Connected subgraphs

Let $C = s_1 \text{--} \dots \text{--} s_k$ be a path of maximal length that never visits the same vertex twice.
Since $G$ is connected and has at least 2 vertices, we have $s_1 \neq s_k$.
Let us show that $G \setminus \{s_1\}$ is connected, which by an analogous reasoning will imply that $G \setminus \{s_k\}$ is also connected.

Let $s$ and $t$ be vertices in $G \setminus \{s_1\}$, and let $s \text{--} \dots \text{--} t$ be a path connecting them in $G$.

• If the path does not pass through $s_1$, then it still exists in $G \setminus \{s_1\}$.
• If the path passes through $s_1$, then each passage through $s_1$ is of the form $s_i \text{--} s_1 \text{--} s_j$ (with $i, j \neq 1$). Indeed, by maximality of $C$, all neighbors of $s_1$ are in $C$. The path can then be modified to avoid $s_1$ by following $C$ to go from $s_i$ to $s_j$, which yields a path in $G \setminus \{s_1\}$ connecting $s$ and $t$.