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

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

If in a tolerance graph the lengths of all intervals are same (unit), then the tolerance graph is called unit tolerance graph [1]. Similarly, a tolerance graph is called a proper tolerance graph [1] if no interval is properly contains in another.

Now, we define fuzzy unit interval graph and fuzzy proper interval graph then that in tolerance representation.

Definition 6.7 A FInvG $\mathscr{G}$ with a fuzzy interval representation is called fuzzy unit interval graph if the lengths of cores and supports are same for all FInvs. Similarly, $\mathscr{G}$ is called fuzzy proper interval graph if cores and supports of FInvs do not properly contain the cores and supports of other FInvs.

Definition 6.8 A FTolG with a tolerance representation is called a fuzzy unit tolerance graph if lengths of cores and length of supports of all FInvs are same.
A FTolG with a tolerance representation is called a fuzzy proper tolerance graph if the core and support of a FInv do not contain the core and support of another FInv.
Clearly, the class of fuzzy unit tolerance graph is a subset of the class of fuzzy proper tolerance graph. In the following, we consider a FG and its fuzzy unit tolerance representation.

## 数学代写|图论代写GRAPH THEORY代考|Fuzzy φ-Tolerance Competition Graph

Fuzzy $\phi$-tolerance competition graph is a combination of FTolG and competition graph. First of all, fuzzy $\phi$-TolComG is defined below. The following results are from [17].

Definition 6.10 (Fuzzy $\phi$-TolCom $G$ ) Let $\phi$ be a given function defined by $\phi$ : $\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$. The fuzzy $\phi$-TolComG $T C_\phi(\overrightarrow{\mathscr{D}})$ defined for a fuzzy digraph $\overrightarrow{\mathscr{D}}=$ $(\mathscr{V}, \sigma, \vec{\mu})$ is an undirected $\mathrm{FG} T C_\phi(\overrightarrow{\mathscr{D}})=\left(\mathscr{V}, \sigma, \mu^{\prime}\right)$ which has the same fuzzy vertex set as in $\overrightarrow{\mathscr{D}}$ and

The notion of edge clique covering [18] is used to characterize the competition graph. A collection of cliques of a graph $G$ is called an edge clique cover (ECC) if every edge of $G$ is in at least one of these cliques. The smallest number of cliques that cover all edges of a graph $G$ is called the ECC number of $G$ and is denoted by $\theta_e(G)$

These concepts have been generalized in $[8,9]$ and lead to another type of ECC called $p$-edge clique cover. A $p$-edge clique cover $(p$-ECC) of a graph $G$ is the family of sets $\left{S_1, S_2, \ldots, S_k\right}$ such that $S_{i_1} \cap S_{i_2} \cap \cdots \cap S_{i_p}$ (the subscripts must be distinct) is either empty or induces a clique of $G$ and these $p$ sets form an ECC of $G$. The $p$-edge clique cover number is the smallest $p$ for which there is a $p$-ECC, and is denoted by $\theta_e^p(G)$.

Again, Brigham et al. [2] extended this definition. Let $\phi$ be a symmetric function such that $\phi: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$, and $T=\left(t_1, t_2, \ldots, t_n\right)$ be an $n$-tuple of non-negative integers (not necessarily distinct). For a graph $G=(V, E), V=\left{a_1, a_2, \ldots, a_n\right}$ a $\phi$ – $T$-edge clique cover ( $\phi$-T-ECC) is a collection $S_1, S_2, \ldots, S_k$ of subsets of $V$ such that $\left(a_i, a_j\right) \in E$ if and only if at least $\phi\left(t_i, t_j\right)$ of the sets $S_i,(i=1,2, \ldots, k)$ contain both $a_i$ and $a_j$. The size of the smallest $\phi$-T -ECC of $G$ taken over all vectors $T$ is called the $\phi$-T-edge clique cover number of $G$ and it is denoted $\theta_\phi(G)$.

## 数学代写图论代写GRAPH THEORY代考|Fuzzy $\varphi$-Tolerance Competition Graph

$\backslash$ left 缺少或无法识别的分隔符 这样 $S_{i_1} \cap S_{i_2} \cap \cdots \cap S_{i_p}$ (下标必须不同) 是空的或归纳出一个团 $G$ 还有这 些 $p$ 集形成一个ECCG. 这 $p-$ 边团票盖数最小 $p$ 为此有一个 $p-\mathrm{ECC}$ ，记为 $\theta_e^p(G)$.

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