Ratio Mathematica Volume 46, 2023

Transversal core of intuitionistic fuzzy
k-partite hypergraphs

Myithili K.K*
Keerthika R†

Abstract

In graph theory, a transversal is a set of vertices incident to every edge
in a graph but in Intuitionistic Fuzzy k-Partite Hypergraph(IFk-PHG),
the transversal is a hyperedge which cuts every hyperedges. In this
article, Intuitionistic Fuzzy Transversal(IFT), minimal IFT, locally
minimal IFT, IFTC(Intuitionistic Fuzzy Transversal Core) of IFk-
PHG has been defined. It has been proved that every IFk-PHG has
a nonempty IFT. Also few of the properties relating to the transversal
of IFk-PHG were discussed.
Keywords: IFT; minimal IFT; locally minimal IFT; IFTC of IFk-
PHG.
2020 AMS subject classifications: 34K36, 57Q65, 05C65,93B20.

1

*Associate Professor (Department of Mathematics(CA),Vellalar College for Women, Erode-
638012, Tamil Nadu, India.); mathsmyth@gmail.com.

†Assistant Professor (Department of Mathematics, Vellalar College for Women, Erode-638012,
Tamil Nadu, India.); keerthibaskar18@gmail.com.
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1076. ISSN: 1592-7415. eISSN: 2282-8214. ©Myithili K.K et.
This paper is published under the CC-BY licence agreement.

202



Myithili K.K, Keerthika R

1 Introduction
Euler was the first author who found graph theory in 1736. The Graph the-

oretical approach is widely used to solve numerous issues in different areas like
computer science, optimization, algebra and number theory. As an application
part, the concept of graph has been extended to hypergraph, an edge with more
than one or two vertices. An idea of graph and hypergraph was popularized by
Berge [1976] in 1976. Fuzzy graph and fuzzy hypergraph concepts are developed
by the authors in J.N.Mordeson and Nair [2000]. In K.T.Atanassov [1999], the
author wrote ideas of Intuitionistic Fuzzy Sets(IFS). According to K.T.Atanassov
[2002], K.T.Atanassov [2012], the researcher putforth ideas of intuitionistic fuzzy
relations and cartesian products are defined.

In Myithili and Keerthika [2020a], the authors proposed the notion of k-partite
hyperedges in IFHGs(Intuitionistic Fuzzy Hypergraphs). Certain operations like
Union, Intersection, Ringsum, Cartesian Product were discussed in Myithili and
Keerthika [2020b] . It has numerous application problems in decision-making. In
Myithili and Parvathi [2015], Myithili and Parvathi [2016], Myithili et al. [2014]
transversals and its properties on intuitionistic fuzzy directed hypergraphs were
discussed.

The authors in Goetschel [1995], Goetschel [1998], Goetschel et al. [1996]
initiated the concepts like fuzzy transversal and fuzzy coloring in fuzzy hyper-
graph. In this article an attempt has been made to analyze the transversal and its
related properties in IFk-PHGs.

2 Symbolic representation
MNMV-membership and non-membership values
FSV-Finite set of vertices
FIFS-family of intuitionistic fuzzy subsets
IFH-intuitionistic fuzzy hyperedge
ONV-Open Neighborhood of the vertex
CNV-Closed Neighborhood of the vertex
ℵ = (∨, Ξ,ψ) - Intuitionistic fuzzy(IF) k-partite hypergraph with edge set Ξ,
vertex set ∨ and k-partite hyperedge ψ
h(ℵ) - Height of IFk-PHG
Fk(ℵ) - Fundamental sequence (FS) of IFk-PHG
c(ℵ) - Core set(CS) of IFk-PHG
Ik(ℵ) - Induced fundamental sequence(IFS) of IFk-PHG

203



Transversal Core of Intuitionistic Fuzzy k-Partite Hypergraphs

ℵ(ai,bi) - (ai,bi)-level of IFk-PHG
(ai,bi) - Edge membership(EM) and non-membership values(ENMV)
T R(ℵ) - minimal intuitionistic fuzzy transversal(MIFT) of IFk-PHG

3 Preliminaries
Definition 3.1. Myithili and Keerthika [2020a] The IFk-PHG ℵ is an ordered
triple ℵ = (∨, Ξ,ψ) where,
•∨ = {g1,g2,g3, · · · ,gn} is a FSV,
• Ξ = {Ξ1, Ξ2, Ξ3, · · · , Ξm} is a FIFS of ∨,
• Ξj = {(gi,ωj(gi),νj(gi)) : ωj(gi),νj(gi) ≥ 0, ωj(gi) + νj(gi) ≤ 1}, 1 ≤ j ≤ m,
• Ξj 6= ∅, 1 ≤ j ≤ m,
•
⋃
j supp(Ξj) = ∨, 1 ≤ j ≤ m.

For all gi ∈ Ξ ∃ k - disjoint sets ψi, i = 1, 2, · · · ,k and no two vertices in the

same set are adjacent such that Ξk =
k⋂
i=1

ψi = ∅

Definition 3.2. Myithili and Keerthika [2020a] Let an IFk-PHG be ℵ = (∨, Ξ,ψ).
The height of IFk-PHG is defined by h(ℵ) = {max(min(ωkij )),max(max(νkij ))}
for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. Also ωkij and νkij are MNMV of the k-partite
hyperedge ψij.

Definition 3.3. Myithili and Keerthika [2020a] Let ℵ be an IFk-PHG. Suppose
ψj,ψk ∈ψ and 0 < δ,ε ≤ 1. The (δ,ε)-level is defined by (ψj,ψk)(δ,ε) = {gi ∈
∨|min(ωδkij (gi)) ≥ δ,max(ν

ε
kij

(gi)) ≤ ε}.

Definition 3.4. Myithili and Keerthika [2020a] Let ℵ be IFk-PHG, ℵai,bi =
〈
∨ai,bi, Ξai,bi

〉
be the (ai,bi)-level of ℵ. The sequence of real numbers {a1,a2, · · · ,ak; b1,b2, · · · ,bk}
3 0 ≤ ai ≤ hω(ℵ) and 0 ≤ bi ≤ hν(ℵ), satisfies:

(i) If a1 < δ ≤ 1 & 0 ≤ ε < b1 then ψδ,ε = ∅,
(ii) If ai+1 ≤ δ ≤ ai; bi ≤ ε ≤ bi+1 then ψδ,ε = ψai,bi ,
(iii) ψai,bi @ ψai+1,bi+1 is fundamental sequence of IFk-PHG and it is denoted

as Fk(ℵ).

Definition 3.5. Myithili and Keerthika [2020a] Let c(ℵ) = {ℵa1,b1,ℵa2,b2, · · · ,ℵak,bk}
be core set of ℵ. The analogous set of (ai,bi)-level hypergraphs is ℵa1,b1 ⊂
ℵa2,b2 ⊂ ··· ⊂ ℵak,bk is said to be ℵ-IFS and it is denoted by Ik(ℵ). The (ak,bk)-
level is known as support level of ℵ. ℵak,bk is known as the support of ℵ.

Definition 3.6. Myithili and Keerthika [2020a] Let ℵ = (∨, Ξ,ψ) & ℵ′ = (∨′, Ξ′,ψ′ )
are IFk-PHGs, ℵ is known as partial IFk-PHG of ℵ′ , if

204



Myithili K.K, Keerthika R

∨
′
=

{
min (supp (ωkij )) |ωkij ∈ ψ

′

max (supp (νkij )) |νkij ∈ ψ
′

the partial IFk-PHG generated by ψ
′

and is represented as ℵ ⊆ ℵ′ . Also, if
ℵ⊆ℵ′ and ℵ 6= ℵ′ exists then ℵ⊂ℵ′ .

Definition 3.7. Myithili and Keerthika [2020a] Let ℵ be the IFk-PHG, c(ℵ) =
{ℵa1,b1,ℵa2,b2, · · · ,ℵak,bk}. ℵ is called as ordered if c(ℵ) is ordered (i.e) ℵa1,b1 ⊂
ℵa2,b2 ⊂ ··· ⊂ ℵak,bk . The IFk-PHG is known as simply ordered if {ℵai,bi|i =
1, 2, · · · ,k} is simply ordered, (i.e) if it is ordered and if ψ ∈ ℵai+1,bi+1\ℵai,bi
then ψ * ℵai,bi .

4 Main results
Definition 4.1. Consider an IFk-PHG ℵ. An IFT T of IFk-PHG is an IF subset
of ∨ with T (ψj,ψk) ∩ A (ψj,ψk) 6= ∅ for each A ∈ ψ where ψj = min(ωkij ) and
ψk = max(νkij ) ∀ 1 ≤ i ≤ m, 1 ≤ j ≤ n. Also ωkij and νkij is the MNMV of kth
partition of jth edge in ith vertex.

Definition 4.2. A minimal IFT T for IFk-PHG be a transversal of ℵ, which sat-
isfies the condition that if T1 ⊂ T , then T1 is not IFT of ℵ.
Note: The set of minimal IFT of IFk-PHG is denoted as T R(ℵ). Always T R(ℵ) 6=
∅.

Example 4.1. An IFH (intuitionistic fuzzy hypergraph) with ∨ = {g1,g2,g3,g4,g5,g6,g7,g8},
Ξ = {Ξ1, Ξ2, Ξ3, Ξ4} has been considered.

205



Transversal Core of Intuitionistic Fuzzy k-Partite Hypergraphs

Figure 1: Intuitionistic Fuzzy Hypergraph

Using the above figure we can construct an IFk-PHG ℵ, with ψ = {ψ1,ψ2,ψ3}
disjoint hyperedges which are represented below as incidence matrix

The minimal IFT of IFk-PHG is attained as follows,

206



Myithili K.K, Keerthika R

The corresponding graph is shown below.

Figure 2: ℵ and minimal IFT of ℵ

Definition 4.3. If T is an IFS with T (ai,bi) as minimal IFT (MIFT) of ℵ(ai,bi)
for each (ai,bi) ∈ (0, 1) then T is called as locally minimal IFT (LMIFT) of
IFk-PHG. The set containing LMIFT of IFk-PHG is written as T R∗(ℵ)

Theorem 4.1. If T is an IFT of ℵ then h(T ) ≥ h(ψj) for ψj ∈ ψ. Also, if T is
the minimal IFT of IFk-PHG, then

207



Transversal Core of Intuitionistic Fuzzy k-Partite Hypergraphs

h(T ) = {max(min(ωkij )),max(max(νkij )) | ωkij,νkij ∈ ψ} = h(ℵ).

Theorem 4.2. Every IFk-PHG has a nonempty IFT.
Note: Every IFT of IFk-PHG contains a MIFT.

Theorem 4.3. If T
′ ∈ T R(ℵ) and for every g ∈ ∨, T ′ (g) ∈ Fk(ℵ), then

Fk(T R(ℵ)) ⊆ Fk(ℵ).

Theorem 4.4. T R(ℵ) is sectionally elementary.

Proof. Let Fk(T R(H )) = a1,a2, · · · ,ak; b1,b2, · · · ,bk. Assume that
T

′ ∈ T R(H ) and some δ,ε ∈ (ai,bi) such that T (ai,bi) ⊂ T (δ,ε). Since
T R(ℵai,bi ) = T R(ℵδ,ε), ∃ some A ∈ T R(H ) 3 A ai,bi = T δ,ε. Then
T δ,ε ⊂ A ai,bi implies the IFS ∨(gi) defined by

∨(gi) =

{
(δ,ε) if x ∈ A ai,bi \ T ai,bi

A (gi) Otherwise

is an IFT of IFk-PHG. Here ∨ < A , implies the contradiction of minimality (CM)
of A .

Theorem 4.5. For every A ∈ T R(ℵ), A a1,b1 is a minimal IFT of ℵa1,b1 .

Proof. For any IFk-PHG ℵ = (∨, Ξ,ψ), consider a minimal IFT T of ℵa1,b1
such that T ⊂ A a1,b1 .
Define the IFS ∨(gi) where

∨(gi) =

{
(a2,b2) if x ∈ A a1,b1 \ T
A (gi) Otherwise

By the above theorem, ∨ is an IFT of IFk-PHG, CM of A .

Definition 4.4. Let ℵ be IFk-PHG. The Intuitionistic Fuzzy Transversal Core
(IFTC) of ℵ is ℵ′ = (∨′, Ξ′,ψ′ ) with the following condition that
(i) min T R(ℵ) = min T R(ℵ′ ),
(ii) ℵ′ = ∪ min T R(ℵ),
(iii) ψ \ ψ′ is exactly the set containing vertices of ℵ which does not belong to
T R(ℵ),where ψ′ is the remaining hyperedge set, after deleting hyperedges that
are correctly contained in another hyperedge.
The remarks of the statement is,
(i) For any IFk-PHG without spike hyperedges, ∃ transversal core which are al-
ways unique.
(ii) The definition also holds good for IFk-PHGs with spike (a hyperedge with
single vertex) hyperedges.

208



Myithili K.K, Keerthika R

Definition 4.5. In IFk-PHG, the ONV gi is the set containing adjacent vertices of
gi except itself in a k-partite hyperedge and is denoted as Nk(gi).

Example 4.2. Consider an IFk-PHG with ∨ = {g1,g2,g3,g4,g5,g6,g7},
Ξ = {Ξ1, Ξ2, Ξ3} where,
Ξ1 = {g1 〈0.5, 0.2〉 ,g2 〈0.3, 0.4〉 ,g3 〈0.6, 0.3〉},
Ξ2 = {g2 〈0.3, 0.4〉 ,g4 〈0.2, 0.5〉 ,g5 〈0.3, 0.4〉},
Ξ3 = {g3 〈0.6, 0.3〉 ,g6 〈0.4, 0.3〉 ,g7 〈0.1, 0.7〉} with
ψ1 = {g1 〈0.5, 0.2〉 ,g4 〈0.2, 0.5〉 ,g7 〈0.1, 0.7〉}, ψ2 = {g2 〈0.3, 0.4〉 ,g6 〈0.4, 0.3〉},
ψ3 = {g3 〈0.6, 0.3〉 ,g5 〈0.3, 0.4〉}
Here g1 and g7 are the ONV g4 in ψ1.

Definition 4.6. In IFk-PHG, the CNV gi is the set containing adjacent vertices of
gi including the vertex in a k-partite hyperedge and is denoted as Nk[gi].

Example 4.3. From the above example it is clear that the Closed Neighborhood
of the vertex g3 is g3 and g5 in ψ3.

Theorem 4.6. In ℵ, the following claims are related
(i) T is an IFT of IFk-PHG,
(ii) T ai,bi∩A ai,bi 6= ∅, for all IFH A ∈ ψ and every (ai,bi) with 0 < ai ≤ hω(ℵ),
0 < bi ≤ hν(ℵ),
(iii) T ai,bi is an IFT of ℵai,bi , for each (ai,bi) with 0 < ai ≤ δ, 0 < bi ≤ ε.

Proof. From the definition, ”A minimal IFT T for IFk-PHG is a transversal
of ℵ, which satisfies the property that if T1 ⊂ T , then T1 is not an IFT of ℵ” the
result is immediate.

Theorem 4.7. For a simple IFk-PHG, T R(T R(ℵ)) = ℵ.

Theorem 4.8. For any IFk-PHG, T R(T R(ℵ)) ⊆ℵ.

Proof. From definition 4.4, ∃ a ℵ′ (partial hypergraph) of a simple IFk-PHG
3 T R(ℵ′ ) = T R(ℵ). From Theorem 4.7, T R(T R(ℵ)) = T R(T R(ℵ′ ))
implies ℵ′ ⊆ℵ.

Theorem 4.9. Let ℵ be an IFk-PHG and suppose that T ∈ T R(ℵ). If ℵ′ ⊆
supp(T ) ⊆ ℵ, then ∃ a hyperedge of IFk-PHG A , (ai,bi) ∈ A represents the
MNMV of A 3
(i) (ai,bi) = h(A ) = h(T ai,bi ) > 0,
(ii) Th(A ) ∩ Ah(A ) = ℵ.

Proof. Let 0 < h(T ai,bi ) ≤ 1 and ψ′ be the set of all IF k-partite hyperedges
where h(τai,bi ) ≥ h(T ai,bi ) for each τ ∈ ψ′ .

209



Transversal Core of Intuitionistic Fuzzy k-Partite Hypergraphs

Since T ai,bi is an IFT of ℵai,bi and ℵ′ ⊆ T ai,bi is nonempty. Further, for
each τ ∈ ψ′ , h(τ) ≥ h(τai,bi ) ≥ h(T ai,bi ) is true. Also, assume that T ai,bi is the
MIFT, then for all τ ∈ ψ′ , h(τ) > h(T ai,bi ) and ∃ℵτ 6= ℵ with ℵτ ∈ τh(τ)∩Th(τ).
Define an IFk-PHG ℵ1 3

ℵ1(U) =




T (U) whenever U 6= ℵ′,
min (h(A )/h(A ) < h(T ai,bi )),max (h(A )/h(A ) < h(T ai,bi ))

whenever U = ℵ′

Hence ℵ1 is an IFT of IFk-PHG and h(ℵ
ai,bi
1 ) < h(T

ai,bi ), It does not meet the
basic requirement of T . For each τ ∈ ψ′ satisfies the first part of the theorem 4.9
and has ℵτ which is not in ℵ with ℵτ ∈ τh(τ) ∩ Th(τ). The procedure is repeated,
and the argument of (i) provides a contradiction and bringing close to the proof.

Theorem 4.10. Let ℵ be an IFk-PHG. Then, ∃T ∈ T R(ℵ) with ℵ′ ⊆ supp(T ) ⊆
ℵ, if and only if for A ∈ ψ it meets the following requirements:
(i) (ai,bi) = h(A ),
(ii) The level cut (aj,bj) of h(A

′
) is not a subhypergraph of the level cut (ai,bi)

of h(A ), for all A
′ ∈ ψ with h(A ′ ) > h(A ),

(iii) The level cut (ai,bi) of h(A ) does not contain any other hyperedge of ℵh(A ),
where (ai,bi) denotes MNMV of A .

Proof. Necessary Part:
(i) Let T ∈ T R(ℵ) and 0 < h(T ai,bi ) ≤ 1. Condition (i) is followed from
Theorem 4.9.
(ii) Suppose that for each A satisfying (i) ∃ A ′ ∈ ψ 3 h(A ′ ) > h(A ) and
A

′

h(A
′
) ⊆ Ah(A ), then ∃ U 6= ℵ

′
, with U ∈ A ′h(A ′) ∩ Th(A ′) ⊆ Ah(A ) ∩ Th(A )

which differs from the concept of Theorem 4.9.
(iii) Assume for each A satisfying (i) and (ii) then ∃ A ′ ∈ ψ so that ∅ 6=
A

′
h(A ) ⊂ Ah(A ). Since A

′
h(A ) 6= ∅ and by (ii), we have h(A

′
) = h(A ) =

(ai,bi).
If (aj,bj) = h(A ′) and A ′′ ∈ ψ such that ∅ 6= A ′′h(A ) ⊂ A ′h(A ) ⊂ Ah(A ).
The process is continued and the chain ends finitely, without loss of abstraction
assume (ai,bi) < h(A ). But, ∃U 6= ℵ

′ 3U ∈ A ′h(A )∩Th(A ) ⊆ Ah(A )∩Th(A ),
which contradicts Theorem 4.9.
Sufficient Part:

Let A ∈ ψ satisfy the condition (i), (ii) and (iii). By condition (i), the process
is trivial. By condition (ii) and (iii) ∃ U ∈ A ′h(A ′) \ Ah(A ) for every A

′ ∈ ψ
3 A ′ 6= A and h(A ′ ) ≥ h(A ). Let ∨A be the set of all vertices of ℵ 3
∨A ∩ Ah(A ) = ∅.

The initial sequence of transversals are constructed. So τs ⊆ ∨ for all s,

210



Myithili K.K, Keerthika R

1 ≤ s < i and τi ⊆∨A ∪∨i. Hence, ∨i ∈ τi for each i. The process is terminated
till it reaches a minimal IFT with (ai,bi) = h(A ) = h(T ai,bi ).

Theorem 4.11. Let ℵ be an IFk-PHG with Fk(ℵ) = {a1,a2, · · · ,ak; b1,b2, · · · ,bk}
so that 0 ≤ ai ≤ hω(ℵ), 0 ≤ bi ≤ hν(ℵ). Also, ℵai,bi ⊆ A

′
, be the elementary

IFk-PHG if and only if h(A
′
) = (ai,bi) and supp(A

′
) is a hyperedge of ℵai,bi .

Then T R(T R(ℵ)) is the partial IFk-PHG of ℵai,bi .

Proof. From Theorem 4.5 and by the construction of minimal IFT, the (ai,bi)-
level hypergraph of T R(ℵ) is T R(ℵai,bi ) which means that (T R(ℵ))ai,bi =
T R(ℵai,bi ). Let τ belongs to T R(T R(ℵ)). From Theorem 4.9, h(τ(∨i)) > 0,
this implies that ∃ T ∈ T R(ℵ) with h(τ(∨i)) = h(T ). From Theorem 4.1,
h(T ) = (max(min(ωkij )),max(max(νkij ))) = h(ℵ) for all minimal IFT T .
Hence τ is elementary with h(ai,bi). Since supp(τ) = τai,bi , Theorem 4.5 suggest
that supp(τ) is the minimal IFT of (T R(ℵ))ai,bi . It is obvious that supp(τ) is a
hyperedge of ℵai,bi . Hence τ is a hyperedge of ℵai,bi .

Theorem 4.12. Let ℵ be an IFk-PHG with ℵai,bi is a simple. Then T R(T R(H )) =
ℵai,bi .

Proof. By the above theorem, T R(T R(ℵ)) ⊆ ℵai,bi . Let τ be an elemen-
tary with h(T ) = (ai,bi) and supp(τ) ∈ ℵai,bi . By Theorem 4.11, supp(τ)
is a minimal IFT of (T R(ℵ))ai,bi . Since each minimal IFT of T R(ℵ) is ele-
mentary by definition of minimal IFT the process ends at (ai,bi)-level and τ ∈
T R(T R(ℵ)).
Hence ℵai,bi ⊆ T R(T R(ℵ)) which implies ℵai,bi = T R(T R(ℵ)).

5 Conclusion
In this article, some interesting concepts like IFT, minimal IFT, locally mini-

mal IFT and IFTC of IFk-PHGs were discussed. It is important to note that IFTC
exists for both spike and non-spike intuitionistic fuzzy k-partite hyperedges. In
future, the authors planned to work on Robotics with multi-task concept as an
application part of IFk-PHG.

References
C. Berge. Graphs and Hypergraphs. North - Holland, New York, 1976.

R. H. Goetschel. Introduction to fuzzy hypergraphs and hebbian structures. Fuzzy
Sets and Systems, 76:113–130, 1995.

211



Transversal Core of Intuitionistic Fuzzy k-Partite Hypergraphs

R. H. Goetschel. Fuzzy colorings of fuzzy hypergraphs. Fuzzy Sets and Systems,
94:185–204, 1998.

R. H. Goetschel, W. L. Craine, and W. Voxman. Fuzzy transversals of fuzzy
hypergraphs. Fuzzy Sets and Systems, 84:235–254, 1996.

J.N.Mordeson and P. Nair. Fuzzy Graphs and Fuzzy Hypergraphs. Physica -
Verlag, New York, 2000.

K.T.Atanassov. Intuitionistic Fuzzy Sets - Theory and Applications. Physica -
Verlag, New York, 1999.

K.T.Atanassov. On index matrix representation of the intuitionistic fuzzy graphs.
Notes on Intuitonistic Fuzzy Set, 4:73–78, 2002.

K.T.Atanassov. On Intuitionistic Fuzzy Sets Theory. Springer, New York, 2012.

K. Myithili and R. Keerthika. Types of intuitionistic fuzzy k-partite hypergraphs.
AIP Conference Proceedings, 2261:030012–1 – 030012–13, 2020a.

K. Myithili and R. Keerthika. Properties of strong and complete intuitionistic
fuzzy k-partite hypergraphs. Turkish Journal of Computer and Mathematics
Education, 11(2):784–791, 2020b.

K. Myithili and R. Parvathi. Transversals of intuitionistic fuzzy directed hyper-
graphs. Notes on Intuitionistic Fuzzy Sets, 21(3):66–79, 2015.

K. Myithili and R. Parvathi. Properties of transversals of intuitionistic fuzzy di-
rected hypergraphs. Advances in Fuzzy Sets and Systems, 21(1):93–105, 2016.

K. Myithili, R. Parvathi, and M. Akram. Certain types of intuitionistic fuzzy
directed hypergraphs. International Journal of Machine Learning and Cyber-
netics, 2:1–9, 2014.

212