Fuzzy graph

FUZZY GRAPH
A Fuzzy graph G(σ, µ)on G*(V,E) is a pair of functions σ : V → [0,1] and µ: V x V
→ [0,1] where for all u, v in V, we have µ(u,v) ≤ min {σ (u), σ (v) }.
σ(u)=0.1
σ(v)=0.2 σ(w)=0.3
µ(u,v)=0.1
µ(u,w)=0.1
µ(v,w)=0.2

The degree of any vertex 𝑢𝑖 of a fuzzy graph is sum of degree of
membership of all those edges which are incident on vertex 𝑢𝑖.And is
denoted by d (𝑢𝑖).
A fuzzy sub-graph H : (τ , υ) is called a fuzzy sub-graph of G=(σ,µ) if
τ(u) ≤σ(u) for all uєV. And υ( u, v)≤µ ( u ,v) all u ,v є V

A fuzzy sub-graph H : (τ , υ) is said to be a spanning fuzzy graph of G=(σ,µ) if τ(u) =σ(u)
for all u. In this case, two graphs have same vertex set, they differ only in the arc weights.
An edge E1 (x,y) of a fuzzy graph is called an effective edge if
µ (x,y) = min {σ (x), σ (y) }.
A fuzzy graph is called an effective fuzzy graph if every edge is an
effective edge.

The degree of any vertex 𝑢𝑖 of an effective fuzzy graph is sum of
degree of membership of all those edges which are incident on
vertex 𝑢𝑖.And is denoted by dE1(𝑢𝑖).
The minimum effective incident degree of a fuzzy graph G is ^ {
dE1 (v) / v ∈ V} . and it is denoted by δE1 (G).
The maximum effective incident degree of a fuzzy graph
G is v { dE1 (v) / v ∈ V} . and it is denoted by ∆E1(G)

The order of a effective fuzzy graph G is O(G)= 𝑢∈𝑉 𝜎(𝑢)
The size of a effective fuzzy graph G is S(G)= 𝑢𝑣∈𝐸1 𝜇(𝑢𝑣).
Let G=(𝜎, 𝜇) be a fuzzy graph on G*=(V,E).If dG(v)=k for all v∈V that
is if each vertex has same degree k, then G is said to be a regular
fuzzy graph of degree k or a k-degree fuzzy graph.

.
For any real number 𝛼,0< 𝛼 ≤ 1, a 𝛼 -path 𝜌 𝛼 in a fuzzy graph G =
(σ,µ) is a sequence of distinct vertices 𝑥0, 𝑥1, 𝑥2 … … … 𝑥 𝑛 such that
σ (𝑥𝑗)≥ 𝛼 , 0≤ 𝑗 ≤ 𝑛, and
µ(𝑥𝑖−1, 𝑥𝑖)≥ 𝛼, 0≤ 𝑖 ≤ 𝑛, here n≥ 0, is called the length of 𝜌 𝛼.In this
case we write 𝜌 𝛼=(𝑥0, 𝑥1, 𝑥2 … … … . . 𝑥 𝑛) and 𝜌 𝛼 is called a (𝑥0, 𝑥 𝑛) –
𝛼 path.

A path P is called effective path if each edge in a path P is an effective edge.
An effective path P is called an effective cycle if x0 = xn and n ≥ 3.
A fuzzy graph G = (σ , µ) is said to be effective connected if there exists an
effective path between every pair of vertices.
A fuzzy tree is an acyclic and connected fuzzy graph.

A fuzzy effective tree is an effective acyclic and effective
connected fuzzy graph.
The fuzzy effective tree T is said to be a fuzzy effective spanning
tree of a fuzzy effective connected graph G if T is an effective sub
graph of an effective fuzzy graph G and T contains all vertices of G.

Fuzzy Domination Number
The complement of a fuzzy graph G=(σ , μ) is a fuzzy graph
𝐺 =( 𝜎, 𝜇)where 𝜎=σ and 𝜇(u ,v )=σ(u) Λ σ(v)-μ(u ,v ) for all u ,v in
V.
The complement of a complement fuzzy graph 𝐺 = ( 𝜎, 𝜇) where 𝜎= 𝜎
and 𝜇(u,v)=𝜎(𝑢) Λ𝜎(𝑣)-𝜇(𝑢, 𝑣) for all u,v in V i.e
𝜇(u,v)= σ(u) Λ σ(v)-( σ(u) Λ σ(v)-μ(u ,v )) for all u,v in V then
𝐺 = G

u(0.8) v(0.5)
0.5
w(0.7) x(0.5)
0.5
0.5
u(0.8) v(0.5)
w(0.7
x(0.5)
0.5
0.5
0.5
𝐺
G
Let G=(σ , μ) be a fuzzy graph on G*(V,E) . A subset D of V is said to be fuzzy
dominating set of G if for every v є V-D .there exists u in D such that. µ(u,v) =σ (u)˄
σ (v).

A fuzzy dominating set D of a fuzzy graph G is called minimal dominating set of G, if for every
vertex v є D ,D-{v} is not a dominating set. The domination number γ (G) is the minimum
cardinality teaken over all minimal dominating sets of vertices of G.
a(0.3)
b(0.2)
c(0.1)
d(0.2)
e(0.2)
0.1
0.10.1
0.2
0.2
0.2
0.2
0.2

Fuzzy Domination Set D={a}
Fuzzy Domination Number=0.3
Two vertices in a fuzzy graph G are said to be fuzzy independent if there
is no strong arc between them.
A subset S of V is said to be fuzzy independent set of G if every two
vertices of S are fuzzy independent.

A fuzzy independent set S of G is said to be maximal
fuzzyindependent, if for every vertex v є V-S, the set S∪{v} is not
a fuzzy independent.
The independence number i(G) is the minimum cardinalities taken
over all maximal independent sets of nodes of G.

a(0.1)
b(0.1)
c(0.2)
d(0.2)
0.1
0.1
0.1
Maximal fuzzy independent={a,c,d} and i(G)=0.5

Fuzzy Global and Factor Domination ,Fuzzy Multiple Domination
Fuzzy Global Domination Number
A fuzzy graph H=(σ,μ) on H*(V,E) is said to have a t-factoring into
factors F(H)= {G1 G2,G3,......Gt}if each fuzzy graph Gi=(σi,μi)such
that σi=σ and the set{μ1,μ2,μ3……..μt}form a partition of μ.
Given a t-factoring F of H, a subset Df⊆V is a fuzzy factor
dominating set if Df is a fuzzy dominating set of Gi, for1≤i≤t.

The fuzzy factor domination number is the minimum cardinality of a fuzzy factor
dominating sets of F(H).and is denoted by γft(F(H)) .
a(0.3)
b(0.2)
c(0.1)
d(0.2)
e(0.2)
H
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2

a(0.3)
b(0.2)
c(0.1)
0.1
d(0.2)
e(0.2)
0.2
0.2
a(0.3)
b(0.2)
c(0.1)d(0.2)
e(0.2)
0.2
0.2
0.1
a(0.3)
b(0.2)
c(0.1)d(0.2)
d(0.2

Fuzzy factor domimating set={a,c,e}
Fuzzy factor domination number=0.6
letG=(σ, μ) be a fuzzy graph on G*(V,E).A subset Dg of V is said to
be fuzzy global Dominating set of G and 𝑮 if for every vєV- Dg
there exists u in Dg such that µ(u,v) =σ (u)˄ σ (v)both G and 𝑮.

A fuzzy global dominating set Dg of a fuzzy graph G is called minimal global
dominating set of G, if for every vertex v є Dg , Dg -{v} is not a dominating set. The global
domination number is the minimum cardinality taken over all minimal dominating sets of
vertices of G. and is denoted by γg(G)
a(0.4)
c(0.4)
b(0.2)d(0.2)
0.2
0.2
0.2
0.2
a(0.4)
b(0.2)
d(0.2)
c(0.4)
0.2
0.4)

Fuzzy global dominating sets {a,d} and{b,c}
Fuzzy global domination number=0.6
For any real number 𝛼,0< 𝛼 ≤ 1, a vertex cover of a fuzzy graph
G=(𝜎, 𝜇) on G*=(V,E) is a set of vertices σ (𝑥𝑗)≥ 𝛼 , 0≤ 𝑗 ≤ 𝑛 that
covers all the edges such that µ(𝑥𝑖−1, 𝑥𝑖)≥ 𝛼, 0≤ 𝑖 ≤ 𝑛, here n≥ 0,

An edge cover of a fuzzy graph is a set of edges µ(𝑥𝑖−1, 𝑥𝑖)≥ 𝛼,
0≤ 𝑖 ≤ 𝑛, that covers all the vertices such that σ (𝑥𝑗)≥ 𝛼 , 0≤
𝑗 ≤ 𝑛. The minimum cardinality of vertex cover is α0(G) and the
minimum cardinality of edge cover isα1 (G).
a(0.2)
d(0.2)
b(0.3)
c(0.4)
0.2
0.2
0.2
0.3
Vertex cover={a,c}
and {b,d}
α0(G)=0.5
α1(G)=0.4

Let G= (σ,µ) be a fuzzy graph . And let D be a subset of V is said to be fuzzy k-
dominating set if for every vertex vєV-D , there exists atleast ‘k’u in D such that
µ(u,v)=σ(u)˄σ(v).
In a fuzzy graph G every vertex in V-D is fuzzy k- dominated, then D is
called a fuzzy k-dominating set.
The minimum cardinality of a fuzzy k-dominating set is
called the fuzzy k-domination number 𝛾k (G).

a(0.2)
d(0.2) g(0.2)
b(0.1)
c(0.1)
e(0.1)
f(0.1)
h(0.1)
0.1
0.1
0.1 0.1
0.1
0.1
0.1
0.1
D={b,c,e,f,h} and V-D= {a,d,g}
D is a fuzzy two dominating set
The fuzzy two domination number= 0.5

Domination in Fuzzy Digraphs
A fuzzy digraph GD= (σD,μD) is a pair of function σD :V→[0,1] and
μD : V×V→[0,1] where μD(u,v)≤ σD (u) Λ σD (v) for u,v є V, σD a
fuzzy set of V,(V× 𝑉, μD ) a fuzzy relation on V and μD is a set of
fuzzy directed edges are called fuzzy arcs.
Let GD= (σD,μD) be a fuzzy digraph of V.IfσD(u)>0, for u in V, then
u is called a vertex of GD.IfσD(u) = 0 for u in V,then u is called an
empty vetex of GD.IfμD(u,v)=0, then (u,v) is called an empty arc of
GD.

Let 𝐺 𝐷1= (𝜎 𝐷1, 𝜇 𝐷1) and 𝐺 𝐷2= (𝜎 𝐷2, 𝜇 𝐷2) be two fuzzy
digraphs of V . Then𝐺 𝐷2= (𝜎 𝐷2, 𝜇 𝐷2) called a fuzzy sub-
digraph of 𝐺 𝐷1= (𝜎 𝐷1, 𝜇 𝐷1) if
𝜎 𝐷2(u) ≤ 𝜎 𝐷1(u)for all u in V and
𝜇 𝐷2(u,v) ≤ 𝜇 𝐷1(u,v) for all u,v in V, then we write
𝐺 𝐷2 ≤ 𝐺 𝐷1.

For any real number𝛼,0< 𝛼 ≤ 1,a fuzzy directed walk from a
vertex 𝜎 𝐷(𝑥𝑖) to 𝜎 𝐷(𝑥𝑗) is an alternating sequence of vertices
and edges, beginning with 𝜎 𝐷(𝑥𝑖) and ending with 𝜎 𝐷(𝑥𝑗) , such
that𝜎 𝐷(𝑥𝑗)≥ 𝛼 , 0≤ 𝑗 ≤ 𝑛, and
𝜇 𝐷(𝑥𝑖−1, 𝑥𝑖)≥ 𝛼, 0≤ 𝑖 ≤ 𝑛, here n≥ 0, 𝑎𝑛𝑑 each edge is oriented
from the vertex preceding it to the vertex following it. No edge in
a fuzzy directed walk appears more than once, but a vertex may
appears more than once, as in the case of fuzzy undirected graphs
.

For any real number 𝛼,0< 𝛼 ≤ 1, a directed 𝛼 -path 𝜌 𝛼 in a
fuzzy digraph 𝐺 𝐷 = (𝜎 𝐷, 𝜇 𝐷) is a sequence of distinct nodes
𝑥0, 𝑥1, 𝑥2 … … … 𝑥 𝑛 such that
𝜎 𝐷(𝑥𝑗)≥ 𝛼 , 0≤ 𝑗 ≤ 𝑛, and
𝜇 𝐷(𝑥𝑖−1, 𝑥𝑖)≥ 𝛼, 0≤ 𝑖 ≤ 𝑛, here n≥ 0, is called the length of
𝜌 𝛼.In this case ,we write 𝜌 𝛼=(𝑥0, 𝑥1, 𝑥2 … … … . . 𝑥 𝑛) and 𝜌 𝛼 is
called a (𝑥0, 𝑥 𝑛) –𝛼 path.

Two vertices in a fuzzy digraphs GD are said to be fuzzy independent
if there is no effective edges between them.
A subset S of Vis said to be fuzzy independent set of GD if every two
vertices of S are fuzzy independent.
The fuzzy independence number β0(GD) is the maximum cardinality of
an independent set in GD.

A subset S of V in a fuzzy digraph is said to be a fuzzy
dominating set of GD if every vertex v Є V -S ,there exists u in S
such that μD (u,v)=σD (u) Λ σD (v).
The fuzzy domination number γ(GD) of a fuzzy digraph GD is the
minimum cardinality of a fuzzy dominating set in GD

a(0.1)
d(0.1)
g(0.1
h(0.2)
b(0.2)
c(0.2)
e(0.2)
f(0.2)
0.1
0.1
0.1
0.1 0.1
0.1
0.1
γ(GD) =0.3

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