For a vertex $u$ in a graph $G$, let $N_G(u) = \{v \in V (G)|\{u, v\} \in E(G)\}$. For $U \subseteq V(G)$, define $G\setminus U$ to be the induced subgraph of $G$ on the vertex set $V (G) \setminus U$.
Let $G$ be a graph. A subset $C \subseteq V(G)$ is a vertex cover of $G$ if for each $e \in E(G)$, $e\cap C \neq \phi$. If $C$ is minimal with respect to inclusion, then $C$ is called minimal vertex cover of $G$, denoted by $\textbf{m}(G)$, is the maximum cardinality of minimal vertex cover of $G$.
What is the relation between $\textbf{m}(G)$ and $\textbf{m}(G \setminus N_G(x))$, for some $x \in V(G)$? In particular, suppose $\deg_G(x)=1$, what is the relation between $\textbf{m}(G)$ and $\textbf{m}(G \setminus N_G(x))$?