$*8.$ Let $G$ be a connected graph with $(G)=k1.$ Prove that $G$ contains a path $x_1{x}_2...x_k$ such that $G-\{x_1,x_2,...,x_k\}$ is also connected.

$[$HINT: $($fill in the details of the following.$)$ Let $x_1{x}_2...x_l$ be a longest path. Then $lk+1($why$?)$ Suppose $G-\{x_1,x_2,...,x_k\}$ is disconnected and let $y_0{y}_{1}dots{y}_m$ be a longest path in the component $C$ not containing $x_{k+1}{x}_{k+2}dots{x}_m.$ Then $d_C(y_0)m($why$?)$ but $y_0$ cannot be joined to $k-m$ of the vertices $x_1,x_2,$dots,$x_k($why$?).$ Now finish it$.]$