the graph with vertex set $V(G)$ and edge set $E(G) \setminus F$. For any vertex $v \in V(G)$, we denote the set $\{x: \{x, v\} \in E(G)\}$ by $N_G(v)$.

For any two nonadjacent vertices $a$ and $b$ of $G$, an $a$-$b$ $separator$ $U$ of $G$ is a set of vertices whose removal disconnects $G$ and $a$ and $b$ are in different connected components of $G-U$. An $a$-$b$ separator $U$ is *minimal* if for any $U' \subset U$, $U'$ is not an $a$-$b$ separator. A set $U \subseteq V(G)$ is a *minimal separator* if $U$ is a minimal $a$-$b$ separator for some nonadjacent vertices $a$ and $b$ of $G$. Two minimal separators $U$ and $U'$ are *parallel* if $G-U$ contains at most one component $H$ where $V(H) \cap U' \neq \emptyset$. We represent the set of all minimal separators of graph $G$ by $\triangle_G$.

Assume that $G$ is connected. A *cut* is a set of edges $F \subseteq E(G)$ whose removal disconnects $G$. A cut $F$ is *minimal* if there does not exist $F' \subset F$ where $G-F'$ is disconnected. Note that if $F$ is minimal, there will exactly be two connected components in $G-F$. Two minimal cuts $F$ and $F'$ 