# Characterizations of Certain Types of Type 2 Soft Graphs

Authors: Khizar Hayat; Bing-Yuan Cao; Muhammad Irfan Ali; Faruk Karaaslan; Zejian Qin
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

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. Recently, type 2 soft graphs have been introduced. Structurally, it is a very efficient model of uncertainty to deal with graph neighbors and applicable in applied intelligence, computational analysis, and decision-making. The present paper characterizes type 2 soft graphs on underlying subgraphs (regular subgraphs, irregular subgraphs, cycles, and trees) of a simple graph. We present regular type 2 soft graphs, irregular type 2 soft graphs, and type 2 soft trees. Moreover, we introduce type 2 soft cycles, type 2 soft cut-nodes, and type 2 soft bridges. Finally, we present some operations on type 2 soft trees by presenting several examples to demonstrate these new concepts.


---

## Body

## 1. Preliminaries and Introduction

A graphG=(X,E) consists of a nonempty set of objects X, called vertices, and a set E of two element subsets of X called edges. Two vertices x and y are adjacent if {x,y}∈E. A loop is an edge that connects a vertex to itself. 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 graph neighborhoods of a vertex x in a graph is the set of all the vertices adjacent to x including x itself. The graph 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 distance between two vertices x and y in a graph is the number of edges in a shortest path, denoted by d(x,y). The radius of a graph is the minimum eccentricity of any vertex x. A graph G=(X,E) is called a tree if it is connected and contains no cycles. Equivalently (and sometimes more useful), a tree is a connected graph on m vertices with exactly m-1 edges. A forest is a disjoint union of trees. The degree of a vertex of a simple graph is the number of edges incident to the vertex. A regular graph is a graph where each vertex has the same number of neighbors. A graph that is not a regular graph is called irregular graph. A graph is called neighborly irregular graph if no two adjacent vertices have the same degree. A complete graph is a graph in which each pair of graph vertices is connected by an edge. For basic definitions of graphs see [1–3].Soft set theory [4], firstly initiated by Molodtsov, is a new mathematical tool for dealing with uncertainties. Some fruitful operations, soft set theory, are presented by Maji et al. [5] and Ali et al. [6]. We refer to Molodtsov’s soft sets as type 1 soft sets (briefly T1SS). Let A be a set of parameters that can have an arbitrary nature (numbers, functions, sets of words, etc.). Let U be a universe and the power set of U is denoted by P(U). The soft set is defined as follows.Definition 1 (see [5]).
A pair(S,A) is called a soft set over U, where S is a mapping given by S:A→P(U).Note that the set of allT1SS over U will be denoted by σU. Many researchers take attention at applicability of soft sets in real and practical problems. In recent years, research on soft set theory has been rapidly developed, and great progress has been achieved, including works of soft sets in graph theory. Ali et al. [7] introduced a representation of graphs based on neighborhoods. Akram et al. introduced the concepts of soft graphs [8, 9] and soft trees [10]. Let G=(X,E) be a simple graph, A be any nonempty set.LetR⊆A×X be an arbitrary relation from A to X. A mapping I from A to P(X) denoted as I:A→P(X) and defined as I(x)={y∈X∣xRy} and a mapping J from A to P(E) denoted as J:A→P(E) and defined as J(x)={uv∈E∣{u,v}⊆I(x)}. Then (I,A) is a T1SS over X and (J,A) is a T1SS over E. The notion of a soft graph is defined as follows.Definition 2 (see [9]).
A4-tuple G≀=(G,I,J,A) is called a soft graph if it satisfies;(i)
G=(X,E) is a simple graph.(ii)
A is a non-empty set of parameters.(iii)
[I,A] is a T1SS over X.(iv)
[J,A] is a T1SS over E.(v)
(I(α),J(α)) for all α∈A, represents a subgraph of G.The soft graphG≀=(G,I,J,A) can also be written as G≀=(G,I,J)={T(α)∣α∈A}, where T(α)=(I(α),J(α))∀α∈A.Definition 3 (see [9, 10]).
LetG≀ be a soft graph of G. Then G≀ is said to be a soft tree (resp., regular soft graph, irregular soft graph, neighborly irregular soft graph, soft cycle) if every T(α) is a tree (resp., regular graph, irregular graph, neighborly irregular graph, and cycle) for all α∈A.A soft graphG≀ is called a regular soft graph of degree r if T(α) is a regular graph of degree r for all α∈A. In the rest of paper soft tree and soft cycle will be written as type 1 soft tree (briefly, T1ST) and type 1 soft cycle (briefly, T1SC), respectively. Some operations of T1STs are defined as follows.Definition 4 (see [10]).
LetG≀=〈I1,J1,A1〉 and G′≀=〈I2,J2,A2〉 be two T1STs of G. Then G′≀ is a type 1 soft subtree of G≀ if(i)
A2⊆A1;(ii)
for eachx∈A2, T2(x) is a subtree of T1(x).Definition 5 (see [10]).
LetG1≀=〈I1,J1,A1〉 and G2≀=〈I2,J2,A2〉 be two T1STs of G. The extended union of G1≀ and G2≀, denoted by G1≀⊔εG2≀=G≀=〈I,J,B〉, where B=A1∪A2, is defined ∀α∈B, as(1)Iα=I1α,ifα∈A1-A2I2α,ifα∈A2-A1I1α∪I2α,ifα∈A1∩A2Jα=J1α,ifα∈A1-A2J2α,ifα∈A2-A1J1α∪J2α,ifα∈A1∩A2
It can be written asG1≀⊔εG2≀={T(α)=(I(α),J(α))∣α∈B}.Definition 6 (see [10]).
LetG1≀=〈I1,J1,A1〉 and G2≀=〈I2,J2,A2〉 be two T1STs in G. The restricted intersection of G1≀=〈I1,J1,A1〉 and G2≀=〈I2,J2,A2〉, denoted by G1≀⊓rG2≀=G≀=〈I,J,C〉, where B=A1∩A1, is defined ∀α∈B, as I(α)=I1(α)∩I2(α), J(α)=J1(α)∩J2(α),∀α∈A1∩A2.
It can be written asG1≀⊓rG2≀={T(α)=(I(α),J(α))∣α∈B}.Definition 7 (see [10]).
LetG1≀=〈I1,J1,A1〉 and G2≀=〈I2,J2,A2〉 be two T1STs 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)=I1(α)∩I2(β), J(α,β)=J1(α)∩J2(β),∀(α,β)∈A1×A2.Definition 8 (see [10]).
LetG1≀=〈I1,J1,A1〉 and G2≀=〈I2,J2,A2〉 be two T1STs 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)=I1(α)∪I2(β), J(α,β)=J1(α)∪J2(β),∀(α,β)∈A1×A2.Definition 9 (see [8]).
LetG≀=〈I,J,A〉 be a soft graph of G. Then the complement of G≀ is denoted by G≀c and defined by G≀c=〈Ic,Jc,A〉, where Ic(α)=I(α) and Jc(α) contain those edges which are not included in J(α).A generalization of soft sets called type 2 soft sets is introduced by Chatterjee et al. [11]. Some results on type 2 soft sets are validated by Yang and Wang [12] and some new operation on typ-2 soft sets are defined by Khizar et al. [13]. Let X be universe set and E be the set of parameters. The definition of type 2 soft set is as follows:Definition 10 (see [11]).
Let(U,E) be a soft universe and σ(U) be the collection of all T1SS over (U,E). Then a mapping :A→σ(U), A⊆E is called a type 2 soft set (briefly T2SS) over (U,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(U) 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”.LetG=(X,E) be a simple graph. 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). The set of neighbors of an element x∈X is denoted by NBx and defined by NBx={z∈X∣xz∈E}. Then NBA=⋃x∈ANBx, where A⊂X. Let a subset R of NBA×X be an arbitrary relation from NBA to X. Let G=(X,E) be a simple graph. Khizar et al. [14] introduced type 2 soft graphs by using T2SSs over X and T2SSs over E.Definition 11 (see [14]).
LetG=(X,E) be a simple graph, A⊂X and Γ(X) be the collection of all T1SS over X. Then a mapping I∗:A→Γ(X) is called a T2SS over X and it is denoted by [I∗,A]. In this case, corresponding to each vertex x∈A, I∗(x) is a T1SS. Thus, for each x∈A, there exists a T1SS, (Ix,NBx) such that I∗(x)=(Ix,NBx) where Ix:NBx→P(X) can be defined as Ix(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 (see [14]).
LetG=(X,E) be a simple graph, A⊂X and Γ(E) be the collection of all T1SS over E. Let [I∗,A] be a VN-type 2 soft set over X. Then a mapping J∗:A→Γ(E) is called a T2SS over E and it is denoted by [J∗,A]. In this case, corresponding to each vertex x∈A, J∗(x) is a T1SS. Thus, for each x∈A, there exists a T1SS, (Jx,NBx) such that J∗(x)=(Jx,NBx) where Jx:NBx→P(E) can be define as Jx(z)={αβ∈E∣{α,β}⊆Ix(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.Definition 13 (see [14]).
A5-tuple G∗=(G,I∗,J∗,A,NBA) is called a type 2 soft graph (briefly, T2SG) if it satisfies following conditions:(i)
G=(X,E) is a simple graph.(ii)
A is a non-empty set of parameters.(iii)
[I∗,A] is a VN-type 2 soft set over X.(iv)
[J∗,A] is a VN-type 2 soft set over E.(v)
T1SS corresponding to (I∗(x),J∗(x))∀x∈A, represents a type 1 soft graph.
AT2SG can also be represented by G∗=〈I∗,J∗,A〉={T∗(x)∣x∈A}, where T∗(x)=(Tx,NBx) such that Tx(z)=(Ix(z),Jx(z)), ∀z∈NBx.


## 2. Certain Types of Type 2 Soft Graphs

In this section, we present regular type 2 soft graphs, irregular type 2 soft graphs, and type 2 soft trees. Moreover, we introduce type 2 soft cycles, type 2 soft cut-nodes, and type 2 soft bridges.Definition 14.
LetG∗=(G,I∗,J∗,A,NBA) be a T2SG of G. Then G∗ is said to be a regular T2SG if T1SG corresponding to every T∗(γ) is a regular T1SG for all γ∈A. A T2SGG∗ is called a regular T2SG of degree r if T1SG corresponding to T∗(γ) is a regular T1SG of degree r for all γ∈A.Example 15.
Consider a graphG=(X,E) as shown in Figure 1. Let A={e9,e10}. It may be written that NBe9={e11,e8} and NBe10={e8,e11}. Let [I∗,A] and [J∗,A] be two T2SSs over X and E respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Define Ie9(z)={y∈X∣zRy⇔5≥d(z,y)≥3}, Je9(z)={αβ∈E∣{α,β}⊆Ie9(z)} for all z∈NBe9⊆X, and Ie10(z)={y∈X∣zRy⇔5≥d(z,y)≥3}, Je10(z)={αβ∈E∣{α,β}⊆Ie10(z)} for all z∈NBe10⊆X. Then T2SSs[I∗,A] and [J∗,A] are as in the following I∗(e9)={e11,{e4,e5,e6,e7}},{e8,{e1,e3,e4,e2}}, J∗e9={{e11,{e4e5,e5e7,e4e6,e7e6}},{e8,{e1e2,e3e2,e3e4,e1e4,e2e4, e1e3}}}, I∗(e10)={e11,{e4,e5,e6,e7}},{e8,{e1,e3,e4,e2}}, J∗e10={{e11,{e4e5,e5e7,e4e6,e7e6}},{e8,{e1e2,e3e2,e3e4, e1e4,e2e4,e1e3}}}. One can check that G∗=(T∗(e9),T∗(e10)) is a regular T2SG of G as shown in Figure 2.Figure 1
Simple graph.Figure 2
G ∗ = ( T ∗ ( e 9 ) , T ∗ ( e 10 ) ).Theorem 16.
LetG be a complete graph. Then every T2SG of G is a regular T2SG of G.Proof.
SupposeG is a complete graph and G∗ be a T2SG of G. Let (Tα,NBα) be a T1SG corresponding to T∗(α) for all α∈A. Then Tα(x)∀x∈NBα is a subgraph of G. Since every subgraph of a complete graph is complete and every complete graph is regular. Therefore, Tα(x)∀x∈NBα is a regular subgraph. This implies that T1SG corresponding to T∗(α) for all α∈A is regular T1SG. Hence G∗ is a regular T2SG of G.Note that ifG∗ is a regular T2SG of G then G need not be a complete graph.Example 17.
In Example15, G∗ is regular T2SG but G is not a complete graph.Definition 18.
LetG∗=(G,I∗,J∗,A,NBA) be a T2SG of G. Then the complement of G∗ is denoted by G∗c and defined by G∗c=(T∗c(z1),T∗c(z2),..,T∗c(zn)) for all x1,x2,..,xn∈A, where T1SG corresponding to T∗c(zi)=(I∗c(zi),J∗c(zi)) is the complement of T1SG corresponding to T∗(zi)=(I∗(zi),J∗(zi)) for all zi∈A(i=1 to n).Example 19.
Consider a graphG=(X,E) as shown in Figure 3. Let A={e2,e5}. It may be written that NBe2={e1,e3,e4} and NBe5={e4,e6}. Let [I∗,A] and [J∗,A] be two T2SSs over X and E respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Define Ie2(z)={y∈X∣zRy⇔d(z,y)≤1}, Je2(z)={αβ∈E∣{α,β}⊆Ie2(z)} for all z∈NBe2⊆X, and Ie5(z)={y∈X∣zRy⇔d(z,y)=2}, Je5(z)={αβ∈E∣{α,β}⊆Ie5(z)} for all z∈NBe5⊆X. Then T2SSs[I∗,A] and [J∗,A] are as follows:(2)I∗e2=e1,e1,e3,e6,e2,e3,e1,e3,e7,e2,e4,e7,e5,e4,e2,J∗e2=e1,e1e2,e2e3,e3e1,e1e6,e3,e1e2,e2e3,e3e1,e3e7,e4,e4e7,e4e5,e4e2,I∗e5=e4,e6,e1,e3,e6,e3,e4,e2,J∗e5=e4,e6e1,e1e3,e6,e3e2,e4e2,Then [I∗c,A] and [J∗c,A] are as follows:(3)I∗ce2=e1,e1,e3,e6,e2,e3,e1,e3,e7,e2,e4,e7,e5,e4,e2,J∗ce2=e1,e2e6,e6e3,e3,e1e7,e2e7,e4,e2e7,e2e5,e7e5,I∗ce5=e4,e6,e1,e3,e6,e3,e4,e2,J∗ce5=e4,e6e3,e6,e3e4.The complement of G∗=(T∗(e2),T∗(e5)) is G∗c=(T∗c(e2),T∗c(e5)) as shown in Figure 4.Figure 3
Simple graph.Figure 4
G ∗ c = ( T ∗ c ( e 2 ) , T ∗ c ( e 5 ) ).Proposition 20.
IfG∗ is a regular T2SG of G then G∗c is a regular T2SG of G.Proof.
LetG∗ be a regular T2SG of G. Let (Tα,NBα) be a T1SG corresponding to T∗(α) for all α∈A. Then Tα(x)∀x∈NBα is a regular subgraph of G. Since complement of a regular graph is regular, Tαc(x)∀x∈NBα is a regular subgraph. This implies that T1SG corresponding to T∗c(α) for all α∈A is regular T1SG. Hence G∗c is a regular T2SG of G.Proposition 21.
LetG be a regular graph. Then every T2SG of G may not be a regular T2SG of G.Example 22.
Consider a regular graphG=(X,E), where X={e1,e2,e3,e4,e5,e6} and E={e1e2,e3e2,e3e4,e4e5,e5e6,e1e6,e1e4,e6e3,e5e2}. Let A={e1,e4}. It may be written that NBe1={e2,e6,e4} and NBe5={e4,e6,e2}. Let [I∗,A] and [J∗,A] be two T2SSs over X and E respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Define Ie1(z)={y∈X∣zRy⇔d(z,y)≤1}, Je1(z)={αβ∈E∣{α,β}⊆Ie1(z)} for all z∈NBe1⊆X, and Ie5(z)={y∈X∣zRy⇔d(z,y)≤1}, Je5(z)={αβ∈E∣{α,β}⊆Ie5(z)} for all z∈NBe5⊆X. Then T2SSs[I∗,A] and [J∗,A] are as follows:(4)I∗e1=e2,e2,e1,e5,e3,e4,e4,e1,e5,e3,e6,e1,e6,e3,e5,J∗e1=e2,e1e2,e2e4,e5e2,e4,e5e4,e4e1,e4e3,e6,e5e6,e6e3,e1e6,I∗e5=e4,e2,e1,e5,e6,e3,e2,e3,e4,e1,e6,e5,e6,e3,e4,e1,e2,e5,J∗e5=e4,e1e2,e3e2,e5e6,e1e6,e6e3,e5e2,e6,e1e2,e3e2,e3e4,e4e5,e1e4,e5e2,e2,e3e4,e4e5,e5e6,e1e6,e1e4,e6e3.Then G∗=(T∗(e1),T∗(e5)) is not a regular T2SG of G as shown in Figure 5.Figure 5
G ∗ = ( T ∗ ( e 1 ) , T ∗ ( e 5 ) ).Definition 23.
LetG∗ be a T2SG of G. Then G∗ is said to be irregular T2SG if T1SG corresponding to T∗(γ) is an irregular T1SG for all γ∈A.Example 24.
Consider a graphG=(X,E) as shown in Figure 6. Let A={e5,e6}. It may be written that NBe5={e1,e6} and NBe6={e5,e7}. Let [I∗,A] and [J∗,A] be two T2SSs over X and E respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Define Ie5(z)={y∈X∣zRy⇔d(z,y)≤1}, Je5(z)={αβ∈E∣{α,β}⊆Ie5(z)} for all z∈NBe5⊆X, and Ie6(z)={y∈X∣zRy⇔d(z,y)≤1}, Je6(z)={αβ∈E∣{α,β}⊆Ie6(z)} for all z∈NBe6⊆X. Then T2SSs[I∗,A] and [J∗,A] are as follows:(5)I∗e5=e1,e1,e4,e5,e2,e6,e5,e6,e7,J∗e5=e1,e1e2,e2e4,e5e1,e1e4,e6,e5e6,e6e7,I∗e6=e5,e6,e1,e5,e7,e3,e4,e6,e7,J∗e6=e5,e6e1,e5e6,e7,e3e4,e7e3,e7e4,e7e6.Then G∗=(T∗(e6),T∗(e5)) is an irregular T2SG of G as shown in Figure 7.Figure 6
Simple graph.Figure 7
G ∗ = ( T ∗ ( e 5 ) , T ∗ ( e 6 ) ).Definition 25.
LetG∗ be a type 2 soft graph of G. Then G∗ is said to be neighborly irregular T1SG if T1SG corresponding to T∗(γ) is an neighborly irregular T1SG for all γ∈A.Example 26.
Consider a graphG=(X,E) as shown in Figure 8. Let A={e1,e6}. It may be written that NBe1={e2,e4} and NBe6={e5,e3}. Let [I∗,A] and [J∗,A] be two T2SSs over X and E respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Define Ie1(z)={y∈X∣zRy⇔d(z,y)≤1}, Je1(z)={αβ∈E∣{α,β}⊆Ie1(z)} for all z∈NBe1⊆X, and Ie6(z)={y∈X∣zRy⇔d(z,y)≤1}, Je6(z)={αβ∈E∣{α,β}⊆Ie6(z)} for all z∈NBe6⊆X. Then T2SSs[I∗,A] and [J∗,A] are as in the following; I∗(e1)={e2,{e1,e3,e2}},{e4,{e4,e1,e3,e5}}, J∗(e1)={e2,{e1e2,e2e3}},{e4,{e4e3,e4e5,e4e2}}, I∗(e6)={e3,{e3,e2,e4,e6}},{e5,{e5,e4,e6}}, J∗(e6)={e3,{e4e3,e3e6,e3e2}},{e5,{e5e4,e5e6}}, Then G∗=(T∗(e1),T∗(e6)) is a neighborly irregular T2SG of G as shown in Figure 9.Figure 8
Simple graph.Figure 9
G ∗ = ( T ∗ ( e 1 ) , T ∗ ( e 6 ) ).Definition 27.
LetG∗ be a T2SG of a simple graph G. Let (Tα,NBα) be a T1SG corresponding to T∗(α) for all α∈A. An ede uv in G∗ is said to be a type 2 soft bridge if its removal disconnect the subgraph Tα(x), ∀x∈NBα.Definition 28.
LetG∗ be a T2SG of a simple graph G. Let (Tα,NBα) be a T1SG corresponding to T∗(α) for all α∈A. An vertex z in G∗ is said to be a type 2 soft cut-vertex if its removal disconnect the subgraph Tα(x), ∀x∈NBα.Example 29.
ConsiderT2SGG∗ defined in Example 24. In the Figure 7, type 2 soft bridges of G∗ are e1e5 in Te5(e1), e7e6,e6e5 in Te5(e6), e5e6,e5e1 in Te6(e5) and e6e7 in Te6(e7). Moreover, type 2 cut-vertices of G∗ are e1 in Te5(e1), e5 in Te5(e6), e6 in Te6(e5) and e7 in Te6(e7).Definition 30.
LetG∗=(G,I∗,J∗,A,NBA) be a T2SG of G. Then G∗ is said to be a type 2 soft tree (briefly, T2ST) if T1SG corresponding to every T∗(γ) is a T1ST for all γ∈A.Example 31.
Consider a graphG=(X,E) as shown in Figure 10. Let A={c,f}⊂X and NBc={b,d}, NBf={e,g}. Let [I∗,A] and [J∗,A] be two neighbor-induced T2SSs over X and E respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. We define Ic(z)={y∈X∣zRy⇔d(z,y)=rad(G)}, Jc(z)={αβ∈E∣{α,β}⊆Ic(z)} for all z∈NBc⊆X and If(z)={y∈X∣zRy⇔d(z,y)=rad(G)}, Jf(z)={αβ∈E∣{α,β}⊆If(z)} for all z∈NBf⊆X. Then T2SSs[I∗,A] and [J∗,A] are as in the following I∗(c)={b,{g,e}},{d,{a,g,f}}, J∗(c)={b,{ge}},{d,{ag,gf}}, I∗(f)={g,{d,b}},{e,{a,b,c}}, J∗(f)={g,{bd}},{e,{ab,bc}}. Therefore, G∗=(T∗(c),T∗(f)) is a T2ST of G as shown in Figure 11. It is also called VN-type 2 soft tree.
Hence,G∗=(T∗(c),T∗(f)) is a T2SG of G. It is also called VN-type 2 soft graph. It may be written that P=⋃x∈ANBx={b,d,e,g}. 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 T2ST is given in Table 1.Table 1
Tabular representation of a type 2 soft tree.
(A↣P)∖X a b c d e f g  c↣b 0 0 0 0 1 0 1 c↣d 1 0 0 0 0 1 1 f↣g 0 1 0 1 0 0 0 f↣e 1 1 1 0 0 0 0  (A↣P)∖X a b b c b d d c d e e f e g g f ag  c↣b 0 0 0 0 0 0 1 0 0 c↣d 0 0 0 0 0 0 0 1 1 f↣g 0 0 1 0 0 0 0 0 0 f↣e 1 1 0 0 0 0 0 0 0Figure 10
Simple graph.Figure 11
G ∗ = ( T ∗ ( c ) , T ∗ ( f ) ).Theorem 32.
Let(Tγ,NBγ) be a T1SS corresponding to T∗(γ)∀γ∈A. Let Tγ(x)∀x∈NBγ be subgraph with n≥3 vertices of G and G∗ a T2SG of G. Then G∗ is not a complete T2SG of G.Proof.
Let(Tγ,NBγ) be a T1SS corresponding to T∗(γ)∀γ∈A. Suppose on the contrary that G is a complete T2SG, then every subgraph Tγ(x), for all x∈NBγ is complete. Suppose α,β be arbitrary nodes of (Tγ(x) and they are connected by an edge αβ. Since Tγ(x) is subgraph with n≥3 vertices of G, then we can always find at least one vertex η which is connected to α by an edge αη and to β by an edge βη, because Tγ(x) is a complete graph. Then there exists a cycle αβη. Therefore, Tγ(x)∀x∈NBγ cannot be a T1ST which contradicts the fact that Tγ(x) is a connected subgraph of T2SG. Hence, G∗ cannot be a complete T2SG.Definition 33.
LetG∗ be a T2SG and (Tγ,NBγ) be a T1ST corresponding to T∗(γ) for any γ∈A. Then G∗ is called type 2 soft forest if Tγ(x) consists of more than one disconnected tree for all x∈NBγ.Definition 34.
LetG∗ be a T2SG of G. Then G∗ is said to be a type 2 soft cycle (briefly, T2SC) if T1SG corresponding to T(γ) is a type 1 soft cycle of G for each γ∈A.Example 35.
Consider a simple graphG=(X,E), where X={e1,e2,e3,e4,e5,e6,e7,e8,e9,e10} and E={e1e2,e2e3,e3e1,e3e4,e5e3,e5e6,e6e4,e7e6,e7e9,e7e8,e9e10, e8e10}. Let A={e9,e8}⊂X. It may be written that NBe9={e10,e7} and NBe8={e7,e10}. Let [I∗,A] and [J∗,A] be two T2SSs over X and E, respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Define Ie9(z)={y∈X∣zRy⇔5≥d(z,y)≥3}, Je9(z)={αβ∈E∣{α,β}⊆Ie9(z)} for all z∈NBe9⊆X, and Ie8(z)={y∈X∣zRy⇔5≥d(z,y)≥3}, Je8(z)={αβ∈E∣{α,β}⊆Ie8(z)} for all z∈NBe8⊆X. Then T2SSs[I∗,A] and [J∗,A] are as follows:(6)I∗e9=e7,e2,e3,e1,e10,e5,e3,e4,e6,J∗e9=e10,e2e3,e3e1,e2e1,e10,e4e3,e3e5,e6e5,e6e4,I∗e8=e7,e2,e3,e1,e10,e5,e3,e4,e6,J∗e8=e10,e2e3,e3e1,e2e1,e10,e4e3,e3e5,e6e5,e6e4.One can check that G∗=(T∗(e4),T∗(e8)) is a T2SC of G as shown in Figure 12. It is also called VN-type 2 soft cycle.Figure 12
G ∗ = ( T ∗ ( e 9 ) , T ∗ ( e 8 ) ).Theorem 36.
IfG∗ is a T2SC of G then G∗ is not a T2ST of G.Proof.
LetG∗ be a T2SC of G. Let (Tγ,NBγ) be a type 1 soft cycle corresponding to T∗(γ)∀γ∈A. By definition, tree does not contain cycle. Then Tγ(x) is not a tree for all x∈NBγ, so that (Tγ,NBγ) is not a type 1 soft tree. Hence G∗ is not a T2ST of G.The converse of above theorem is not true in general; i.e., ifG∗ is not a T2ST then G need not be a T2SC. The following example illustrates it.Example 37.
Consider a graphG=(X,E) as shown in Figure 13.
Let={a,b}⊂X. Then NBa={b,c,d}, NBb={a,c,d}. Let [I∗,A] and [J∗,A] be two T2SSs over X and E, respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Let Ia(α)={y∈X∣αRy⇔d(α,y)≤1}, Ja(α)={uv∈E∣{u,v}⊆Ia(α)}∀α∈NBa and Ib(β)={y′∈X∣βRy′⇔d(β,y′)=1}, Jb(β)={u′v′∈E∣{u′,v′}⊆Ib(β)}∀β∈NBb. Then(7)I∗a=b,a,b,d,c,c,a,b,c,e,d,a,b,d,e,J∗a=b,ab,ad,ac,bc,bd,c,ab,ac,ce,bc,d,ab,ad,de,db,I∗b=a,b,c,d,c,a,b,e,d,a,b,e,J∗b=a,bc,bd,c,ab,d,abFigure 14 shows the respective T1SGs corresponding to T∗(a)=(I∗(a),J∗(a)) and T∗(b)=(I∗(b),J∗(b)) respectively. One can check that Ta(b)=(Ia(b),Ja(b)), Ta(c)=(Ia(c),Ja(c)), Ta(d)=(Ia(d),Ja(d)), Tb(c)=(Ib(c),Jb(c)) and Tb(d)=(Ib(d),Jb(d)) are not trees. This implies that G∗=(T∗(a),T∗(b)) is not a T2ST of G. But G∗ is not a T2SC of G.Figure 13
Simple graph.Figure 14
G ∗ = ( T ∗ ( a ) , T ∗ ( b ) ).Proposition 38.
EveryT2SC of G is a regular T2SG of G.Proof.
Suppose thatG∗ is a T2SC. Let (Tγ,NBγ) be a T1SC corresponding to T∗(γ) for any γ∈A. Then Tγ(x) is a cycle for all x∈NBα. Since cycle is closed path and each vertex has degree 2, this implies that Tγ(x) is a regular graph for all x∈NBγ. Therefore (Tγ,NBγ) is regular T1SG. Since γ was taken to be arbitrary, thus it holds for all γ∈A. Hence G∗ is a regular T2SG of G.


## 3. Operations on Type 2 Soft Trees

In this section, we present type 2 soft subtree ofT2ST, union, intersection, OR operation, and AND operation of T2ST.Definition 39.
LetG1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 be two T2STs of G. Then G2∗ is a type 2 soft subtree of G1∗ if(i)
A2⊆A1,(ii)
for eachx∈A2, T1ST corresponding to T2∗(x)=(I2∗(x),J2∗(x)) is a type 1 soft subtree of T1ST corresponding to T1∗(x)=(I1∗(x),J1∗(x)).Example 40.
Consider a simple graphG=(X,E), where X={e1,e2,e3,e4,e5,e6,e7} and E={e1e2,e2e3,e3e4,e5e4,e5e6,e6e7,e7e1}. Let A={e2,e4}, B={e2,e4,e7}. It may be written that NBe2={e3,e1}, NBe4={e3,e5} and NBe7={e1,e6}.
Let[I∗,A] and [J∗,A] be two T2SSs over X and E, respectively, such that I∗(x)=(Ix,NBx) and J∗(x)=(Jx,NBx) for all x∈A. Define Ie2(z)={y∈X∣zRy⇔d(z,y)≤1}, Je1(z)={αβ∈E∣{α,β}⊆Ie1(z)} for all z∈NBe2⊆X, Ie4(z)={y∈X∣zRy⇔d(z,y)≤1}, Je4(z)={αβ∈E∣{α,β}⊆Ie4(z)} for all z∈NBe4⊆X. Then T2SSs[I∗,A] and [J∗,A] are as follows:(8)I∗e2=e1,e1,e2,e7,e3,e3,e2,e4,J∗e2=e1,e1e2,e7e1,e3,e2e3,e3e4,I∗e4=e3,e4,e2,e3,e5,e6,e5,e4,J∗e4=e3,e3e4,e3e2,e5,e6e5,e5e4.
ThenG∗=(T∗(e2),T∗(e4)) is a T2ST of G as shown in Figure 15.
Let[I′∗,B] and [J′∗,B] be two T2SSs over X and E, respectively, such that I′∗(x)=(Jx′,NBx) and J′∗(x)=(Jx′,NBx) for all x∈B. Define(9)Ie2′z=y∈X∣zRy⇔dz,y≤1,Je2′z=αβ∈E∣α,β⊆Ie2′zfor  allz∈NBe2⊆X,Ie4′z=y∈X∣zRy⇔dz,y≤2,Je4′z=αβ∈E∣α,β⊆Ie4′zfor  allz∈NBe4⊆X,Ie7′z=y∈X∣zRy⇔dz,y≤1,Je7′z=αβ∈E∣α,β⊆Ie7′zfor  allz∈NBe7⊆X.
ThenT2SS[I′∗,B] and [J′∗,B] are as follows:(10)I′∗e2=e1,e1,e2,e7,e3,e3,e2,e4,J′∗e2=e1,e1e2,e7e1,e3,e2e3,e3e4,I′∗e4=e3,e4,e2,e3,e1,e5,e5,e6,e5,e4,e7,e3,J′∗e4=e3,e1e2,e2e3,e3e4,e5e4,e5,e3e4,e4e5,e5e6,e6e7,I′∗e7=e1,e1,e2,e7,e6,e5,e6,e7,J′∗e7=e1,e1e2,e7e1,e6,e5e6,e6e7,
ThenG′∗=(T′∗(e2),T′∗(e4),T′∗(e7)) is a T2ST of G as shown in Figure 16.
One can check thatA⊂B and T∗(e2)⊆~T′∗(e2), T∗(e4)⊆~T′∗(e4). Hence G∗ is a type 2 subtree of G′∗.Figure 15
G ∗ = ( T ∗ ( e 2 ) , T ∗ ( e 4 ) ).Figure 16
G ′ ∗ = ( T ′ ∗ ( e 2 ) , T ′ ∗ ( e 4 ) , T ′ ∗ ( e 7 ) ).Theorem 41.
LetG1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 be two T2STs of G. Then G2∗ is a type 2 soft subtree of G1∗ if and only if I2∗(x)⊆~I1∗(x) and J2∗(x)⊆~J1∗(x) for all x∈A2.Proof.
SupposeG2∗ is a type 2 soft subtree of G1∗. Then, by the definition of type 2 soft subtree,(i)
A2⊆A1,(ii)
For eachx∈A2, T1ST corresponding to T2∗(x)=(I2∗(x),J2∗(x)) is a type 1 soft subtree of T1ST corresponding to T1∗(x)=(I1∗(x),J1∗(x)).
SinceT1ST corresponding to T2∗(x) is a type 1 soft subtree of T1ST corresponding to T1∗(x) for all x∈A2. Then I2∗(x)⊆~I1∗(x) and J2∗(x)⊆~J1∗(x) for all x∈A2.
Conversely, given thatI2∗(x)⊆~I1∗(x) and J2∗(x)⊆~J1∗(x) for all x∈A2. As G1∗ is a T2ST of G, T1SS corresponding to T1∗(x) is a T1ST of G for all x∈A1. Also, G2∗ is a T2ST of G, T1SS corresponding to T2∗(x) is a T1ST of G for all x∈A2. This implies that T1ST corresponding to T2∗(x) is a type 1 soft subtree of T1ST corresponding to T1∗(x) for all x∈A2. Thus, G2∗ is a type 2 soft subtree of G1∗.Definition 42.
LetG1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 be two T2STs of G. The union of G1∗ and G2∗, is denoted by G1∗⊔G2∗=G∗=〈I∗,J∗,C〉, where C=A1∪A2 is defined ∀α∈C, as(11)I∗α=I1∗α,ifα∈A1-A2I2∗α,ifα∈A2-A1I1∗α∪~I2∗α,ifα∈A1∩A2
whereI1∗(α)∪~I2∗(α) for all α∈A1∩A2 refers to the usual type 1 soft union between the respective T1ST corresponding to I1∗(α) and I2∗(α), respectively. And,(12)J∗α=J1∗α,ifα∈A1-A2J2∗α,ifα∈A2-A1J1∗α∪~J2∗α,ifα∈A2∩A1
whereJ1∗(α)∪~J2∗(α) for all α∈A2∩A1 refers to the usual type 1 soft extended union between the respective T1ST corresponding to J1∗(α) and J2∗(α), respectively.It can be written asG1∗⊔G2∗={T∗(α)=(I∗(α),J∗(α))∣α∈C}.Theorem 43.
LetG1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 be two T2STs of G with A1∩A2=∅. Then G1∗⊔G2∗ is a T2ST of G.Proof.
Theunion of G1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 is defined as G1∗⊔G2∗=G∗=〈I∗,J∗,C〉 where C=A1∪A2 for all α∈C,(13)I∗α=I1∗α,ifα∈A1-A2I2∗α,ifα∈A2-A1I1∗α∪~I2∗α,ifα∈A1∩A2
whereI1∗(α)∪~I2∗(α) for all α∈A1∩A2 refers to the usual type 1 soft union between the respective T1SS corresponding to I1∗(α) and I2∗(α), respectively. And,(14)J∗α=J1∗α,ifα∈A1-A2J2∗α,ifα∈A2-A1J1∗α∪~J2∗α,ifα∈A1∩A2
whereJ1∗(α)∪~J2∗(α) for all α∈A1∩A2 refers to the usual type 1 soft extended union between the respective T1SS corresponding to J1∗(α) and J2∗(α), respectively.
SinceG1∗ is a T2ST of G. Then T1ST corresponding to (I1∗(x),J1∗(x)) is a T1ST of G for all x∈A1-A2.
SinceG2∗ is a T2ST of G. Then T1ST corresponding to (I2∗(x),J2∗(x)) is a T1ST of G for all x∈A2-A1.
It is given thatA1∩A2=∅. Thus, G1∗⊔G2∗=G∗=〈I∗,J∗,A1∪A2〉 is a T2ST of G.IfA1∩A2≠∅ then union of two T2ST may not be a T2ST as it can be seen in the following example.Example 44.
Consider a graphG defined in Example 40. Let A={e2,e4},B={e2,e7}. It may be written that NBe2={e3,e1}, NBe4={e3,e5} and NBe7={e1,e6}.
Let[I∘∗,A] and [J∘∗,A] be two T2SSs over X and E, respectively, such that I∘∗(x)=(Ix∘,NBx) and J∘∗(x)=(Jx∘,NBx) for all x∈A. Define Ie2∘(z)={y∈X∣zRy⇔d(z,y)≤1}, Je2∘(z)={αβ∈E∣{α,β}⊆Ie2∘(z)} for all z∈NBe2⊆X and Ie4∘(z)={y∈X∣zRy⇔d(z,y)≤1}, Je4∘(z)={αβ∈E∣{α,β}⊆Ie4∘(z)} for all z∈NBe4⊆X. Then T2SSs[I∘∗,A] and [J∘∗,A] are as follows:(15)I∘∗e2=e1,e1,e2,e7,e3,e3,e2,e4,J∘∗e2=e1,e1e2,e7e1,e3,e2e3,e3e4,I∘∗e4=e3,e4,e2,e3,e5,e6,e5,e4,J∘∗e4=e3,e3e4,e3e2,e5,e6e5,e5e4.Then G∗=(T∘∗(e2),T∘∗(e4)) is a T2ST of G.
Let[I′∗,B] and [J′∗,B] be two T2SSs over X and E, respectively, such that I′∗(x)=(Ix′,NBx) and J′∗(x)=(Jx′,NBx) for all x∈B. Define Ie2′(z)={y∈X∣zRy⇔d(z,y)≥2}, Je1′(z)={αβ∈E∣{α,β}⊂Ie2′(z)} for all z∈NBe2⊆X and Ie7′(z)={y∈X∣zRy⇔d(z,y)≥1}, Je7′(z)={αβ∈E∣{α,β}⊂Ie7′(z)} for all z∈NBe7⊆X. Then T2SSs[I′∗,B] and [J′∗,B] are as follows:(16)I′∗e2=e1,e4,e5,e3,e6,e7,J′∗e2=e1,e4e5,e3,e6e7,I′∗e7=e1,e3,e4,e5,e6,e6,e1,e2,e3,e4,J′∗e7=e1,e3e4,e4e5,e5e6,e6,e1e2,e2e3,e3e4.Then G′∗=(T′∗(e2),T′∗(e4),T′∗(e7)) is a T2ST of G. By the definition of union of T2STs, (17)I∗e4=I∘∗e4andJ∗e4=J∘∗e4,e4∈A-B,I∗e7=I′∗e7andJ∗e7=J′∗e7,e7∈B-A,I∗e2=I∘∗e2∪I′∗e2andJ∗e2=J∘∗e2∪~J′∗e2,e2∈B∩A.By routine calculations, it is easy to see that T1SGs corresponding to T∗(e4)=T∘∗(e4) and T∗(e7)=T′∗(e7) are T1STs. But T1SG corresponding to T∗(e2)=T∘∗(e2)∪~T′∗(e2) is a disconnected type 2 soft forest, as shown in Figure 17. Therefore, G1∗⊔G2∗=G∗=〈I∗,J∗,A∪B〉 is not a T2ST of G.Figure 17
T ∗ ( e 2 ).Lemma 45.
LetG1∗=〈S1∗,T1∗,A1〉 and G2∗=〈S2∗,T2∗,A2〉 be two T2STs of G. If A1∩A2=∅, then their union is a T2ST of G.Definition 46.
LetG1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 be two T2STs of G. The intersection of G1∗ and G2∗, denoted by G1∗⊓G2∗=G∗=〈I∗,J∗,C〉, where C=A∩B is defined as I∗(α)=I1∗(α)∩~I2∗(α), for all α∈A∩B, where I1∗(α)∩~I2∗(α) for all α∈A∩B refers to the usual type 1 soft intersection between the respective T1SS corresponding to I1∗(α) and I2∗(α), respectively. And J∗(α)=J1∗(α)∩~J2∗(α) for all α∈A∩B where J1∗(α)∩~J2∗(α) for all α∈A∩B refers to the usual type 1 soft intersection between the respective T1SS corresponding to J1∗(α) and J2∗(α), respectively.It can be written asG1∗⊓G2∗={T∗(α)=(I∗(α),J∗(α))∣α∈C}.The intersection of twoT2STs may not be T2ST as it can be seen in the following example.Example 47.
Consider a simple graphG shown in Figure 18. Let A={a,h} and B={b,h}. It may be written that NBa={v,b}, NBh={v,g} and NBb={a,c}.
Let[I∘∗,A] and [J∘∗,A] be two T2SSs over X and E, respectively, such that I∘∗(x)=(Ix∘,NBx) and J∘∗(x)=(Jx∘,NBx) for all x∈A. Define Ia∘(z)={y∈X∣zRy⇔d(z,y)≤1}, Ja∘(z)={αβ∈E∣{α,β}⊂Ia∘(z)} for all z∈NBa⊆X and Ih∘(z)={y∈X∣zRy⇔d(z,y)≤1}, Jh∘(z)={αβ∈E∣{α,β}⊂Ih∘(z)} for all z∈NBh⊆X. Then T2SSs[I∘∗,A] and [J∘∗,A] are as follows:(18)I∘∗a=b,a,b,c,v,b,v,h,d,J∘∗a=b,ab,bc,v,vb,vh,vd,I∘∗h=v,a,d,h,v,g,g,f,h,J∘∗h=v,vd,va,vh,g,gf,gh.Then G∗=(T∘∗(a),T∘∗(g)) is a T2ST of G.
Let[I′∗,B] and [J′∗,B] be two T2SSs over X and E, respectively, such that I′∗(x)=(Ix′,NBx) and J′∗(x)=(Jx′,NBx) for all x∈B. Define Ib′(z)={y∈X∣zRy⇔d(z,y)≥3}, Jb′(z)={αβ∈E∣{α,β}∈Ib′(z)} for all z∈NBb⊆X and Ih′(z)={y∈X∣zRy⇔1≤d(z,y)≤3}, Jh′(z)={αβ∈E∣{α,β}∈Ih′(z)} for all z∈NBh⊆X. Then T2SSs[I′∗,B] and [J′∗,B] are as follows:(19)I′∗b=a,e,f,g,c,f,g,h,J′∗b=a,ef,fg,c,fg,gh,I′∗h=v,e,f,g,d,a,b,h,c,g,d,e,f,h,v,a,J′∗h=v,ef,fg,ed,vd,dc,g,ef,ed,vh,va,vd.Then G′∗=(T′∗(b),T′∗(h)) is a T2ST of G. By the definition of intersection of T2ST, (20)I∗h=I∘∗h∩~I′∗handJ∗h=J∘∗h∩~J′∗h,h=B∩A.By routine calculations, it is easy to see that T1SG corresponding to T∗(h)=T∘∗(h)∩~T′∗(h) is a disconnected T1SG, as shown in Figure 19. Therefore, G1∗⊓G2∗=G∗=〈I∗,J∗,A∪B〉 is not a T2ST of G.Figure 18
Simple graph.Figure 19
G 1 ∗ ⊓ G 2 ∗ = G ∗ = 〈 I ∗ , J ∗ , A ∩ B 〉.Definition 48.
LetG1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 be two T2STs of G. The AND operation of G1∗ and G2∗, denoted by G1∗⋀G2∗=G∗=〈I∗,J∗,A×B〉 defined by I∗(γ,η)=I1∗(γ)∧~I2∗(η), J∗(γ,η)=J1∗(γ)∧~J2∗(η) for all (γ,η)∈A×B, where I1∗(γ)∧~I2∗(η) for all (γ,η)∈A×B refers to the usual type 1 soft AND operation between the respective T1SS corresponding to I1∗(γ) and I2∗(η) respectively and J1∗(γ)∧~J2∗(η) for all (γ,η)∈A×B refers to the usual type 1 soft AND operation between the respective T1SS corresponding to J1∗(γ) and J2∗(η) respectively.Example 49.
Consider a simple graphG=(X,E), where X={e1,e2,e3,e4,e5,e6,e7,e8} and E={e1e2,e2e3,e3e4,e5e4,e5e6,e6e7,e7e8,e8e1}. Let A={e1,e8},B={e5}. It may be written that NBe1={e8,e2}, NBe8={e7,e1} and NBe5={e4,e6}.
Let[I∘∗,A] and [J∘∗,A] be two T2SSs over X and E, respectively, such that I∘∗(x)=(Ix∘,NBx) and J∘∗(x)=(Jx∘,NBx) for all x∈A. Define Ie1∘(z)={y∈X∣zRy⇔d(z,y)≤1}, Je1∘(z)={αβ∈E∣{α,β}⊂Ie1∘(z)} for all z∈NBe1⊆X and Ie8∘(z)={y∈X∣zRy⇔d(z,y)≤1}, Je8∘(z)={αβ∈E∣{α,β}⊂Ie8∘(z)} for all z∈NBe8⊆X. Then T2SSs[I∘∗,A] and [J∘∗,A] are as follows:(21)I∘∗e1=e2,e1,e2,e3,e8,e1,e8,e7,J∘∗e1=e2,e1e2,e2e3,e8,e1e8,e8e7,I∘∗e8=e1,e8,e1,e7,e7,e6,e7,e8,J∘∗e8=e1,e2e1,e1e8,e8e7,e7e6,e7,e1e8,e7e8,e6e7,e6e5.Then G∗=(T∘∗(e1),T∘∗(e8)) is a T2ST of G.
Let[I′∗,B] and [J′∗,B] be two T2SSs over X and E, respectively, such that I′∗(x)=(Jx′,NBx) and J′∗(x)=(Jx′,NBx) for all x∈B. Define Ie5′(z)={y∈X∣zRy⇔d(z,y)≥1}, Je5′(z)={αβ∈E∣{α,β}⊂Ie5′(z)} for all z∈NBe5⊆X. Then T2SSs[I′∗,B] and [J′∗,B] are as follows:(22)I′∗e5=e6,e4,e2,e3,e1,e5,e8,e7,e4,e6,e5,e1,e2,e3,e8,e7,J′∗e5=e6,e1e2,e2e3,e3e4,e5e4,e1e8,e7e8,e4,e3e2,e2e1,e7e6,e6e5,e5e4,e1e8,e5e4,e7e8,
ThenG′∗=(T′∗(e5)) is a T2ST of G. The AND operation on G∗ and G′∗ is defined as in the following:(23)I∗e1,e5=I∘∗e1∧~I′∗e5=e2,e6,e2,e3,e1,e2,e4,e1,e3,e2,e8,e6,e1,e8,e7,e8,e4,e1,e8,e7,J∗e1,e5=J∘∗e1∧~J′∗e5=e2,e6,e2e3,e2e1,e2,e4,e1e2,e3e2,e8,e6,e1e8,e8e7,e8,e4,e1e8,e8e7,I∗e8,e5=I∘∗e8∧~I′∗e5=e1,e6,e7,e8,e1,e1,e4,e1,e8,e7,e7,e6,e8,e7,e7,e4,e6,e7,e8,J∗e8,e5=J∘∗e8∧~J′∗e5=e1,e6,e7e8,e1e8,e1,e4,e1e8,e7e8,e7,e6,e8e7,e7,e4,e6e7,e8e7.
TheAND operation on G∗ and G′∗ is shown in Figure 20.Figure 20
AND operation.Definition 50.
LetG1∗=〈I1∗,J1∗,A1〉 and G2∗=〈I2∗,J2∗,A2〉 be two T2STs of G. The OR operation of G1∗ and G2∗, denoted by G1∗⋁G2∗=G∗=〈I∗,J∗,A×B〉 defined by I∗(γ,η)=I1∗(γ)∨~I2∗(η), J∗(γ,η)=J1∗(γ)∨~J2∗(η) for all (γ,η)∈A×B, where I1∗(γ)∨~I2∗(η) for all (γ,η)∈A×B refers to the usual type 1 soft OR operation between the respective T1SS corresponding to I1∗(γ) and I2∗(η) respectively and J1∗(γ)∨~J2∗(η) for all (γ,η)∈A×B refers to the usual type 1 soft OR operation between the respective T1SS corresponding to J1∗(γ) and J2∗(η) respectively.Example 51.
Consider a simple graphG=(X,E) defined in Example 49. Let A={e3,e5}, B={e6}. It may be written that NBe3={e4,e2}, NBe6={e7,e5} and NBe5={e4,e6}.
Let[I∘∗,A] and [J∘∗,A] be two T2SSs over X and E respectively, such that I∘∗(x)=(Ix∘,NBx) and J∘∗(x)=(Jx∘,NBx) for all x∈A. Define Ie3∘(z)={y∈X∣zRy⇔d(z,y)≤1}, Je3∘(z)={αβ∈E∣{α,β}⊂Ie3∘(z)} for all z∈NBe3⊆X, Ie5∘(z)={y∈X∣zRy⇔d(z,y)≥3}, Je5∘(z)={αβ∈E∣{α,β}⊂Ie5∘(z)} for all z∈NBe5⊆X. Then T2SSs[I∘∗,A] and [J∘∗,A] are as follows:(24)I∘∗e3=e2,e1,e2,e3,e4,e3,e4,e5,J∘∗e3=e2,e1e2,e2e3,e4,e3e4,e4e5,I∘∗e5=e4,e8,e1,e7,e6,e1,e2,e3,J∘∗e5=e4,e8e7,e1e8,e6,e1e2,e2e3.Then G∗=(T∘∗(e2),T∘∗(e4)) is a T2ST of G.
Let[I′∗,B] and [J′∗,B] be two T2SSs over X and E, respectively, such that I′∗(x)=(Ix′,NBx) and J′∗(x)=(Jx′,NBx) for all x∈B. Define Ie6′(z)={y∈X∣zRy⇔d(z,y)≤1}, Je6′(z)={αβ∈E∣{α,β}⊂Ie6′(z)} for all z∈NBe6⊆X. Then T2SSs[I′∗,B] and [J′∗,B] are as follows:(25)I′∗e6=e7,e4,e2,e3,e1,e5,e5,e1,e2,e3,e8,e7,J′∗e6=e7,e1e2,e2e3,e3e4,e5e4,e5,e3e2,e2e1,e1e8,e7e8,
ThenG′∗=(T′∗(e6)) is a T2ST of G. The OR operation on G∗ and G′∗ is defined as in the following:(26)I∗e3,e6=I∘∗e3∧~I′∗e6=e2,e7,e4,e2,e3,e1,e5,e2,e5,e7,e2,e3,e1,e8,e4,e7,e4,e2,e3,e1,e5,e4,e5,e4,e2,e3,e1,e5,e7,e8,J∗e3,e6=J∘∗e3∧~J′∗e6=e2,e7,e1e2,e2e3,e3e4,e5e4,e2,e5,e1e2,e2e3,e1e8,e7e8,e4,e7,e1e2,e3e2,e4e3,e4e5,e4,e5,e1e2,e2e3,e3e4,e5e4,e1e8,e8e7,I∗e5,e6=I∘∗e5∧~I′∗e6=e4,e7,e4,e2,e3,e1,e5,e7,e8,e4,e5,e7,e2,e3,e1,e8,e6,e7,e1,e2,e3,e4,e5,e6,e5,e1,e2,e3,e8,e7,J∗e8,e5=J∘∗e8∧~J′∗e5=e4,e7,e7e8,e1e8,e4e5,e3e4,e3e2,e1e2,e4,e5,e1e2,e2e3,e1e8,e8e7,e6,e7,e1e2,e2e3,e3e4,e4e5,e6,e5,e1e2,e2e3,e7e8,e1e8.
TheOR operation on G∗ and G′∗ is shown in Figure 21.Figure 21
OR operation.


## 4. Conclusion

In above study, we have characterized type 2 soft graphs on underlying subgraphs (regular subgraphs, irregular subgraphs, cycles, trees) of a simple graph. We have presented regular type 2 soft graphs, irregular type 2 soft graphs, and type 2 soft trees. Moreover, we have introduced type 2 soft cycles, type 2 soft cut-nodes, and type 2 soft bridges. Finally, we have presented some operations on type 2 soft trees by presenting several examples to demonstrate these new concepts. In future work, we will extend our work in following areas of research:(i)
Applications of type 2 soft graphs in computer networks and social networks(ii)
Fuzzy type 2 soft graphs and their applications.



---

*Source: 8535703-2018-09-20.xml*