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# 数学代写|图论代考GRAPH THEORY代写|Math7410 Partial Intuitionistic Fuzzy Labeling Tree

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## 数学代写|图论代写GRAPH THEORY代考|Partial Intuitionistic Fuzzy Labeling Tree

Partial IFLT is another type of labeling tree.
Definition $9.45$ A connected IFLG $\mathscr{G}$ is said to be a partial IFLT if $\mathscr{G}$ has a spanning subgraph $F$ that is a tree and for every $\operatorname{arc}(p, q)(\notin F)$ of $\mathscr{G}, C O N N_{1 \mathscr{G}}(p, q)>$ $\mu_1(p, q)$ and $C O N N_{2 S}(p, q)<\mu_2(p, q)$.

When a graph $\mathscr{G}$ is disconnected and the above condition holds for all components of $\mathscr{G}$, then $\mathscr{G}$ is called a partial fuzzy forest.
Following is a characterization of partial IFLT.
Theorem $9.33$ A connected IFLG $\mathscr{G}$ is a partial IFLT if and only if for any cycle $\mathscr{C}$ in $\mathscr{G}$, there is an arc $e=(p, q)$ with $\mu_1(e)$ CONN $\mathrm{N}_{2(\mathscr{S}-e)}(p, q)$.

Proof Suppose $\mathscr{G}$ is a connected IFLG. If there is no cycle, then $\mathscr{G}$ is obviously a tree and also a partial fuzzy tree. If there is cycle in $\mathscr{G}$, let $(p, q)$ be an arc of $\mathscr{C}$ with the least membership and greatest non-membership values in $\mathscr{G}$. Remove the arc $(p, q)$ from $\mathscr{G}$. If $\mathscr{G}$ has another cycle, repeat the process. Not at each step no previously removed arc is strongest the arc being presently deleted. When the graph $\mathscr{G}$ does not contain any cycle, then the subgraph is a tree $F$. Suppose the $\operatorname{arc}(p, q)$ is not in $F$. Then $(p, q)$ is one of the arcs removed in the process to construct $F$. Since $F$ is a tree and $(p, q)$ is the arc having least membership and greatest non-membership values among all the arcs of a cycle in $\mathscr{G}$, it follows that there is a path between $a$ and $b$ whose membership value is greater than $\mu_1(p, q)$ and non-membership value is less than $\mu_2(p, q)$ and this does not cover $(p, q)$ or any other edges removed before. If that path covers arcs that were removed later, then the path can be further detached and so on. This process gives a path belonging entirely the arcs of $F$. Thus $\mathscr{G}$ is a partial IFLT.

## 数学代写|图论代写GRAPH THEORY代考|Partial Intuitionistic Fuzzy Labeling Tree

Partial IFLT is another type of labeling tree.
Definition $9.45$ A connected IFLG $\mathscr{G}$ is said to be a partial IFLT if $\mathscr{G}$ has a spanning subgraph $F$ that is a tree and for every $\operatorname{arc}(p, q)(\notin F)$ of $\mathscr{G}, C O N N_{1 \mathscr{G}}(p, q)>$ $\mu_1(p, q)$ and $C O N N_{2 \mathscr{G}}(p, q)<\mu_2(p, q)$.

When a graph $\mathscr{G}$ is disconnected and the above condition holds for all components of $\mathscr{G}$, then $\mathscr{G}$ is called a partial fuzzy forest.
Following is a characterization of partial IFLT.
Theorem $9.33$ A connected IFLG $\mathscr{G}$ is a partial IFLT if and only if for any cycle $\mathscr{C}$ in $\mathscr{G}$, there is an arc $e=(p, q)$ with $\mu_1(e)$ $C O N N_{2(\mathscr{S}-e)}(p, q)$.

Proof Suppose $\mathscr{G}$ is a connected IFLG. If there is no cycle, then $\mathscr{G}$ is obviously a tree and also a partial fuzzy tree. If there is cycle in $\mathscr{G}$, let $(p, q)$ be an arc of $\mathscr{C}$ with the least membership and greatest non-membership values in $\mathscr{G}$. Remove the arc $(p, q)$ from $\mathscr{G}$. If $\mathscr{G}$ has another cycle, repeat the process. Not at each step no previously removed arc is strongest the arc being presently deleted. When the graph $\mathscr{G}$ does not contain any cycle, then the subgraph is a tree $F$. Suppose the arc $(p, q)$ is not in $F$. Then $(p, q)$ is one of the arcs removed in the process to construct $F$. Since $F$ is a tree and $(p, q)$ is the arc having least membership and greatest non-membership values among all the arcs of a cycle in $\mathscr{G}$, it follows that there is a path between $a$ and $b$ whose membership value is greater than $\mu_1(p, q)$ and non-membership value is less than $\mu_2(p, q)$ and this does not cover $(p, q)$ or any other edges removed before. If that path covers arcs that were removed later, then the path can be further detached and so on. This process gives a path belonging entirely the arcs of $F$. Thus $\mathscr{G}$ is a partial IFLT.

Conversely, suppose $\mathscr{G}$ is a partial IFLT and $\mathscr{P}$ is cycle, then for some arc $e=$ $(p, q)$ of $\mathscr{P}$ does not lie on $F$. Thus, by definition, $\mu_1(e)C O N N_{2(\mathscr{G}-e)}(p, q)>C O N N_{2 \mathscr{}}(p, q)$

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