Ratio Mathematica Volume 46, 2023

On fz- domination number of fuzzy graphs

Lekha A*
Parvathy K.S†

Abstract

Given a fuzzy graph G = (V,µ,σ), the fz- domination number, γfz(G),
is the least scalar cardinality of an fz- dominating set of G. In this ar-
ticle, we examine several features of fz-domination number of fuzzy
graphs as a result of various fuzzy graph operations. We find bounds
for the fz-domination number of a few graph products and look at the
requirements for the sharpness of these bounds.
Keywords: fuzzy graph; fz-dominating sets; fz-domination number;
graph operations
2020 AMS subject classifications:05C69, 05C72. 1

*Maharaja’s Technological Institute, Thrissur, Kerala, India. alekharemesh@gmail.com,
lekha.a.res@smctsr.ac.in.

†St. Mary’s College, Thrissur, Kerala, India. parvathy.math@gmail.com (Corresponding Au-
thor).
1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20,
2023. DOI: 10.23755/rm.v46i0.1078. ISSN: 1592-7415. eISSN: 2282-8214. ©Lekh A et al.
This paper is published under the CC-BY licence agreement.

213



Lekh A and Parvathy K.S

1 Introduction
Since the initial introduction of fuzzy graphs by Rosenfeld [1975], a large

number of researchers have studied the subject. The notion of domination in
fuzzy graphs was first proposed by Somasundaram and Somasundaram [1998].
Somasundaram [2005], Gani and Chandrasekaran [2006], Manjusha and Sunitha
[2015], Bhutani and Sathikala [2016] also studied domination in fuzzy graphs.
Mordeson and Chang-Shyh [1994] developed operations of fuzzy graphs that are
comparable to those in crisp graphs.

Different variations of domination in fuzzy graphs found in literature do not
consider the situations where we need to take all the non-zero edges incident at a
vertex into consideration. These definitions use either the effective or the strong
edges of the fuzzy graph.But our model of fuzzy domination in fuzzy graphs
[2022] takes into account all the non-zeroedges incident at a vertex, even if they
are small in strength. Further most variations of domination in fuzzy graphs found
in literature do not considerthe fuzzy subsets of the vertex set, instead considered
the crisp subsets of the fuzzy vertex set.But while considering fuzzy graphs and
their subset problems it is more apt toconsider fuzzy subsets of the vertex set
than their crisp subsets. By taking all these into consideration we defined fz-
domination in fuzz graphs[2022].

We, Lekha and Parvathy [2022] developed fz-domination in fuzzy graphs,
which coincides with fractional domination in crisp graphs presented by Hedet-
niemi and Wimer [1987] and explored by Hedetniemi and Mynhardt [1990]. In
this article, we examine the effects of several graph operations on fz-domination.

For basic definitions, terminology and notation in fuzzy graphs we refer to
Mordeson and Nair [2000].

Definition 1.1. (Lekha and Parvathy [2022]). Given a fuzzy graph G = (V,µ,σ),
a fuzzy subset µ′ of µ is defined as an fz-dominating set of G, if for every v ∈ V ,

µ′(v) +
∑
x∈V

(
σ(x,v)∧µ′(x)

)
≥ µ(v).

A fuzzy subset µ′ is a minimal fz-dominating set, if µ′′ ⊂ µ′ is not an fz-
dominating set.

Definition 1.2. (Lekha and Parvathy [2022]) Fuzzy domination number or fz-
domination number of G, denoted by γfz(G), is defined as

γfz(G) = min {|µ′| : µ′ is a minimal fz-dominating set of G}

Example 1.1. For F = (µ,σ) shown in Fig. 1,

µ1 = {(x,0.1),(y,0.5),(z,0.2)}

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On fz- domination number of fuzzy graphs

µ2 = {(x,0.6),(y,0),(z,0.6)}

µ3 = {(x,0.4),(y,0.2),(z,0.4)}

µ4 = {(x,0.5),(y,0.1),(z,0.5)}

are all minimal fz-dominating sets of F. µ1 is a minimum fz-dominating set and
γfz(F) = 0.8.

Figure 1: Fuzzy graph, F

2 fz- domination in union of fuzzy graphs

Let G = (V1,µ1,σ1) and H = (V2,µ2,σ2). G∪H = (V,µ,σ) where

V = V1 ∪V2

µ(u) = µ1(u) if u ∈ V1 \V2
= µ2(u) if u ∈ V2 \V1
= µ1(u)∨µ2(u) if u ∈ V1 ∩V2

and
σ(u,v) = σ1(u,v) if u ∈ V1 \V2,v ∈ V1

= σ2(u,v) if u ∈ V2 \V1,v ∈ V2
= σ1(u,v)∨σ2(u,v) if u,v ∈ V1 ∩V2
= 0 otherwise

The following theorem gives a general upper bound for the fz-domination number
of union of two fuzzy graphs.

Theorem 2.1. For any two non- trivial fuzzy graphs G and H,

γfz(G∪H) ≤ γfz(G) + γfz(H)

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Lekh A and Parvathy K.S

Proof. Consider the fuzzy graphs G = (V1,µ1,σ1) and H = (V2,µ2,σ2). Let µ′1
and µ′2 be the minimum fz-dominating sets of G and H respectively. Let the fuzzy
subset µ′ of V be defined by

µ′(u) = µ′1(u) if u ∈ V1 \V2
= µ′2(u) if u ∈ V2 \V1
= µ′1(u)∨µ

′
2(u) if u ∈ V1 ∩V2

Now let v ∈ V .
Case (i)
If v ∈ V1 \V2, then

µ(v) = µ1(v)

≤
(
µ′1(v) +

∑
x∈V1

σ1(x,v)∧µ′1(x)
)

= µ′(v) +
∑
x∈V

σ(x,v)∧µ′(x).

Case (ii)
If v ∈ V2 \V1, then

µ(v) = µ2(v)

≤
(
µ′2(v) +

∑
x∈V2

σ2(x,v)∧µ′2(x)
)

= µ′(v) +
∑
x∈V

σ(x,v)∧µ′(x).

Case (iii)
If v ∈ V1 ∩V2
µ(v) = µ1(v)∨µ2(v)

≤
(
µ′1(v) +

∑
x∈V1

σ1(x,v)∧µ′1(x)
)
∨
(
µ′2(v) +

∑
x∈V2

σ2(x,v)∧µ′2(x)
)

≤ (µ′1(v)∨µ
′
2(v)) +

( ∑
x∈V1\V2

(σ1(x,v)∧µ′1(x)) +
∑

x∈V2\V1

(σ2(x,v)∧µ′2(x))

+
∑

x∈V1∩V2

(σ1(x,v)∨σ2(x,v))∧ (µ′1(x)∨µ
′
2(x))

)
≤ µ′(v) +

∑
x∈V

σ(x,v)∧µ′(x).

Thus µ′ is an fz-dominating set of G∪H and µ′(v) ≤ µ′1(v) + µ′2(v).
Hence |µ′| ≤ |µ′1|+ |µ′2|.
Thus, γfz(G∪H) ≤ γfz(G) + γfz(H).

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On fz- domination number of fuzzy graphs

Remark 2.1. Obviously equality holds in the above theorem if the vertex sets of
G and H are disjoint. The following example shows that equality may hold even
if they are not disjoint. For the graphs in Fig. 2, γfz(G) = 0.5, γfz(H) = 0.6 and
γfz(G∪H) = 1.1 so that γfz(G∪H) = γfz(G) + γfz(H).

Figure 2: Fuzzy Graphs G, H and G∪H

3 fz- domination in join of fuzzy graphs
Let G = (V1,µ1,σ1) and H = (V2,µ2,σ2) whose vertex sets are disjoint. The

join G +H is defined by G +H = (V,µ,σ) where V = V1 ∪V2,

µ(u) = µ1(u) if u ∈ V1
= µ2(u) if u ∈ V2

and
σ(u,v) = σ1(u,v) if u,v ∈ V1

= σ2(u,v) if u,v ∈ V2
= µ1(u)∧µ2(v) if u ∈ V1 and v ∈ V2

Theorem 3.1. For any two non- trivial fuzzy graphs G and H whose vertex sets
are disjoint,

γfz(G +H) ≤ max{γfz(G),γfz(H)}

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Lekh A and Parvathy K.S

Proof. Let G = (V1,µ1,σ1) and H = (V2,µ2,σ2) be two fuzzy graphs such that
V1 ∩V2 = φ. Let γfz(G) ≥ γfz(H) and let µ′1 be a minimum fz-dominating set of
G.
Define µ′ ⊂ µ by

µ′(u) = µ′1(u) if u ∈G
= 0 if u ∈H

Let m be such that m = max{µ2(u);u ∈H}.
Now m ≤ γfz(H) ≤ γfz(G) implies that µ′ is an fz-dominating set of G +H.
Hence,

γfz(G +H) ≤ max{γfz(G),γfz(H)}

In the following discussion M, m1 and m2 denote the maximum membership
value of a vertex in G +H, G and H respectively.

Observation 3.1. It is possible that

γfz(G +H) ≤ min{γfz(G),γfz(H)}

For example, if M ≤ γfz(H) ≤ γfz(G), then µ′ ⊂ µ defined by

µ′(u) = µ′2(u) if u ∈H
= 0 if u ∈G

is an fz-dominating set of G +H and hence γfz(G +H) ≤ γfz(H). Here equality
occurs if M = γfz(H). The following example shows that strict inequality can
also occur in this relation.

Example 3.1. Consider the fuzzy graphs G1, G1 and G1 +G2 given in Fig.3.
Here γfz(G1) = 1.6, γfz(G2) = 1, M = 0.8, γfz(G1 +G2) = 0.9 so that

γfz(G1 +G2) < γfz(G2)

Observation 3.2. If γfz(H) ≤ M ≤ |µ2|, then γfz(G +H) = M.

Claim: Define µ′′2 ⊃ µ′2 in H such that |µ′′2| = M. Then, µ′′2 is an fz-dominating
set of G+H. Hence γfz(G+H) ≤ M. Also, since there is a vertex of membership
value M in G +H, we get γfz(G +H) = M.

Observation 3.3. If m1 ≤ |µ2| and m2 ≤ |µ1|, then γfz(G +H) ≤ m1 + m2.

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On fz- domination number of fuzzy graphs

Figure 3: Fuzzy Graphs G1, G2 and G1 +G2

Claim: Define µ′1 ⊂ µ1 in G such that |µ′1| = m2 and µ′2 ⊂ µ2 in H such that
|µ′2| = m1. Now µ′ defined by

µ′(u) = µ′1(u) if u ∈G
= µ′2(u) if u ∈H

is an fz- dominating set in G +H. Hence γfz(G +H) ≤ m1 + m2.

Observation 3.4. If |µ2| ≤ M, then µ′ ⊂ µ in G +H defined by

µ′(u) = µ2(u) if u ∈H
= max{0,µ′1(u)−|µ2|} if u ∈G

is an fz- dominating set in G + H. Then, γfz(G + H) ≤ γfz(G) −n|µ2| where n
is the number of vertices u ∈G having µ1(u) ≥ |µ2|.

Observation 3.5. If γfz(H) ≤ M ≤ γfz(G), then µ′ ⊂ µ in G +H defined by

µ′(u) = µ′2(u) if u ∈H
= max{0,µ′1(u)−γfz(H)} if u ∈G

is an fz- dominating set in G +H. Then, γfz(G +H) ≤ γfz(G)−nγfz(H) where
n is the number of vertices u ∈G having µ1(u) ≥ γfz(H).

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Lekh A and Parvathy K.S

4 fz- domination in corona of fuzzy graphs
The corona of G = (V1,µ1,σ1) and K1 = (u,µ2(u)) is the fuzzy graph G◦K1

obtained by attaching a copy of K1 to each vertex vi ∈ V1 such that σ(vi,ui) =
µ1(vi)∧µ2(ui) where ui represents the vertex in the copy of K1 corresponding to
vi ∈ V1.

Observation 4.1. The following two results are obvious.

1. γfz(G ◦K1) ≥ γfz(G)

2. γfz(G ◦K1) ≥ nµ2(u)

Figure 4: Fuzzy graph G ◦K1

Remark 4.1. The example below shows that equality may occur in observation
4.1(a).
Consider G ◦K1 in figure 4. µ′ = {(a, 13),(b,

1
3
),(c, 1

3
),(d, 1

3
)} is a minimum fz-

dominating set of G and γfz(G) = 43 . µ
′ is an fz-dominating set of G ◦K1 also.

Therefore, γfz(G ◦K1) ≤ 43 = γfz(G).
On the other hand from observation 4.1(a), γfz(G ◦K1) ≥ γfz(G).
Thus we get γfz(G ◦K1) = γfz(G).

Theorem 4.1. γfz(G ◦K1) ≤ γfz(G) + nµ2(u) where n = |V1|.

Proof. Let µ be the fuzzy subset of G ◦K1 and µ′1 be a minimum fz-dominating
set of G. Let µ′ ⊂ µ be such that

µ′(v) = µ′1(v) if v ∈ V1
= µ2(v) otherwise

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On fz- domination number of fuzzy graphs

Then µ′ is an fz-dominating set of G ◦K1 and

|µ′| = |µ′1|+ n|µ2|
= γfz(G) + nµ2(u)

Therefore γfz(G ◦K1) ≤ γfz(G) + nµ2(u)

Theorem 4.2. If µ2(u) ≥ µ1(v) for all v ∈ V1, then

γfz(G ◦K1) = nµ2(u)

Proof. It is clear that γfz(G ◦K1) ≥ nµ2(u)
Consider µ′ where

µ′(v) = 0 if v ∈ V1
= µ2(v) if v = u

Then, µ′ is fz-dominating set of G ◦K1.
Therefore, γfz(G ◦K1) ≤ |µ′| = nµ2(u).
Hence γfz(G ◦K1) = nµ2(u)

Figure 5: G′ ◦K′1

Remark 4.2. The condition µ2(u) ≥ µ1(v) for all v ∈ V1 is not necessary to get
γfz(G ◦K1) = nµ2(u). For example, consider G′ ◦K′1 given in figure 5.
Here, µ2(u) < µ1(v) for all v ∈ V1. Now γfz(G ◦K1) ≥ nµ2(u) implies that
γfz(G′ ◦K′1) ≥ 2.
Also µ′ = {(a, 1

2
),(b, 1

2
),(c, 1

2
),(d, 1

2
)} is fz-dominating set of G′ ◦K′1.

Hence γfz(G′ ◦K′1) = 2 = nµ2(u)

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Lekh A and Parvathy K.S

5 fz- domination in Cartesian product
Let G1 = (V1,µ1,σ1) and G2 = (V2,µ2,σ2). The Cartesian product is the

fuzzy graph G12G2 = (V,µ1 ×µ2,σ1 ×σ2) where V = V1 ×V2,

(µ1 ×µ2)(a,b) = µ1(a)∧µ2(b)

and
(σ1 ×σ2)

(
(a1,b2),(a2,b2)

)
= µ1(a1)∧σ2(b1,b2) if a1 = a2
= σ1(a1,a2)∧µ2(b1) if b1 = b2
= 0 otherwise

Theorem 5.1. For any two nontrivial fuzzy graphs G and H,

γfz(G2H) ≤ min{nγfz(G),mγfz(H)},

m, n are the number of vertices with nonzero membership values in G and H
respectively.

‘

Proof. Let G = (V1,µ1,σ1) and H = (V2,µ2,σ2) where

V1 = {(u1,µ1(u1)),(u2,µ1(u2)), ...,(um,µ1(um))}

and
V2 = {(v1,µ2(v1)),(v2,µ2(v2)), ...,(vn,µ2(vn))}

G2H = (V,µ,σ) where V = V1 ×V2, µ(u,v) = µ1(u)∧µ2(v) and

σ
(
(ui,vj),(u

′
i,v

′
j)
)
= σ1(ui,u

′
i) if vj = v

′
j

= σ2(vj,v
′
j) if ui = u

′
i

= 0 otherwise

Let Gj denotes the fuzzy sub-graph of G2H induced by V1 ×vj ⊂ V1 ×V2. Then,

V (Gj) = {(u1,vj),(u2,vj), ...,(um,vj)}

µ(ui,vj) = µ1(ui)∧µ2(vj) ≤ µ1(ui)
and

σ
(
(ui,vj),(u

′
i,vj)

)
= min{σ1(ui,u′i),µ2(vj)}≤ σ(ui,u

′
i)

Claim: γfz(Gj) ≤ γfz(G). Define µ′j on Gj as µ′j(ui,vj) = µ′1(ui)∧µ2(vj)
Consider (ui,vj) ∈Gj. µ′1 is an fz-dominatng set of G implies that

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On fz- domination number of fuzzy graphs

µ1(ui) ≤ µ′1(ui) +
∑

uk∈G σ1(uk,ui)∧µ
′
1(uk).

Hence,

µ1(ui)∧µ2(vj) ≤ µ′1(ui)∧µ2(vj) +
∑
uk∈G

σ1(uk,ui)∧µ′1(uk)∧µ2(vj)

≤ µ′j(ui,vj) +
∑
uk∈G

σ((uk,vj),(ui,vj))∧µ′j(uk,vj)

That is,

µ(ui,vj) ≤ µ′j(ui,vj) +
∑
uk∈G

(σ((uk,vj),(ui,vj))∧µ′j(uk,vj)

Thus we get µ′j is an fz- dominating set of Gj for j = 1,2, ...,n
Also |µ′j| ≤ |µ′1| shows that γfz(Gj) ≤ γfz(G) for j = 1.2....,n
Hence

γfz(G2H) ≤ nγfz(g)
Similarly,

γfz(G2H) ≤ mγfz(H)
Thus we get,

γfz(G2H) ≤ min{nγfz(G),mγfz(H)}

In the previous theorem, equality might hold. For G,H and G2H given in
Fig. 6, γfz(G) = 0.2, γfz(H) = 0.2 and γfz(G2H) = 0.4 so that γfz(G2H) =
min{nγfz(g),mγfz(H)}

Figure 6: Fuzzy graph G,H and G2H

V. G. Vizing presented the following conjecture regarding the Cartesian prod-
uct of crisp graphs in 1968.

γ(G2H) ≥ γ(G)γ(H), for every pair of finite crisp graphs G and H.

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Lekh A and Parvathy K.S

Possibly the most significant unsolved issue in the field of domination the-
ory is Vizing’s conjecture. Here, we investigate the applicability of Vizing’s like
inequality to fz-dominantion in fuzzy graphs.

Vizing’s conjecture is said to be satisfied by a fuzzy graph G, if γfz(G2H) ≥
γfz(G)γfz(H) for every fuzzy graph H.
Definition 5.1. A fuzzy graph H = (V1,µ1,σ1) is known as a partial fuzzy sub-
graph of G = (V,µ,σ) induced by V1 if V1 ⊂ V , µ1(u) = µ(u) if u ∈ V1, 0
otherwise and σ1(u,v) = σ(u,v)∧µ(u)∧µ(v) for all u,v ∈ V .
Definition 5.2. The spanning fuzzy subgraph of G = (V,µ,σ) is the partial fuzzy
subgraph G′ = (V1,µ′,σ′) where V = V1 and µ = µ′

If G′ is a spanning fuzzy subgraph of the fuzzy graph G, then γfz(G′) ≥
γfz(G)).
Theorem 5.2. If G satisfies Vizing’s Conjecture and G′ is a spanning fuzzy sub-
graph of G such that γfz(G′) = γfz(G), then G′ also satisfies Vizing’s Conjecture.
Proof. G′2H is a spanning fuzzy sub- graph of G2H for every fuzzy graph H.
Hence

γfz(G′2H) ≥ γfz(G2H)
≥ γfz(G)γfz(H) = γfz(G′)γfz(H)

The example below illustrates that in general this inequality does not hold for
fz-domination in fuzzy graphs.

Example 5.1. Consider the fuzzy graphs G = (V1,µ1,σ1) and H = (V2,µ2,σ2)
given in figure 7. For G, V1 = {(a,1),(b,1),(c,1)}, σ1(a,b) = σ1(b,c) =
1,σ1(a,c) = 0. For H, V2 = {(u,1),(v,1),(w,1)}, σ2(u,v) = σ2(v,w) =
1,σ2(u,w) = 0. µ′1 = {(a,0.8),(b,0.6),(c,0.8)} is a minimum fz-dominating set
of G.
Hence γfz(G) = 2.2. Similarly γfz(H) = 2.2
Now µ′ = {((a,u),0.6),((a,v),0.4),((a,w),0.6)},((b,u),0.4),((b,v),0.2),
((b,w),0.4),((c,u),0.6),((c,v),0.4),((c,w),0.6) is a minimum fz-dominating set
of G2H. Hence γfz(G2H) = 4.2
Here

γfz(G2H) < γfz(G)γfz(H)
There are fuzzy graphs for which

1. γfz(G2H) < γfz(G)γfz(H)

2. γfz(G2H) = γfz(G)γfz(H)

3. γfz(G2H) > γfz(G)γfz(H)

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On fz- domination number of fuzzy graphs

Figure 7: Fuzzy Graphs G, H and G2H

6 Conclusions

Graph operations are techniques for creating new graphs from ones that al-
ready exist, and they are crucial in the design and analysis of large networks. In
this article, we investigate various characteristics of the fz-domination number of
fuzzy graphs under the influence of some graph operations. It is possible to derive
bounds for the fz-domination number of the union, join, corona, and Cartesian
product of fuzzy graphs. Through examples, the sharpness of these bounds are
demonstrated and the factors that contribute to the sharpness are examined.

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