Capitol Tech U Doctoral Presentation - April 2024.pptx
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Fuzzy graph
1.
2. FUZZY GRAPH
A Fuzzy graph G(ฯ, ยต)on G*(V,E) is a pair of functions ฯ : V โ [0,1] and ยต: V x V
โ [0,1] where for all u, v in V, we have ยต(u,v) โค min {ฯ (u), ฯ (v) }.
ฯ(u)=0.1
ฯ(v)=0.2 ฯ(w)=0.3
ยต(u,v)=0.1
ยต(u,w)=0.1
ยต(v,w)=0.2
3. The degree of any vertex ๐ข๐ of a fuzzy graph is sum of degree of
membership of all those edges which are incident on vertex ๐ข๐.And is
denoted by d (๐ข๐).
A fuzzy sub-graph H : (ฯ , ฯ ) is called a fuzzy sub-graph of G=(ฯ,ยต) if
ฯ(u) โคฯ(u) for all uัV. And ฯ ( u, v)โคยต ( u ,v) all u ,v ั V
4. A fuzzy sub-graph H : (ฯ , ฯ ) is said to be a spanning fuzzy graph of G=(ฯ,ยต) if ฯ(u) =ฯ(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
ยต (x,y) = min {ฯ (x), ฯ (y) }.
A fuzzy graph is called an effective fuzzy graph if every edge is an
effective edge.
5. The degree of any vertex ๐ข๐ of an effective fuzzy graph is sum of
degree of membership of all those edges which are incident on
vertex ๐ข๐.And is denoted by dE1(๐ข๐).
The minimum effective incident degree of a fuzzy graph G is ^ {
dE1 (v) / v โ V} . and it is denoted by ฮดE1 (G).
The maximum effective incident degree of a fuzzy graph
G is v { dE1 (v) / v โ V} . and it is denoted by โE1(G)
6. The order of a effective fuzzy graph G is O(G)= ๐ขโ๐ ๐(๐ข)
The size of a effective fuzzy graph G is S(G)= ๐ข๐ฃโ๐ธ1 ๐(๐ข๐ฃ).
Let G=(๐, ๐) be a fuzzy graph on G*=(V,E).If dG(v)=k for all vโV 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.
7. .
For any real number ๐ผ,0< ๐ผ โค 1, a ๐ผ -path ๐ ๐ผ in a fuzzy graph G =
(ฯ,ยต) is a sequence of distinct vertices ๐ฅ0, ๐ฅ1, ๐ฅ2 โฆ โฆ โฆ ๐ฅ ๐ such that
ฯ (๐ฅ๐)โฅ ๐ผ , 0โค ๐ โค ๐, and
ยต(๐ฅ๐โ1, ๐ฅ๐)โฅ ๐ผ, 0โค ๐ โค ๐, here nโฅ 0, is called the length of ๐ ๐ผ.In this
case we write ๐ ๐ผ=(๐ฅ0, ๐ฅ1, ๐ฅ2 โฆ โฆ โฆ . . ๐ฅ ๐) and ๐ ๐ผ is called a (๐ฅ0, ๐ฅ ๐) โ
๐ผ path.
8. 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 โฅ 3.
A fuzzy graph G = (ฯ , ยต) 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.
9. 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.
10. Fuzzy Domination Number
The complement of a fuzzy graph G=(ฯ , ฮผ) is a fuzzy graph
๐บ =( ๐, ๐)where ๐=ฯ and ๐(u ,v )=ฯ(u) ฮ ฯ(v)-ฮผ(u ,v ) for all u ,v in
V.
The complement of a complement fuzzy graph ๐บ = ( ๐, ๐) where ๐= ๐
and ๐(u,v)=๐(๐ข) ฮ๐(๐ฃ)-๐(๐ข, ๐ฃ) for all u,v in V i.e
๐(u,v)= ฯ(u) ฮ ฯ(v)-( ฯ(u) ฮ ฯ(v)-ฮผ(u ,v )) for all u,v in V then
๐บ = G
11. 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
๐บ
G
Let G=(ฯ , ฮผ) 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 ั V-D .there exists u in D such that. ยต(u,v) =ฯ (u)ห
ฯ (v).
12. A fuzzy dominating set D of a fuzzy graph G is called minimal dominating set of G, if for every
vertex v ั D ,D-{v} is not a dominating set. The domination number ฮณ (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
13. 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.
14. A fuzzy independent set S of G is said to be maximal
fuzzyindependent, if for every vertex v ั V-S, the set Sโช{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.
16. Fuzzy Global and Factor Domination ,Fuzzy Multiple Domination
Fuzzy Global Domination Number
A fuzzy graph H=(ฯ,ฮผ) on H*(V,E) is said to have a t-factoring into
factors F(H)= {G1 G2,G3,......Gt}if each fuzzy graph Gi=(ฯi,ฮผi)such
that ฯi=ฯ and the set{ฮผ1,ฮผ2,ฮผ3โฆโฆ..ฮผt}form a partition of ฮผ.
Given a t-factoring F of H, a subset DfโV is a fuzzy factor
dominating set if Df is a fuzzy dominating set of Gi, for1โคiโคt.
17. The fuzzy factor domination number is the minimum cardinality of a fuzzy factor
dominating sets of F(H).and is denoted by ฮณft(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
19. Fuzzy factor domimating set={a,c,e}
Fuzzy factor domination number=0.6
letG=(ฯ, ฮผ) be a fuzzy graph on G*(V,E).A subset Dg of V is said to
be fuzzy global Dominating set of G and ๐ฎ if for every vัV- Dg
there exists u in Dg such that ยต(u,v) =ฯ (u)ห ฯ (v)both G and ๐ฎ.
20. A fuzzy global dominating set Dg of a fuzzy graph G is called minimal global
dominating set of G, if for every vertex v ั 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 ฮณg(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)
21. Fuzzy global dominating sets {a,d} and{b,c}
Fuzzy global domination number=0.6
For any real number ๐ผ,0< ๐ผ โค 1, a vertex cover of a fuzzy graph
G=(๐, ๐) on G*=(V,E) is a set of vertices ฯ (๐ฅ๐)โฅ ๐ผ , 0โค ๐ โค ๐ that
covers all the edges such that ยต(๐ฅ๐โ1, ๐ฅ๐)โฅ ๐ผ, 0โค ๐ โค ๐, here nโฅ 0,
22. An edge cover of a fuzzy graph is a set of edges ยต(๐ฅ๐โ1, ๐ฅ๐)โฅ ๐ผ,
0โค ๐ โค ๐, that covers all the vertices such that ฯ (๐ฅ๐)โฅ ๐ผ , 0โค
๐ โค ๐. The minimum cardinality of vertex cover is ฮฑ0(G) and the
minimum cardinality of edge cover isฮฑ1 (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}
ฮฑ0(G)=0.5
ฮฑ1(G)=0.4
23. Let G= (ฯ,ยต) be a fuzzy graph . And let D be a subset of V is said to be fuzzy k-
dominating set if for every vertex vัV-D , there exists atleast โkโu in D such that
ยต(u,v)=ฯ(u)หฯ(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 ๐พk (G).
25. Domination in Fuzzy Digraphs
A fuzzy digraph GD= (ฯD,ฮผD) is a pair of function ฯD :Vโ[0,1] and
ฮผD : VรVโ[0,1] where ฮผD(u,v)โค ฯD (u) ฮ ฯD (v) for u,v ั V, ฯD a
fuzzy set of V,(Vร ๐, ฮผD ) a fuzzy relation on V and ฮผD is a set of
fuzzy directed edges are called fuzzy arcs.
Let GD= (ฯD,ฮผD) be a fuzzy digraph of V.IfฯD(u)>0, for u in V, then
u is called a vertex of GD.IfฯD(u) = 0 for u in V,then u is called an
empty vetex of GD.IfฮผD(u,v)=0, then (u,v) is called an empty arc of
GD.
26. Let ๐บ ๐ท1= (๐ ๐ท1, ๐ ๐ท1) and ๐บ ๐ท2= (๐ ๐ท2, ๐ ๐ท2) be two fuzzy
digraphs of V . Then๐บ ๐ท2= (๐ ๐ท2, ๐ ๐ท2) called a fuzzy sub-
digraph of ๐บ ๐ท1= (๐ ๐ท1, ๐ ๐ท1) if
๐ ๐ท2(u) โค ๐ ๐ท1(u)for all u in V and
๐ ๐ท2(u,v) โค ๐ ๐ท1(u,v) for all u,v in V, then we write
๐บ ๐ท2 โค ๐บ ๐ท1.
27. For any real number๐ผ,0< ๐ผ โค 1,a fuzzy directed walk from a
vertex ๐ ๐ท(๐ฅ๐) to ๐ ๐ท(๐ฅ๐) is an alternating sequence of vertices
and edges, beginning with ๐ ๐ท(๐ฅ๐) and ending with ๐ ๐ท(๐ฅ๐) , such
that๐ ๐ท(๐ฅ๐)โฅ ๐ผ , 0โค ๐ โค ๐, and
๐ ๐ท(๐ฅ๐โ1, ๐ฅ๐)โฅ ๐ผ, 0โค ๐ โค ๐, here nโฅ 0, ๐๐๐ 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
.
28. For any real number ๐ผ,0< ๐ผ โค 1, a directed ๐ผ -path ๐ ๐ผ in a
fuzzy digraph ๐บ ๐ท = (๐ ๐ท, ๐ ๐ท) is a sequence of distinct nodes
๐ฅ0, ๐ฅ1, ๐ฅ2 โฆ โฆ โฆ ๐ฅ ๐ such that
๐ ๐ท(๐ฅ๐)โฅ ๐ผ , 0โค ๐ โค ๐, and
๐ ๐ท(๐ฅ๐โ1, ๐ฅ๐)โฅ ๐ผ, 0โค ๐ โค ๐, here nโฅ 0, is called the length of
๐ ๐ผ.In this case ,we write ๐ ๐ผ=(๐ฅ0, ๐ฅ1, ๐ฅ2 โฆ โฆ โฆ . . ๐ฅ ๐) and ๐ ๐ผ is
called a (๐ฅ0, ๐ฅ ๐) โ๐ผ path.
29. 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 ฮฒ0(GD) is the maximum cardinality of
an independent set in GD.
30. A subset S of V in a fuzzy digraph is said to be a fuzzy
dominating set of GD if every vertex v ะ V -S ,there exists u in S
such that ฮผD (u,v)=ฯD (u) ฮ ฯD (v).
The fuzzy domination number ฮณ(GD) of a fuzzy digraph GD is the
minimum cardinality of a fuzzy dominating set in GD