainty and vagueness in 

robot configuration. Also an algorithm designed to identify the nearest robot index to seize the 

fugitives when an intrusion is detected in a large area. Future work will explore the intuitionistic 

fuzzy hesitancy threshold hypergraphs and its application in real life scenario. 

 

References 

[1] Atanassov K T, Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 20 (1986), 87–96. 

[2] Atanassov K T, On Intuitionistic Fuzzy Graphs and Intuitionistic Fuzzy Relations, Proceedings of the 

6th IFSA World Congress, Sao Paulo, Brazil, 1(1995), 551–554. 

[3] Atanassov K T, Intuitionistic Fuzzy Sets: Theory and Applications, Studies in Fuzziness and Soft 

Computing, Physica-Verlag, Heidelberg, New York, 2012. 

[4] Atanassov K T, Parvathi R and Karunambigai MG, Operations on In- tuitionistic Fuzzy Graphs, IEEE 

International Conference on Fuzzy Sys- tems, Korea, (2009), 20-24. 

[5] Berge C, Graphs and Hypergraphs, North - Holland, New York, 1976. 

[6] Chvatal.V, Hammer. P. L, Set-packing Problems and Threshold Graphs, Technical Report, University of 

Waterloo, Canada, 1973. 

[7] Muhammad Akram, Musavarah Sarwar, Rajab Ali Borzooei, A Novel Decision-making Approach based 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

22 
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on Hypergraphs in Intuitionistic Fuzzy Environment, Journal of Intelligent & Fuzzy Systems, 35, (2018), 1905- 

1922. 

[8] Myithili K K, Parvathi R and Akram M, Certain Types of Intuitionistic Fuzzy Directed Hypergraphs, 

International Journal of Machine Learning and Cybernetics, Springer-Verlag Berlin Heidelberg, 7(2) (2016), 

287-295. 

[9] Parvathi R, Karunambigai M G, Intuitionistic Fuzzy Graphs, Proceed- ings of 9th Fuzzy days International 

Conference on Computational In- telligence, Advances in Soft Computing: Computational Intelligence, 

Theory and Applications, Springer-Verlag, New York, 20(2006), 139– 150. 

[10] Parvathi R, Thilagavathi S and Karunambigai M G, Intuitionistic Fuzzy Hypergraphs, Bulgarian Academy 

of Science, Cybernetics and Informa- tion Technologies, 9(2) (2009), 46–53. 

[11] Parvathi R, Thilagavathi S, Intuitionistic Fuzzy Directed Hypergraphs, Advances in Fuzzy Sets and 

Systems, Pushpa Publishing House, 14(1) (2013), 39–52. 

[12] Lanzhen Yang, Hua Mao, Intuitionistic Fuzzy Threshold Graphs, Journal of Intelligent and Fuzzy Systems, 

36 (2019), 6641–6651. 

[13] Zadeh L A, Fuzzy sets, Information and Control, 8 (1965), 338–353