# A New Type-2 Soft Set: Type-2 Soft Graphs and Their Applications

Authors: Khizar Hayat; Muhammad Irfan Ali; Bing-Yuan Cao; Xiao-Peng Yang
Journal: Advances in Fuzzy Systems
 (2017)

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## Abstract

The correspondence between a vertex and its neighbors has an essential role in the structure of a graph. Type-2 soft sets are also based on the correspondence of primary parameters and underlying parameters. In this study, we present an application of type-2 soft sets in graph theory. We introduce vertex-neighbors based type-2 soft sets overX (set of all vertices of a graph) and E (set of all edges of a graph). Moreover, we introduce some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. Finally, we describe an application of type-2 soft graphs in communication networks and present procedure as an algorithm.


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## Body

## 1. Preliminaries and Introduction

A graphG=(X,E) consists of a nonempty set of objects X, called vertices, and a set E of pair of the element of X, called edges. Two vertices x and y are adjacent if xy∈E. A simple graph is an unweighted, undirected graph containing no multiple edges or graph loops. A graph G′=(X′,E′) is said to be a subgraph of G=(X,E) if X′⊆X and E′⊆E. The neighborhoods of a vertex x in a graph are the set of all the vertices adjacent to x including x itself. The neighbors of a vertex x in a graph are the set of all the vertices adjacent to x excluding x itself. The eccentricity of the vertex x is the maximum distance from x to any vertex. The radius of a graph is the minimum eccentricity of any vertex xGraph theory is rapidly moving into the mainstream of mathematics, mainly because of its applications in diverse fields which include electrical engineering (communications networks and coding theory), computer science (algorithms and computations), and operations research (scheduling). In many cases, it is important to deal with uncertainties using fuzzy sets and logics. The fuzzy graph is a weighted graph which gives a normalized relational strength over a fuzzy subset of a set [1]. Kauffman [2] gave the fundamental definition of a fuzzy graph on the basis of Zadeh’s fuzzy relations [3]. Rosenfeld [1] introduced another elaborated definition of fuzzy graphs. Subsequently, Bhattacharya [4] gave some useful results on fuzzy graphs and Mordeson and Nair [5] investigated some operations on fuzzy graph theory. In recent decades, fuzzy graphs were studied by many researchers. Akram et al. extended the concept of fuzzy graphs to bipolar fuzzy graphs [6], intuitionistic fuzzy hypergraphs [7], intuitionistic fuzzy graphs structures [8], and interval valued fuzzy graph [9]. Samanta et al. [10–12] studied vague graphs, fuzzy planar graphs, completeness and regularity of generalized fuzzy graph, and irregular bipolar fuzzy graphs.Soft set theory [13], firstly initiated by Molodtsov, is a new mathematical tool for dealing with uncertainties. Indeed, Molodtsov’s approach shows the applicability of soft sets to several fields and provides some fundamental results subsequently augmented by works like Maji et al. [14] and Aktaş and Çağman [15], among others. Ali et al. [16] presented some new operations in soft set theory and based on the analysis of several operations on soft sets Sezgin and Atagün [17] studied the theoretical aspect of the soft set theory.LetA be a set of parameters that can have an arbitrary nature (numbers, functions, sets of words, etc.). Let X be a universal set and the power set of X is denoted by P(X). The formal definition of soft is defined as follows.Definition 1 (see [14]).
A pair(S,A) is called a soft set over X, where S is a mapping given by S:A→P(X).
We refer to Molodtsov’s soft sets as type-1 soft sets (briefly T1SS). Note that the set of all T1SS overX will be denoted by σX. We refer to [13, 16, 18] for basic notions of soft set theory.
The applications of the soft set are progressing rapidly and many researchers are focusing on real and practical problems. Interestingly, some progress of soft sets has developed with graph theory. Ali et al. [19] introduced a new representation of graphs based on neighborhoods and soft sets. Akram and Nawaz presented the concepts of soft graphs [20, 21] and fuzzy soft graphs [22]. Let G=(X,E) be a simple graph; let A be any nonempty set. Let R⊆A×X be an arbitrary relation from A to X. A mapping S from A to P(X) is denoted as S:A→P(X) and defined as S(x)={y∈X∣xRy} and a mapping T from A to P(E) is denoted as T:A→P(E) and defined as T(x)={uv∈E∣{u,v}⊆S(x)}. Then (S,A) is a T1SS over X and (T,A) is a T1SS over E. The notion of a soft graph is defined as follows.Definition 2 (see [21]).
A4-tuple G∘=(G,S,T,A) is called a soft graph if it satisfies the following conditions:(i)
G=(X,E) is a simple graph.(ii)
A is a nonempty set of parameters.(iii)
[S,A] is a T1SS over X.(iv)
[T,A] is a T1SS over E.(v)
(S(x),T(x)), for all x∈A, represents a subgraph of G.
The soft graphG∘=(G,S,T,A) can also be written as G∘=〈G,S,T〉={H(x)∣x∈A}.
In the rest of the paper, soft graph will be written as type-1 soft graph (briefly, T1SG). Some operations of T1SG are defined as follows.Definition 3 (see [20]).
LetG∘=〈S1,T1,A1〉 and G′∘=〈S2,T2,A2〉 be two T1SG of G. Then G′∘ is a type-1 soft subgraph of G∘ if(i)
A2⊆A1,(ii)
for eachx∈A2, H2(x) is a subgraph of H1(x).Definition 4 (see [20]).
LetG1∘=〈S1,T1,A1〉 and G2∘=〈S2,T2,A2〉 be two T1SG of G. The extended union of G1∘ and G2∘, denoted by G1∘⊔εG2∘=G∘=〈I,J,C〉, where C=A1∪A2, is defined, ∀α∈C, as(1)Iα=S1α,ifα∈A1-A2S2α,ifα∈A2-A1S1α∪S2α,ifα∈A1∩A2Jα=T1α,ifα∈A1-A2T2α,ifα∈A2-A1T1α∪T2α,ifα∈A1∩A2.
It can be written asG1∘⊔εG2∘={H(α)=(I(α),J(α))∣α∈C}.Definition 5 (see [20]).
LetG1∘=〈S1,T1,A1〉 and G2∘=〈S2,T2,A2〉 be two T1SG in G. The restricted intersection of G1∘=〈S1,T1,A1〉 and G2∘=〈S2,T2,A2〉, denoted by G1∘⊓rG2∘=G∘=〈I,J,C〉, where C=A1∩A2, is defined, ∀α∈C, as I(α)=S1(α)∩S2(α), J(α)=T1(α)∩T2(α).
It can be written asG1∘⊓rG2∘={H(x)=(I(x),J(x))∣x∈C}.Definition 6 (see [20]).
LetG1∘=〈S1,T1,A1〉 and G2∘=〈S2,T2,A2〉 be two T1SG in G. The AND operation of G1∘ and G2∘, denoted by G1∘∧G2∘=G∘=〈I,J,C〉, where C=A1×A2, is defined, ∀(α,β)∈A1×A2, as I(α,β)=S1(α)∩S2(β), J(α,β)=T1(α)∩T2(β).Definition 7 (see [20]).
LetG1∘=〈S1,T1,A1〉 and G2∘=〈S2,T2,A2〉 be two T1SG in G. The OR operation of G1∘ and G2∘, denoted by G1∘∨G2∘=G∘=〈I,J,C〉, where C=A1×A2, is defined, ∀(α,β)∈A1×A2, as I(α,β)=S1(α)∪S2(β), J(α,β)=T1(α)∪T2(β).Definition 8 (see [20]).
LetG1∘=〈S1,T1,A1〉 and G2∘=〈S2,T2,A2〉 be two T1SG of G1 and G2, respectively, such that A1∩A2=∅. TheCartesian product of G1∘ and G2∘ is denoted and defined by G1∘⋊G2∘=G∘=〈K,A1×A2〉, ∀(α,β)∈A1×A2, K(α,β)=H1(α)×H2(β), where H1(α) and H2(β) are the subgroups of G1 and G2, respectively.
Note thatK(α,β)=H1(α)×H2(β) denotes the Cartesian product of two crisp subgraphs. That is, G1∘⋊G2∘={K(α,β)∣(α,β)∈A1×A2}.
For basic definitions of graph and soft graph, see [20, 21, 23–25].
In order to deal with associations between parameters, Chatterjeea et al. [26] propose the concept of type-2 soft set which is a generalization of Molodtsov’s soft set. It involves parameterization over an already parameterized set and hence has more freedom and efficiency compared to usual soft sets (termed as type-1 soft sets) in handling impreciseness. In fact, type-2 soft sets are a new approach to managing uncertainty. Interestingly, Chatterjeea et al. [26] investigate some basic operations on type-2 soft sets and use the model of type-2 soft sets in decision making problems. Recently, distance, entropy, and similarity measures of type-2 soft sets are introduced by Chatterjeea et al. [27].
LetX be universe set and let E be the set of parameters.Definition 9 (see [26]).
Let(X,E) be a soft universe and let σ(X) be the collection of all T1SS over (X,E). Then a mapping S:A→σ(X), A⊆E is called a type-2 soft set (briefly T2SS) over (X,E) and it is denoted by [S∗,A]. In this case, corresponding to each parameter α∈A, S∗(α) is a T1SS. Thus, for each α∈A, there exists a T1SS, (Sα,Lα) such that S∗(α)=(Sα,Lα), where Sα:Lα→P(X) and Lα⊂E. In this case, we refer to the parameter set A as the “primary set of parameters,” while the set of parameters ∪Lα is known as the “underlying set of parameters.”
In this study, we present an application of type-2 soft set in graph theory. We introduce vertex-neighbors based type-2 soft sets overX (set of all vertices of a graph) and E (set of all edges of a graph). Moreover, we introduce some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. We describe an application of type-2 soft graphs in communication networks and present procedure as an algorithm.


## 2. Type-2 Soft Graphs

The behavior and the selections of initial parameters and associated parameterized sets corresponding to initial parameters make the study of type-2 soft sets very interesting. Motivated by this, we study type-2 soft sets in graph theory and related properties. In a graph, every vertex regarding its location consists of a special relationship with its neighbors. Suppose that a company makes an analysis on the flow of transportation in two cities “C1” and “C2” of a country. We consider edge (line segment (link) between locations, that, is road) and vertex (location on the transportation network of interest (node), that is, town and road intersections). The company must make a survey on the flow of transportation from neighbor cities to control transportation in “C1” and “C2” and obtain a graphical representation to analyze the flow of transportation. The vertex-neighbors correspondence has an essential role in the structure of a graph. The type-2 soft set is also based on the correspondence of initial parameters and underlying parameters. The set of all neighbors of an element x∈X is defined as follows.Definition 10.
LetG=(X,E) be a simple graph. The set of neighbors of an element x∈X is denoted by NBx and defined by NBx={z∈X∣xz∈E}.
LetG=(X,E) be a simple graph and A⊆X. Then NBA=⋃x∈ANBx. Let a subset R of NBA×X be an arbitrary relation from NBA to X. The set of all T1SS over X is denoted by Γ(X) and the set of all T1SS over E is denoted by Γ(E).Definition 11.
LetG=(X,E) be a simple graph; let A⊂X and Γ(X) be the collection of all T1SS over X. Then a mapping S∗:A→Γ(X) is called a T2SS over X and it is denoted by [S∗,A]. In this case, corresponding to each vertex x∈A, S∗(x) is a T1SS such that S∗(x)=(Sx,NBx), where Sx:NBx→P(X) can be defined as Sx(z)={y∈X∣zRy}, for all z∈NBx⊆X, and NBx is the set of all neighbors of x∈A. This T2SS is also called a vertex-neighbors induced type-2 soft set (briefly, VN-type-2 soft set) over X.Definition 12.
LetG=(X,E) be a simple graph; let A⊂X and Γ(E) be the collection of all T1SS over E. Let [S∗,A] be a VN-type-2 soft set over X. Then a mapping T∗:A→Γ(E) is called a T2SS over E and it is denoted by [T∗,A]. In this case, corresponding to each vertex x∈A, T∗(x) is a T1SS such that T∗(x)=(Tx,NBx), where Tx:NBx→P(E) can be defined as Tx(z)={αβ∈E∣{α,β}⊆Sx(z)}, for all z∈NBx⊆X, and NBx is the set of all neighbors of x∈A. This T2SS is also called a VN-type-2 soft set over E.
As it is mentioned in the above definitions,S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx)∀x∈A are T1SS over X and over E, respectively. If (Sx(z),Tx(z))∀z∈NBx represents a subgraph of G, then by the definition of T1SG (G,Sx,Tx,NBx) will be a T1SG of G. This T1SG can also be represented by G,Sx,Tx={Hx(z)∣z∈NBx}.


### 2.1. Type-2 Soft Graph

A 5-tupleG∗=(G,S∗,T∗,A,NBA) is called a type-2 soft graph (briefly, T2SG) if it satisfies the following conditions:(i)
G=(X,E) is a simple graph.(ii)
A is a nonempty set of parameters.(iii)
[S∗,A] is a VN-type-2 soft set over X.(iv)
[T∗,A] is a VN-type-2 soft set over E.(v)
T1SS corresponding to(S∗(x),T∗(x))∀x∈A, represents a type-1 soft graph.A T2SG can also be represented byG∗=〈S∗,T∗,A〉={H∗(x)∣x∈A}, where H∗(x)=(Hx,NBx) such that Hx(z)=(Sx(z),Tx(z)), ∀z∈NBx. The set of all T2SG of G is denoted by TSG(G).Although the above representation of graphs provides a new framework for the applications of graph theory, it is restricted to the information from neighbors. In this regard, a likeness between parameterized vertices appears in some cases such as distinction between T2SG〈S∗,T∗,A〉 and 〈S′∗,T′∗,A〉 is limited toward respective T1SG. However, T2SG is more general and hence a more efficient approach in graph theory as compared to T1SG.Definition 13.
A T2SGG∗=(G,S∗,T∗,A,NBA) is said to be vertex-induced if H∗(γ)=(S∗(γ),T∗(γ))=S∗(γ) for all γ∈A.Definition 14.
A T2SGG∗=(G,S∗,T∗,A,NBA) is said to be edge-induced if H∗(γ)=(S∗(γ),T∗(γ))=T∗(γ) for all γ∈A.Example 15.
Consider a graphG=(X,E) as shown in Figure 1.
LetA={a,b}⊂X. Then NBa={b,c,d}, NBb={a,c,d}. Let [S∗,A] and [T∗,A] be two T2SS over X and E, respectively, such that S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx) for all x∈A. Let Sa(α)={y∈X∣αRy⇔d(α,y)≤1}, Ta(α)={uv∈E∣{u,v}⊆Sa(α)}∀α∈NBa and Sb(β)={y′∈X∣βRy′⇔d(β,y′)=1}, Tb(β)={u′v′∈E∣{u′,v′}⊆Sb(β)}∀β∈NBb. Then(2)S∗a=ba,b,d,c,ca,b,c,e,da,b,d,e,T∗a=bab,ad,ac,bc,bd,cab,ac,ce,bc,dab,ad,de,db,S∗b=ab,c,d,ca,b,e,da,b,e,T∗b=abc,bd,cab,dab.Figure 2 shows the respective T1SG corresponding to H∗(a)=(S∗(a),T∗(a)) and H∗(b)=(S∗(b),T∗(b)), respectively. Hence, G∗=(H∗(a),H∗(b)) is a T2SG of G. It is also called VN-type-2 soft graph.Figure 1
Undirected graph.Figure 2
G
∗
=
(
H
∗
(
a
)
,
H
∗
(
b
)
).Example 16.
Consider a crisp graphG=(X,E) such that X={a,b,c,u,v} and E={ab,cu,cb,av,uv}. Let A={e1,e2,e3} be a nonempty set of parameters and NBe1={e2,e4}, NBe2={e1,e6}, and NBe3={e4,e5}. We may define the pair of T2SS ([S∗,A],[T∗,A]) as follows:(3)H∗e1=S∗e1,T∗e1=e2a,v,e4b,c,e2av,e4bc,H∗e2=S∗e2,T∗e2=e1a,v,b,e6a,v,u,e1av,ab,e6av,vu,H∗e3=S∗e3,T∗e3=e4b,c,u,e5v,u,e4bc,cu,e5vu.Then the subgraphs ofG corresponding to parameters (vertices) e1, e2, and e3 are shown in Figure 3. Hence, G∗=(H∗(e1),H∗(e2),H∗(e3)) is a T2SG of G. Consider P=⋃x∈ANBx={e1,e2,e4,e5,e6}. We may symbolize α∈NBx, as (x↣α) and denote a set of associations of A, as (A↣P)={(x↣α)∣α∈NBx}. Then tabular representation of T2SG is given in Table 1.Table 1
Tabular representation of a type-2 soft graph.
A
↣
P
X
E
a
b
c
u
v
a
b
c
u
c
b
a
v
u
v
e1↣e2
1
0
0
0
1
0
0
0
1
0
e1↣e4
0
1
1
1
0
0
1
1
0
0
e2↣e1
1
1
0
0
1
1
0
0
1
0
e2↣e6
1
0
0
1
1
0
0
0
1
1
e3↣e4
0
1
1
1
0
0
1
1
0
0
e3↣e5
0
0
0
1
1
0
0
0
0
1Figure 3
G
∗
=
(
H
∗
(
e
1
)
,
H
∗
(
e
2
)
,
H
∗
(
e
3
)
).


## 2.1. Type-2 Soft Graph

A 5-tupleG∗=(G,S∗,T∗,A,NBA) is called a type-2 soft graph (briefly, T2SG) if it satisfies the following conditions:(i)
G=(X,E) is a simple graph.(ii)
A is a nonempty set of parameters.(iii)
[S∗,A] is a VN-type-2 soft set over X.(iv)
[T∗,A] is a VN-type-2 soft set over E.(v)
T1SS corresponding to(S∗(x),T∗(x))∀x∈A, represents a type-1 soft graph.A T2SG can also be represented byG∗=〈S∗,T∗,A〉={H∗(x)∣x∈A}, where H∗(x)=(Hx,NBx) such that Hx(z)=(Sx(z),Tx(z)), ∀z∈NBx. The set of all T2SG of G is denoted by TSG(G).Although the above representation of graphs provides a new framework for the applications of graph theory, it is restricted to the information from neighbors. In this regard, a likeness between parameterized vertices appears in some cases such as distinction between T2SG〈S∗,T∗,A〉 and 〈S′∗,T′∗,A〉 is limited toward respective T1SG. However, T2SG is more general and hence a more efficient approach in graph theory as compared to T1SG.Definition 13.
A T2SGG∗=(G,S∗,T∗,A,NBA) is said to be vertex-induced if H∗(γ)=(S∗(γ),T∗(γ))=S∗(γ) for all γ∈A.Definition 14.
A T2SGG∗=(G,S∗,T∗,A,NBA) is said to be edge-induced if H∗(γ)=(S∗(γ),T∗(γ))=T∗(γ) for all γ∈A.Example 15.
Consider a graphG=(X,E) as shown in Figure 1.
LetA={a,b}⊂X. Then NBa={b,c,d}, NBb={a,c,d}. Let [S∗,A] and [T∗,A] be two T2SS over X and E, respectively, such that S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx) for all x∈A. Let Sa(α)={y∈X∣αRy⇔d(α,y)≤1}, Ta(α)={uv∈E∣{u,v}⊆Sa(α)}∀α∈NBa and Sb(β)={y′∈X∣βRy′⇔d(β,y′)=1}, Tb(β)={u′v′∈E∣{u′,v′}⊆Sb(β)}∀β∈NBb. Then(2)S∗a=ba,b,d,c,ca,b,c,e,da,b,d,e,T∗a=bab,ad,ac,bc,bd,cab,ac,ce,bc,dab,ad,de,db,S∗b=ab,c,d,ca,b,e,da,b,e,T∗b=abc,bd,cab,dab.Figure 2 shows the respective T1SG corresponding to H∗(a)=(S∗(a),T∗(a)) and H∗(b)=(S∗(b),T∗(b)), respectively. Hence, G∗=(H∗(a),H∗(b)) is a T2SG of G. It is also called VN-type-2 soft graph.Figure 1
Undirected graph.Figure 2
G
∗
=
(
H
∗
(
a
)
,
H
∗
(
b
)
).Example 16.
Consider a crisp graphG=(X,E) such that X={a,b,c,u,v} and E={ab,cu,cb,av,uv}. Let A={e1,e2,e3} be a nonempty set of parameters and NBe1={e2,e4}, NBe2={e1,e6}, and NBe3={e4,e5}. We may define the pair of T2SS ([S∗,A],[T∗,A]) as follows:(3)H∗e1=S∗e1,T∗e1=e2a,v,e4b,c,e2av,e4bc,H∗e2=S∗e2,T∗e2=e1a,v,b,e6a,v,u,e1av,ab,e6av,vu,H∗e3=S∗e3,T∗e3=e4b,c,u,e5v,u,e4bc,cu,e5vu.Then the subgraphs ofG corresponding to parameters (vertices) e1, e2, and e3 are shown in Figure 3. Hence, G∗=(H∗(e1),H∗(e2),H∗(e3)) is a T2SG of G. Consider P=⋃x∈ANBx={e1,e2,e4,e5,e6}. We may symbolize α∈NBx, as (x↣α) and denote a set of associations of A, as (A↣P)={(x↣α)∣α∈NBx}. Then tabular representation of T2SG is given in Table 1.Table 1
Tabular representation of a type-2 soft graph.
A
↣
P
X
E
a
b
c
u
v
a
b
c
u
c
b
a
v
u
v
e1↣e2
1
0
0
0
1
0
0
0
1
0
e1↣e4
0
1
1
1
0
0
1
1
0
0
e2↣e1
1
1
0
0
1
1
0
0
1
0
e2↣e6
1
0
0
1
1
0
0
0
1
1
e3↣e4
0
1
1
1
0
0
1
1
0
0
e3↣e5
0
0
0
1
1
0
0
0
0
1Figure 3
G
∗
=
(
H
∗
(
e
1
)
,
H
∗
(
e
2
)
,
H
∗
(
e
3
)
).


## 3. Operations on Type-2 Soft Graphs

In this section, we present type-2 soft subgraph of a T2SG,union,intersection,OR operation,AND operation, andCartesian product of T2SG.Definition 17.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2SG of G. Then G2∗ is a type-2 soft subgraph of G1∗ if(i)
A2⊆A1,(ii)
for eachx∈A2, T1SG corresponding to H2∗(x)=(S2∗(x),T2∗(x)) is a type-1 soft subgraph of T1SG corresponding to H1∗(x)=(S1∗(x),T1∗(x)).Example 18.
Consider a graphG=(X,E) as shown in Figure 4.
LetA={a,c,f} and B={c,f}. Then NBa={g,b}, NBc={b,d}, and NBf={g,e}. Let [S∗,A] and [T∗,A] be two T2SS over X and E, respectively, such that S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx) for all x∈A, where Sx(z)={y∈X∣zRy⇔2≤d(z,y)≤3} and Tx(z)={αβ∈E∣{α,β}⊆Sx(z)} for all z∈NBx⊆X. Now, T2SS [S∗,A] and [T∗,A] are as in the following:(4)S∗a=bg,e,f,gb,c,d,T∗a=beg,fe,gf,gbc,cd,db,S∗c=bg,e,f,da,g,f,T∗c=beg,fd,gf,dag,gf,S∗f=gb,c,d,ea,b,c,T∗f=gbc,cd,bd,eab,bc.Thus, G∗=(H∗(a),H∗(c),H∗(f)) is a T2SG as shown in Figure 5.
Let[S′∗,B] and [T′∗,B] be two T2SS over X and E, respectively, such that S′∗(x)=(Sx′,NBx) and T′∗(x)=(Tx′,NBx) for all x∈B, where Sx′(z)={y∈X∣zRy⇔d(z,y)=rad(G)}, and Tx′(z)={αβ∈E∣{α,β}⊆Sx′(z)} for all z∈NBx⊆X. Then T2SS [S′∗,B] and [T′∗,B] are as in the following:(5)S′∗c=bg,e,da,g,f,T′∗c=bge,dag,gf,S′∗f=gd,b,ea,b,c,T′∗f=gbd,eab,bc.Therefore, G′∗=(H′∗(c),H′∗(f)) is a T2SG of G as shown in Figure 6.
One can easily check thatG′∗=(H′∗(c),H′∗(f)) is a type-2 soft subgraph of G∗=(H∗(a),H∗(c),H∗(f)).Figure 4
Undirected graph.Figure 5
G
∗
=
(
H
∗
(
a
)
,
H
∗
(
c
)
,
H
∗
(
f
)
).Figure 6
G
′
∗
=
(
H
′
∗
(
c
)
,
H
′
∗
(
f
)
).Theorem 19.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2SG of G. Then G2∗ is a type-2 soft subgraph of G1∗ if and only if S2∗(x)⊆~S1∗(x) and T2∗(x)⊆~T1∗(x) for all x∈A2.Proof.
SupposeG2∗ is a T2SG of G1∗. Then by the definition of T2SG, (i)
A2⊆A1,(ii)
for eachx∈A2, T1SG corresponding to H2∗(x)=(S2∗(x),T2∗(x)) is a type-1 soft subgraph of T1SG corresponding to H1∗(x)=(S1∗(x),T1∗(x)).Since T1SG corresponding to H2∗(x) is a type-1 soft subgraph of T1SG corresponding to H1∗(x) for all x∈A2. Then S2∗(x)⊆~S1∗(x) and T2∗(x)⊆~T1∗(x) for all x∈A2.
Conversely,S2∗(x)⊆~S1∗(x) and T2∗(x)⊆~T1∗(x) for all x∈A2. As G1∗ is a T2SG of G, T1SS corresponding to H1∗(x)=(S1∗(x),T1∗(x)) is a T1SG of G for all x∈A1. Also, G2∗ is a T2SG of G; T1SS corresponding to H2∗(x)=(S2∗(x),T2∗(x)) is a T1SG of G for all x∈A2. This implies that T1SG corresponding to H2∗(x) is a type-1 soft subgraph of T1SG corresponding to H1∗(x) for all x∈A2. Thus, G2∗ is a type-2 soft subgraph of G1∗.Definition 20.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2SG of G. Theunion of G1∗ and G2∗, denoted by G1∗⊔G2∗=G∗=〈I∗,J∗,C〉, where C=A1∪A2, is defined, ∀α∈C, as(6)I∗α=S1∗α,ifα∈A1-A2S2∗α,ifα∈A2-A1S1∗α∪~S2∗α,ifα∈A1∩A2,where S1∗(α)∪~S2∗(α) for all α∈A1∩A2 refers to the usual type-1 soft union between the respective T1SS corresponding to S1∗(α) and S2∗(α), respectively. And,(7)J∗α=T1∗α,ifα∈A1-A2T2∗α,ifα∈A2-A1T1∗α∪~T2∗α,ifα∈A1∩A2,where T1∗(α)∪~T2∗(α) for all α∈A1∩A2 refers to the usual type-1 soft union between the respective T1SS corresponding to T1∗(α) and T2∗(α), respectively.
It can be written asG1∗⊔G2∗={H∗(α)=(I∗(α),J∗(α))∣α∈C}.Example 21.
Consider a graphG=(X,E) as shown in Figure 7
LetA={a,e} and B={b,e}. Then NBa={v,b}, NBb={a,c}, and NBe={f,d}. Let [S∗,A] and [T∗,A] be two T2SS over X and E, respectively, such that S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx) for all x∈A, where Sx(z)={y∈X∣zRy⇔d(z,y)≤1} and Tx(z)={αβ∈E∣{α,β}⊆Sx(z)} for all z∈NBx⊆X. Now, T2SS [S∗,A] and [T∗,A] are as in the following:(8)S∗a=ba,b,c,va,d,v,u,T∗a=bab,bc,vva,vd,vu,S∗e=fd,e,f,u,dd,e,f,v,c,T∗e=ffu,fd,fe,de,dde,df,fe,dv.Thus, G∗=(H∗(a),H∗(e)) is a T2SG of G.
Let[S′∗,B] and [T′∗,B] be two T2SS over X and E, respectively, such that S′∗(x)=(Sx′,NBx) and T′∗(x)=(Tx′,NBx) for all x∈B, where Sx′(z)={y∈X∣zRy⇔1≤d(z,y)≤rad(G)}, and Tx′(z)={αβ∈E∣{α,β}⊆Sx′(z)} for all z∈NBx⊆X. Then T2SS [S′∗,B] and [T′∗,B] are as in the following:(9)S′∗b=av,u,d,b,c,cd,f,e,v,b,a,T′∗b=avu,dv,dc,cb,cdf,ef,ed,vd,ab,av,S′∗e=fe,d,u,v,c,de,f,u,v,a,c,b,T′∗e=fed,dv,dc,uv,def,uv,fu,va,cb,ab.Therefore, G′∗=(H′∗(b),H′∗(e)) is a T2SG of G.
Thenunion of G∗=〈S∗,T∗,B〉 and G′∗=〈S′∗,T′∗,B〉 is G∗∪~G′∗=〈I∗,J∗,C〉, where C=A∪B and(10)I∗a=ba,b,c,va,d,v,u,J∗a=bab,bc,vva,vd,vu,I∗b=av,u,d,b,c,cd,f,e,v,b,a,J∗b=auv,dv,dc,cb,cdf,ef,ed,vd,ab,av,I∗e=fe,f,u,d,v,c,dX,J∗e=fed,dv,dc,uv,fe,fd,fu,dE.Theunion of G∗ and G′∗ is shown in Figure 8.
Note that, in the definition of T2SG, subgraphs in T1SG corresponding toH∗(x) may be connected or disconnected subgraphs of a simple graph. However, in this section we are now focusing only on connected subgraphs of a simple graph.Figure 7
Simple graph.Figure 8
Theunion of G∗ and G′∗.Theorem 22.
LetG1∗=〈S∗,T∗,A1〉 and G2∗=〈S′∗,T′∗,A2〉 be two T2SG of G with A1∩A2≠∅. Let T1SS corresponding to S∗(γ) and S′∗(η) be (Sγ,NBγ) and (Sη′,NBη), respectively. If for each α∈A1∩A2, Sα(x)∩Sα′(x)≠∅∀x∈NBα, then theunion of G1∗ and G2∗ is a T2SG of G.Proof.
Theunion of G1∗=〈S∗,T∗,A1〉 and G2∗=〈S′∗,T′∗,A2〉 is defined as G1∗⊔G2∗=G∗=〈I∗,J∗,C〉, where C=A1∪A2 for all α∈C,(11)I∗α=S∗α,ifα∈A1-A2S′∗α,ifα∈A2-A1S∗α∪~S′∗α,ifα∈A1∩A2,where S∗(α)∪~S′∗(α) for all α∈A1∩A2 refers to the usual type-1 soft union between the respective T1SS corresponding to S∗(α) and S′∗(α), respectively. And,(12)J∗α=T∗α,ifα∈A1-A2T′∗α,ifα∈A2-A1T∗α∪~T′∗α,ifα∈A1∩A2,where T∗(α)∪~T′∗(α) for all α∈A1∩A2 refers to the usual type-1 soft extended union between the respective T1SS corresponding to T∗(α) and T′∗(α), respectively.
G
1
∗ is connected T2SG of G. Then T1SS corresponding to (S∗(α),T∗(α)) is a connected T1SG of G for all α∈A1-A2.
G
2
∗ is connected T2SG of G. Then T1SS corresponding to (S′∗(α),T′∗(α)) is a connected T1SG of G for all α∈A2-A1.
Letα∈A1∩A2, (I∗(α),J∗(α))=(S∗(α)∪~S′∗(α),T∗(α)∪~T′∗(α)). Respective T1SS corresponding to (S∗(α),T∗(α)) and (S′∗(α),T′∗(α)) are connected T1SG of G. Given that Sαx∩Sα′x≠∅∀x∈NBα, this implies that T1SS corresponding to (I∗(α),J∗(α)) is connected T1SG of G. As α was taken to be an arbitrary element, theorem holds for all α∈A∩B. Thus, G1∗⊔G2∗=G∗=I∗,J∗,A1∪A2 is a T2SG of G.Lemma 23.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2SG of G. If A1∩A2=∅, then their union is a T2SG of G.Definition 24.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2SG of G. Theintersection of G1∗ and G2∗, denoted by G1∗⊓G2∗=G∗=〈I∗,J∗,C〉, where C=A1∩A2, is defined by I∗(α)=S1∗(α)∩~S2∗(α) and J∗(α)=T1∗(α)∩~T2∗(α) for all α∈A1∩A2, where (i)
S1∗(α)∩~S2∗(α) for all α∈A1∩A2 refers to the usual type-1 soft intersection between the respective T1SS corresponding to S1∗(α) and S2∗(α), respectively,(ii)
T1∗(α)∩~T2∗(α) for all α∈A1∩A2 refers to the usual type-1 soft intersection between the respective T1SS corresponding to T1∗(α) and T2∗(α), respectively.
It can be written asG1∗⊓G2∗={H∗(α)=(I∗(α),J∗(α))∣α∈C}.Example 25.
Consider T2SGG∗ and G′∗ defined in Example 18. Theintersection between G∗ and G′∗ is equal to G′∗ as shown in Figure 6.Definition 26.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2SG of G. The AND operation of G1∗ and G2∗, denoted by G1∗⋀G2∗=G∗=〈I∗,J∗,A×B〉 defined by I∗(γ,η)=S1∗(γ)∧~S2∗(η), J∗(γ,η)=T1∗(γ)∧~T2∗(η) for all (γ,η)∈A×B, where S1∗(γ)∧~S2∗(η) for all (γ,η)∈A×B, refers to the usual type-1 soft AND operation between the respective T1SS corresponding to S1∗(γ) and S2∗(η), respectively, and T1∗(γ)∧~T2∗(η) for all (γ,η)∈A×B refers to the usual type-1 soft AND operation between the respective T1SS corresponding to T1∗(γ) and T2∗(η), respectively.Example 27.
Consider graph that is shown in Figure7. Let A={a,c} and B={b}. Then NBa={v,b}, NBb={a,c}, and NBc={b,d}. Let [S∗,A] and [T∗,A] be two T2SS over X and E, respectively, such that S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx) for all x∈A, where Sx(z)={y∈X∣zRy⇔d(z,y)≤1} and Tx(z)={αβ∈E∣{α,β}⊆Sx(z)} for all z∈NBx⊆X. Now, T2SS [S∗,A] and [T∗,A] are as in the following:(13)S∗a=ba,b,c,va,d,v,u,T∗a=bab,bc,vva,vd,vu,S∗c=dd,e,f,v,c,ba,b,c,T∗c=dfe,fd,de,dv,dc,bab,bc.Then G∗=〈S∗,T∗,A〉 is T2SG of G.
Let[S′∗,B] and [T′∗,B] be two T2SS over X and E, respectively, such that S′∗(x)=(Sx′,NBx) and T′∗(x)=(Tx′,NBx) for all x∈B, where Sx′(z)={y∈X∣zRy⇔1≤d(z,y)≤rad(G)} and Tx′(z)={αβ∈E∣{α,β}⊆Sx′(z)} for all z∈NBx⊆X. Then T2SS [S′∗,B] and [T′∗,B] are as in the following:(14)S′∗b=av,u,d,b,c,cd,f,e,v,b,a,T′∗b=avu,dv,dc,cb,cdf,ef,ed,vd,ab,av.Then G∗′=〈S′∗,T′∗,B〉 is T2SG of G.
TheAND operation of G∗ and G′∗ is G∗∧G′∗=〈I∗,J∗,A×B〉, where A×B={(a,b),(c,b)} and(15)I∗a,b=v,av,u,d,v,ca,v,d,b,ab,c,b,cb,a,I∗c,b=d,av,d,c,d,cd,e,f,v,b,ac,b,b,cb,a,J∗a,b=v,avu,vd,v,cav,vd,b,abc,b,cba,J∗c,b=d,avd,dc,d,cdf,ef,de,dv,b,abc,b,cba.The AND operation of G∗ and G′∗ is shown in Figure 9.Figure 9
A
N
D operations on G∗ and G∗′.Theorem 28.
LetG1∗=〈S∗,T∗,A1〉 and G2∗=〈S′∗,T′∗,A2〉 be two T2SG of G. Let T1SS corresponding to S∗(γ) and S′∗(η) be (Sγ,NBγ) and (Sη′,NBη) for all (γ,η)∈A1×A2, respectively, and let T1SS corresponding to T∗(γ) and T′∗(η) be (Tγ,NBγ) and (Tη′,NBη) for all (γ,η)∈A1×A2, respectively. For each (γ,η)∈A1×A2, Sγ(x)∩Sη′(y)≠∅ and Tγ(x)∩Tη′(y)≠∅∀(x,y)∈NBγ×NBη. Then G1∗∧G2∗ is a T2SG of G.Proof.
TheAND operation of G1∗ and G2∗ is denoted by G1∗∧G2∗=G∗=〈I∗,J∗,A1×A2〉 and defined by I∗(γ,η)=S∗(γ)∧~S′∗(η)and J∗(γ,η)=T∗(γ)∧~T′∗(η) for all (γ,η)∈A1×A2, where S∗(γ)∧~S′∗(η) for all (γ,η)∈A1×A2 refers to the usual type-1 soft AND operation between the respective T1SS corresponding to S∗(γ) and S′∗(η), respectively, and T∗(γ)∧~T′∗(η) for all (γ,η)∈A1×A2 refers to the usual type-1 soft AND operation between the respective T1SS corresponding to T∗(γ) and T′∗(η), respectively.
G
1
∗ is connected T2SG of G. Then T1SS corresponding to (S∗(γ),T∗(γ)) is a connected T1SG of G for all γ∈A1.
G
2
∗ is connected T2SG of G. Then T1SS corresponding to (S′∗(η),T′∗(η)) is a connected T1SG of G for all η∈A2.
Let(γ,η)∈A1×A2. It may be mentioned here that Iγ,η(x,y)=Sγ(x)∩Sη′(y), Jγ,η(x,y)=Tγ(x)∩Tη′(y), such that (Iγ,η(x,y),Jγ,η(x,y))=(Sγ(x)∩Sη′(y),Tγ(x)∩Tη′(y))∀(x,y)∈NBγ×NBη. Respective T1SS corresponding to (S∗(γ),T∗(η)) and (S′∗(γ),T′∗(η)) are connected T1SG of G. Given that Sγ(x)∩Sη′(y)≠∅ and Tγ(x)∩Tη′(y)≠∅∀(x,y)∈NBγ×NBη, this implies that (I(γ,η)(x,y),J(γ,η)(x,y))∀(x,y)∈NBγ×NBη is a T1SG of G. Hence, T1SS corresponding to (I∗(γ,η),J∗(γ,η)) is connected T1SG of G. As (γ,η) was taken to be an arbitrary element, theorem holds for all (γ,η)∈A×B.
Thus,G1∗⋀G2∗=G∗=〈I∗,J∗,A1×A2〉 is a T2SG of G.Definition 29.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2SG of G. The OR operation of G1∗ and G2∗ is denoted by G1∗⋁G2∗=G∗=〈I∗,J∗,A×B〉 and defined by I∗(γ,η)=S1∗(γ)∨~S2∗(η) and J∗(γ,η)=T1∗(γ)∨~T2∗(η) for all (γ,η)∈A×B, where S1∗(γ)∨~S2∗(η) for all (γ,η)∈A×B refers to the usual type-1 soft OR operation between the respective T1SS corresponding to S1∗(γ) and S2∗(η), respectively, and T1∗(γ)∨~T2∗(η) for all (γ,η)∈A×B refers to the usual type-1 soft OR operation between the respective T1SS corresponding to T1∗(γ) and T2∗(η), respectively.Example 30.
Consider graph(X,E) as shown in Figure 10.
LetA={e,c} and B={a}. Then NBa={f,b}, NBe={f,h}, and NBc={g,b}. Let [S∗,A] and [T∗,A] be two T2SS, respectively, over X and E such that S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx) for all x∈A, where Sx(z)={y∈X∣zRy⇔d(z,y)≤1} and Tx(z)={αβ∈E∣{α,β}⊆Sx(z)} for all z∈NBx⊆X. Now, T2SS [S∗,A] and [T∗,A] are as in the following:(16)S∗e=fa,b,f,u,v,e,he,i,g,h,v,u,S∗c=bb,a,f,c,v,gg,h,v,u,c,i,T∗e=fab,bf,fv,fu,fe,fa,bv,uv,heh,ih,hv,hu,hg,ug,gi,vu,gv,T∗c=bab,fa,fb,bc,bv,vf,ggi,gc,gv,gu,gh,hu,uv,vh,hi.
ThenG∗=〈S∗,T∗,A〉 is T2SG of G.
Let[S′∗,B] and [T′∗,B] be two T2SS, respectively over X and E such that S′∗(x)=(Sx′,NBx) and T′∗(x)=(Tx′,NBx) for all x∈B, where Sx′(z)={y∈X∣zRy⇔d(z,y)=2} and Tx′(z)={αβ∈E∣{α,β}⊆Sx′(z)} for all z∈NBx⊆X. Then T2SS [S′∗,B] and [T′∗,B] are as in the following:(17)S′∗a=fh,g,c,be,h,g,u,T′∗a=fgh,gc,bgh,hu,he.
ThenG′∗=〈S′∗,T′∗,B〉 is T2SG of G.
TheOR operation of G∗ and G′∗ is G∗∨G′∗=〈I∗,J∗,A×B〉, where A×B={(e,a),(c,a)} and(18)I∗e,a=f,fa,b,f,u,v,e,g,h,c,f,ba,b,u,v,e,f,h,g,h,fe,i,g,h,v,u,c,h,be,i,g,h,v,uI∗c,a=b,fb,a,v,c,f,h,g,b,bb,a,v,c,f,e,h,g,u,g,fg,h,v,u,c,i,g,bg,h,v,c,u,i,eJ∗e,a=f,fab,bf,af,fe,eh,hg,gc,bc,bv,fv,vu,fu,uh,ug,vg,vh,f,bab,bf,af,fe,eh,hg,bv,fv,vu,fu,uh,ug,vg,vh,h,feh,hg,gc,vu,uh,ug,vg,vh,gi,ih,h,beh,hg,vu,uh,ug,vg,vh,gi,ihJ∗c,a=b,fab,bf,af,hg,gc,bc,bv,fv,vg,vh,b,bab,bf,af,fe,eh,hg,gc,bc,bv,fv,vu,fu,uh,ug,vg,vh,g,fgh,hi,gi,gc,gv,gu,hu,hv,uv,g,bgh,hi,gi,gc,gv,gu,hu,hv,uv,eh
TheOR operation of G∗ and G′∗ is shown in Figure 11.Figure 10
Simple graph.Figure 11
O
R operations on G∗ and G∗′.Theorem 31.
LetG1∗=〈S∗,T∗,A1〉 and G2∗=〈S′∗,T′∗,A2〉 be two T2SG of G. Let T1SS corresponding to S∗(γ) and S′∗(η) are (Sγ,NBγ) and (Sη′,NBη) for all (γ,η)∈A1×A2, respectively. For each (γ,η)∈A1×A2, Sγ(x)∩Sη′(y)≠∅∀(x,y)∈NBγ×NBη. Then G1∗⋁G2∗ is a T2SG of G.Proof.
OmittedDefinition 32.
LetG1∗=〈S1∗,T1∗,A〉 and G2∗=〈S2∗,T2∗,B〉 be two T2SG over G1 and G2, respectively, such that A∩B=∅ and NBγ∩NBη=∅∀γ∈A,η∈B. TheCartesian product of G∗ and G′∗ is denoted by G1∗⊠G2∗ and defined as G1∗⊠G2∗=〈K∗,A×B〉, where K∗(γ,η)=H1∗(γ)⋊H2∗(η) for all (γ,η)∈A×B refers to the usual type-1 soft graphCartesian product between the respective T1SS corresponding to H1∗(γ) and H1∗(η), respectively.Example 33.
Consider two graphs(X1,E1) and (X2,E2) as shown in Figures 12 and 13, respectively. Let A={a3,a8} and B={b1}. Then NBa3={a2,a4}, NBa8={a1,a5}, and NBb1={b2,b6}. Let [S∗,A] and [T∗,A] be two T2SS, respectively, over X1 and E1 such that S∗(x)=(Sx,NBx) and T∗(x)=(Tx,NBx) for all x∈A, where Sx(z)={y∈X1∣zRy⇔d(z,y)=rad(G1)} and Tx(z)={αβ∈E1∣{α,β}⊆Sx(z)} for all z∈NBx⊆X1. Now, T2SS [S∗,A] and [T∗,A] are as in the following:(19)S∗a8=a1a4,a5,a3,a5a1,a2,a3,S∗a3=a2a4,a5,a8,a7,a4a1,a2,a8,T∗a8=a1a4a5,a4a3,a5a1a2,a2a3,T∗a3=a2a4a5,a5a8,a5a7,a4a1a2,a1a8.
ThenG∗=〈S∗,T∗,A〉 is T2SG of G1.
Let[S′∗,B] and [T′∗,B] be two T2SS, respectively, over X2 and E2 such that S′∗(x)=(Sx′,NBx) and T′∗(x)=(Tx′,NBx) for all x∈B, where Sx′(z)={y∈X2∣zRy⇔d(z,y)>1} and Tx′(z)={αβ∈E2∣{α,β}⊆Sx′(z)} for all z∈NBx⊆X2. Then T2SS [S′∗,B] and [T′∗,B] are as in the following:(20)S′∗b1=b6b2,b4,b2b5,b6,T′∗b1=b6b2b4,b2b6b5.
ThenG′∗=〈S′∗,T′∗,B〉 is T2SG of G2.
TheCartesian product of G∗ and G′∗ is G∗⋊G′∗=〈K∗,A×B〉, where A×B={(a3,b1)(a8,b1)} and(21)I∗a8,b1=a1,b6a4,b2,a5,b2,a3,b2,a4,b4,a5,b4,a3,b4,a5,b6a1,b2,a2,b2,a3,b2,a1,b4,a2,b4,a3,b4,a5,b2a1,b5,a2,b5,a3,b5,a1,b6,a2,b6,a3,b6,a1,b2a4,b5,a5,b5,a3,b5,a4,b6,a5,b6,a3,b6J∗a8,b1=a1,b6a4,b2,a5,b2,a4,b2,a3,b2,a5,b2,a3,b2,a4,b4,a5,b4,a4,b4,a3,b4,a5,b4,a3,b4,a4,b2,a4,b4,a5,b2,a5,b4,a3,b2,a3,b4,a5,b6a1,b2,a2,b2,a1,b2,a3,b2,a2,b2,a3,b2,a1,b4,a2,b4,a1,b4,a3,b4,a2,b4,a3,b4,a1,b2,a1,b4,a2,b2,a2,b4,a3,b2,a3,b4,a5,b2a1,b5,a2,b5,a1,b5,a3,b5,a2,b5,a3,b5,a1,b6,a2,b6,a1,b6,a3,b6,a2,b6,a3,b6,a1,b5,a1,b6,a2,b5,a2,b6,a3,b5,a3,b6,a1,b2a4,b5,a5,b5,a4,b5,a3,b5,a5,b5,a3,b5,a4,b6,a5,b6,a4,b6,a3,b6,a5,b6,a3,b6,a4,b5,a4,b6,a5,b5,a5,b6,a3,b5,a3,b6,H∗(a8,b1) is shown in Figure 14. Similarly, we can compute H∗(a3,b1) and show the respective graphs.Figure 12
Simple graph(X1,E1).Figure 13
Simple graph(X2,E2).Figure 14
H
∗
(
a
8
,
b
1
).Theorem 34.
LetQ be theCartesian product of two simple graphs G1 and G2. Let G1∗ and G2∗ be two T2SG of G1 and G2, respectively. Then G1∗⊠G2∗=〈K∗,A×B〉 is a T2SG of Q.Proof.
TheCartesian product of two T2SG G1∗ and G2∗ is G1∗⊠G2∗=〈K∗,A×B〉, where K∗(γ,η)=H1∗(γ)⋊H2∗(η) for all (γ,η)∈A×B refers to the usual type-1 soft graphsCartesian product between the respective T1SS corresponding to H1∗(γ) and H1∗(η), respectively.
Given thatG1∗ is a T2SG of G1, then T1SG corresponding to H1∗(γ)∀γ∈A is a connected T1SG of G1. Also, given that G2∗ is a T2SG of G2, then T1SG corresponding to H2∗(η)∀η∈B is a connected T1SG of G2. This implies that T1SG corresponding to K∗(γ,η)∀(γ,η)∈A×B is aCartesian product of two connected T1SG. Therefore, T1SG corresponding to K∗(γ,η) is connected T1SG of Q, for all (γ,η)∈A×B. Thus, G1∗⊠G2∗=〈K∗,A×B〉 is a T2SG of Q.


## 4. An Application of Type-2 Soft Graphs

It is well known that graph theory has many applications in computer networks and communication networks. In this manner, our approach is noteworthy to enhance the study of networks. In this section, we contemplated an application of T2SG in communication networks.It is necessary to explain some new notions of T2SS that will be used in this section. Let[S∗,A] and [S′∗,B] be two T2SS over U. Let, for each (γk,ηk′)∈A×B, (Sγk,NBγk) and (Sηk′′,NBηk′) be the T1SS corresponding to S∗(γk) and S′∗(ηk′), respectively. A correlation table of T2SS is an m×n table in which the entry skιkȷ′ is a resultant of Sγk(xι)∩Sηk′′(xȷ) for all (xι,xȷ)∈NBγk×NBηk′ (k=1,2,…,m′ and k′=1,2,…,n′). There are m′·n′=q, partitions in correlation table, where each partition represents an AND operation on respective T1SS corresponding to S∗(γk) and S′∗(ηk′) for all (γk,ηk′)∈A×B. The R-union in a partition is the row-wise union of entries and the C-intersection in a partition is the column-wise intersection of entries.Consider a communication network of 14 communications towers (see Figure15), where each tower is represented as a vertex and each communication link between towers is represented with an edge in a resulting graph. The communication among the towers in the network is done by using two types of messages: Hello (used for sending information to neighbors) and Data (used for data transmission). There are two types of towers in the network: (i) working as a base station and working in subnetworks and (ii) working in subnetworks. Hello, messages are only considered by one-hop neighbors, while the Data messages are representing sending and gathering information from subnetworks and therefore must be transmitted in a multihop manner towards the base stations and towards the subnetworks.Figure 15
Communication network.Two communications companies are using this network of towers, and each one uses a set of base stations for communication: the first company is using towersa, e as base stations, while the second company is using towers c, g as base stations. Let A={a,e} and B={c,g}. For any base station l messaging is as follows:(i)
l sends Hello messages and their neighbors NBl send these messages as Data messages towards the set of all towers in the network.(ii)
Also, the neighbors ofl gather messages from the whole network and then send these messages as Hello messages to l.To analyze the weak service (messaging), both companies make a survey on their sets of base stations. The region (towers and the links between towers) of weak service of A and B may be defined in the form of T2SG.The region of weak service forA is as follows: for base station a∈A, region of weak messaging is defined as a T1SS, Sa(z)={y∈X∣yRz⇔d(y,z)≤2} and Ta(z)={uv∈E∣{u,v}⊆Sa(z)}∀z∈NBa.For base statione∈A, region of weak messaging is defined as a T1SS, Se(z′)={y′∈X∣y′Rz′⇔d(y′,z′)=1} and Te(z′)={u′v′∈E∣{u′,v′}⊆Se(z′)}∀z′∈NBe. Then T2SS are as follows:(22)S∗a=wa,b,d,h,k,u,j,v,w,ba,b,c,d,k,u,w,j,h,e,S∗e=db,c,h,e,u,fe,i,h,g,T∗a=wab,bd,aw,bk,wk,wu,wv,uv,uk,ud,uj,uh,kj,hd,bab,bc,bd,cd,dh,aw,bk,ku,uh,de,ud,uw,uj,wk,kj,T∗e=dbc,hu,fhi,gi.One can check that G∗=(G,S∗,T∗,A,NBA) is a T2SG of G.The region (towers and the links between towers) of weak service forB is as follows: for base station c∈B, region of weak messaging is defined as a T1SS, Sc′(z)={y∈X∣yRz⇔1≤d(y,z)≤2} and Tc′(z)={uv∈E∣{u,v}⊆Sc′(z)}∀z∈NBc.For base stationg∈B, region of weak messaging is defined as a T1SS, Sg′(z′)={y′∈X∣y′Rz′⇔1≤d(y′,z′)≤2} and Tg′(z′)={u′v′∈E∣{u′,v′}⊆Sg′(z′)}∀z′∈NBg. Then T2SS are as follows:(23)S′∗c=ba,w,c,u,k,d,e,h,j,da,b,c,w,v,u,j,i,k,f,h,e,S′∗g=ig,f,e,h,d,j,u,k,fe,d,h,u,g,i,j,T′∗c=baw,wk,wu,uk,ud,uj,uh,kj,dc,de,hd,dfi,fe,ab,bc,bk,aw,uv,uh,uj,ij,fh,kw,vw,hi,jk,uk,uw,T′∗g=ied,du,ku,gf,fe,hd,hf,hu,uj,kj,fuj,ud,ed,hd,hi,ig,ji,uh.One can check thatG′∗=(G,S′∗,T′∗,B,NBB) is a T2SG of G.To analyze the region (towers and the links between towers) in which both the sets of base stations are incorporated for very fragile service in the communication network, first, construct correlation tables of(A∧XB) and (A∧EB) for vertices and edges, respectively. Let (γk,ηk′)∈A×B. In the correlation Table 2 entry skιkȷ′ is a resultant of Sγk(xι)∩Sηk′′(xȷ) and in the correlation Table 3 entry tkιkȷ′ is a resultant of Tγk(xι)∩Tηk′′(xȷ) for all (xι,xȷ)∈NBγk×NBηk′ (k=1,2 and k′=1,2). There are 2·2=4, partitions in the correlation Table 2, where each partition represents an AND operation on respective T1SS corresponding to S∗(γk) and S′∗(ηk′), respectively. Also, there are 2·2=4, partitions in the correlation Table 3, where each partition represents an AND operation on the respective T1SS corresponding to T∗(γk) and T′∗(ηk′), respectively.Table 2
(
A
∧
X
B
) correlation table.
B
A
S
∗
(
a
)
S∗(e)
S
a
(
w
)
S
a
(
b
)
S
e
(
d
)
S
e
(
f
)
S′∗(c)
S
c
′
(
b
)
{
a
,
w
,
u
,
d
,
h
,
k
,
j
}
{
a
,
c
,
d
,
k
,
u
,
w
,
j
,
h
,
e
}
{
c
,
h
,
e
,
u
}
{
e
,
h
}
S
c
′
(
d
)
{
a
,
b
,
k
,
u
,
j
,
h
,
v
,
w
}
{
a
,
b
,
c
,
k
,
u
,
w
,
j
,
h
,
e
}
{
b
,
c
,
h
,
e
,
u
}
{
e
,
i
,
h
}
S′∗(g)
S
g
′
(
i
)
{
d
,
u
,
k
,
h
,
j
}
{
e
,
d
,
u
,
k
,
h
,
j
}
{
e
,
u
,
h
}
{
e
,
h
,
g
}
S
g
′
(
f
)
{
u
,
d
,
j
,
h
}
{
e
,
u
,
d
,
j
,
h
}
{
u
,
h
,
e
}
{e,i,h,g}Table 3
(
A
∧
E
B
) correlation table.
B
A
T∗(a)
T
∗
(
e
)
T
a
(
w
)
T
a
(
b
)
T
e
(
d
)
T
e
(
f
)
T′∗(c)
T
c
′
(
b
)
a
w
,
w
k
,
w
u
,
u
k
,
u
d
,
u
j
,
u
h
,
k
j
,
h
d
a
w
,
w
k
,
w
u
,
u
k
,
d
c
,
h
d
,
d
e
,
u
h
,
u
j
,
k
j
,
u
d
{
h
u
}
∅
T
c
′
(
d
)
a
b
,
a
w
,
b
k
,
w
k
,
w
u
,
w
v
,
u
v
,
u
k
,
u
j
,
u
h
,
k
j
a
b
,
b
c
,
a
w
,
b
k
,
k
u
,
u
h
,
w
u
,
w
k
,
u
j
,
k
j
{
h
u
,
b
c
}
{
h
i
}
T′∗(g)
T
g
′
(
i
)
{
u
d
,
k
u
,
k
j
,
u
h
,
u
j
,
h
d
}
{
d
e
,
u
k
,
u
d
,
h
d
,
j
u
,
u
h
,
k
j
}
{
h
u
}
∅
T
g
′
(
f
)
{
u
j
,
u
d
,
u
h
,
h
d
}
{
u
j
,
u
h
,
h
d
,
d
e
}
{
h
u
}
{
h
i
,
g
i
}TakeR-union of each partition in Table 2 and represent it in Table 4. Similarly, compute Table 5 from Table 3.Table 4
R
-union on A∧XB correlation table.
R-union of S∗(a)∧~S′∗(c)
R-union of S∗(e)∧~S′∗(c)
{
a
,
w
,
u
,
d
,
h
,
k
,
j
,
c
,
e
}
{
e
,
h
,
c
,
u
}
{
a
,
b
,
w
,
k
,
u
,
j
,
h
,
v
,
c
,
e
}
{
e
,
h
,
c
,
u
,
b
,
i
}
R-union of S∗(a)∧~S′∗(g)
R-union of S∗(e)∧~S′∗(g)
{
e
,
d
,
u
,
k
,
j
,
h
}
{e,u,h,g}
{
u
,
d
,
j
,
h
,
e
}
{u,h,e,i,g}Table 5
R-union on A∧EB correlation table.
R-union of T∗(a)∧~T′∗(c)
R-union of T∗(e)∧~T′∗(c)
{
a
w
,
w
k
,
w
u
,
u
k
,
u
d
,
u
j
,
u
h
,
h
d
,
d
c
,
d
e
}
{
h
u
}
{
a
b
,
a
w
,
b
k
,
w
k
,
w
u
,
w
v
,
u
v
,
u
k
,
u
j
,
u
h
,
k
j
,
b
c
}
{
h
u
,
b
c
,
h
i
}
R-union of T∗(a)∧~T′∗(g)
R-union of T∗(e)∧~T′∗(g)
{
u
k
,
u
d
,
d
e
,
j
k
,
u
h
,
h
d
,
u
j
}
{
h
u
}
{
u
j
,
u
d
,
u
h
,
h
d
,
d
e
}
{
h
u
,
g
i
,
h
i
}Next, takeC-intersection on each partition in Table 4 and represent it in Table 6. Similarly, compute Table 7 from Table 5.Table 6
C-intersection of R-union of A∧XB correlation table.
C-intersection of R-union of S∗(a)∧~S∗(c)
C-intersection of R-union of S∗(e)∧~S′∗(c)
{
a
,
w
,
u
,
h
,
k
,
j
,
c
,
e
}
{
e
,
h
,
c
,
u
}
C-intersection of R-union of S∗(a)∧~S′∗(g)
C-intersection of R-union of S∗(e)∧~S′∗(g)
{
u
,
d
,
e
,
j
,
h
}
{
u
,
h
,
e
,
g
}Table 7
C
-intersection of R-union of A∧EB correlation table.
C-intersection of R-union of T∗(a)∧~T′∗(c)
C-intersection of R-union of T∗(e)∧~T′∗(c)
{
a
w
,
w
k
,
u
w
,
u
k
,
u
j
,
u
h
}
{
h
u
}
C-intersection of R-union of T∗(a)∧~T′∗(g)
C-intersection of R-union of T∗(e)∧~T′∗(g)
{
u
d
,
u
j
,
h
d
,
u
h
,
d
e
}
{
h
u
}TakeR-union on Table 6 and represent it in 2×1 matrix of vertices. Sequentially, obtain a 2×1 matrix of edges from Table 7.Select the subgraph that is involved in extremely poor communications for both companies integratedly. One can easily check that the verticesu, h, e, and j appear maximum times (twice) in 2×1 matrix of vertices and the edges uj and uh appear twice in 2×1 matrix of edges. Thus, the very fragile communication for both companies integratedly is shown in Figure 16. Eventually, both companies are incorporated to fix the fragile communication on resultant subgraph.(24)R-UnionTable6=a,w,u,h,k,j,c,eu,d,e,j,h,g,R-UnionTable7=aw,wk,uw,uk,uj,uhud,uj,hd,uh,de.Figure 16
Resultant graph.We now present the general procedure of our method which is used in our application from the following algorithm.Algorithm
(i)
Input the two sets of base stations in a communication network.
(ii)
Define T2SG on each set of base stations, according to the weak communication messaging, that is,Hello messages andData messages.
(iii)
TakeAND operation on T2SG represents it in the correlation tables:
(1)
Table1 represents AND operation on two T2SS of the vertices and it contains m′·n′ partitions, where each partition is an AND operation of respective TISS corresponding to S∗(γk) and S′∗(ηk′), respectively, (k=1,2,…,m′,k′=1,2,…,n′).
(2)
Table2 represents AND operation on two T2SS of the edges and it contains m′·n′ partitions, where each partition is an AND operation of respective TISS corresponding to T∗(γk) and T′∗(ηk′), respectively, (k=1,2,…,m′,k′=1,2,…,n′).
(iv)
TakeR-union on each partition of Table 1, representing it in Table 3. Similarly, compute Table 4 from Table 2.
(v)
TakeC-intersection on each partition in Table 3, representing it in Table 5. Similarly, compute Table 6 from Table 4.
(vi)
TakeR-union on Table 5 represents an n′×1 matrix of vertices. Similarly, obtain an n′×1 matrix of edges from Table 6.
(vii)
Find the vertices that appear maximum “n∗” times in the n′×1 matrix of vertices, and then find the edges that appear “n∗” times in the n′×1 matrix of edges. Note that n∗≤n′.
(viii)
By using vertices and edges that are obtained from step (vii), represent a graph of very unwell communications for both sets of base stations integratedly. It may be a connected or disconnected graph.Finally, both companies incorporated to fix the fragile communication on resultant subgraph. Similarly, we can present an algorithm for more than two sets of base stations in a network.


## 5. Conclusion

In the above study, we have presented an application of type-2 soft sets in graph theory. We have introduced vertex-neighbors based type-2 soft sets overX and E. We have introduced some operations on type-2 soft graphs. We have described a practical application of type-2 soft graphs in communication networks and present procedure as an algorithm. The type-2 soft graphs are more general and hence a more efficient tool as compared to type-1 soft graphs. The idea presented in this paper has practical applications in the social networks and computer networks.



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*Source: 6162753-2017-10-18.xml*