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# 数学代写|图论代考GRAPH THEORY代写|MATH913 Fuzzy Tolerance and Fuzzy Tolerance Graph

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## 数学代写|图论代写GRAPH THEORY代考|Fuzzy Tolerance and Fuzzy Tolerance Graph

Let $\mathbb{I}=\left{I_1, I_2, \ldots, I_n\right}$ be a set of closed real intervals and $T=\left{T_1, T_2, \ldots, T_n\right}$ be another set of positive real numbers. For each interval, let us consider a vertex.
A graph $G=(V, E)$ along with two sets $I$ and $T$ is said to be a tolerance graph if, for each $I_j$, there is a vertex $a_j$ in $\mathscr{V}$ and there is an edge in $E$ between two vertices $a_i$ and $a_j$ if and only if
$$\left|I_{a_i} \cap I_{a_j}\right| \geq \min \left{T_{a_i}, T_{a_j}\right},$$
where $\left|I_{a_i}\right|$ is the length of the interval $I_{a_i}$. The pair $I$ and $T$ is denoted by $\langle I, T\rangle$ and is called tolerance representation of the graph $G$. For a given tolerance representation, the corresponding graph is unique. But, the converse is not true, i.e., for a given tolerance graph there exists many different tolerance representations.

A tolerance representation $\langle I, T\rangle$ is said to be bounded if $T_a \leq\left|I_a\right|$ for all $a \in V$. A tolerance graph is called a bounded tolerance graph if it has a bounded tolerance representation.

By imposing the conditions on the length of the tolerances, we obtained two well-known intersection graphs. If all the tolerances $T_x$ are equal and equal to a fixed positive constant $c$, then the tolerance graph becomes interval graph [12, 14]. If $T_a=$ $\left|I_a\right|$ for all $a \in V$, then we obtain the permutation graph $[12,14]$ (or, equivalently, the interval containment graph). Therefore, one can conclude that interval graphs and permutation graphs are all bounded tolerance graphs.

So, there is a set of intervals for each tolerance graph. Therefore, before going to define FTolG, we introduce a fuzzy interval (FInv).
The following is the definition of FInv given in [5].

## 数学代写|图论代写GRAPH THEORY代考|Fuzzy Bounded Tolerance Graph

Let $G$ be a tolerance graph and its tolerance representation be $\langle I, T\rangle$. If $T_j \leq\left|I_j\right|$ for every $j=1,2, \ldots, n$, then $G$ is called a [10] bounded tolerance graph.

Here, we extended the concept of (crisp) bounded tolerance graph into fuzzy bounded tolerance graph as follows.

Definition 6.5 Let $\mathscr{I}=\left{\mathscr{I}_1, \mathscr{I}_2, \ldots, \mathscr{I}_n\right}$ be a finite set of FInvs defined on a real line and let the corresponding fuzzy tolerances be $\mathscr{T}=\left{\mathscr{T}_1, \mathscr{T}_2, \ldots, \mathscr{T}_n\right}$. If FInv $\mathscr{I}_i$ with core $c\left(\mathscr{I}_i\right)$ and support $s\left(\mathscr{I}_i\right)$ has tolerance with core $c\left(\mathscr{T}_i\right)$ and support $s\left(\mathscr{T}_i\right)$ such that $c\left(\mathscr{I}_i\right) \geq c\left(\mathscr{T}_i\right)$ and $s\left(\mathscr{I}_i\right) \geq s\left(\mathscr{T}_i\right)$ of a FTolG, then the corresponding representation is said to be fuzzy bounded tolerance representation and the FG is called fuzzy bounded tolerance graph.

It may be noted that if $\langle\mathscr{I}, \mathscr{T}\rangle$ is a fuzzy bounded tolerance representation then no $\operatorname{FInv} \mathscr{I}_v, v \in \mathscr{V}$ is a fuzzy number.

Theorem 6.1 If $\mathscr{G}$ is a FInvG, then $\mathscr{G}$ is a FTolG with constant core and constant support of tolerance.

Proof Let $\mathscr{G}$ be a FInvG and the FInv for the vertex $a$ is $\mathscr{I}_a$. Let the cores of FInvs $\mathscr{I}_a$ and $\mathscr{I}_b$ be denoted by $c\left(\mathscr{I}_a\right)$ and $c\left(\mathscr{I}_b\right)$ and that of supports be $s\left(\mathscr{I}_a\right)$ and $s\left(\mathscr{I}_b\right)$. Also, $c\left(\mathscr{I}_a \cap \mathscr{I}_b\right)=c\left(\mathscr{I}_a\right) \wedge c\left(\mathscr{I}_b\right)$ and $s\left(\mathscr{I}_a \cap \mathscr{I}_b\right)=s\left(\mathscr{I}_a\right) \wedge s\left(\mathscr{I}_b\right)$. Let $\kappa_1$ and $\kappa_2$ be positive real numbers such that $\kappa_1<\left|c\left(\mathscr{I}_a \cap \mathscr{I}_b\right)\right|$ and $\kappa_2<\left|s\left(\mathscr{I}_a \cap \mathscr{I}_b\right)\right|$ for all $a, b \in \mathscr{V}$ with $\kappa_1 \leq \kappa_2$.

Thus, the intervals $\left{\mathscr{I}_v \mid v \in \mathscr{V}\right}$ together with tolerances with core $k_1$ and support $k_2$ give a FTol representation.

## 数学代写|图论代写GRAPH THEORY代考|Fuzzy Tolerance and Fuzzy Tolerance Graph

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