Suppose that $v$ is a vertex of degree $1$ in a connected graph $G$ and that $e$ is the edge incident on $v$. Let $G′$ be the sub- graph of G obtained by removing $v$ and $e$ from $G$. Must $G′$ be connected? Why?
If I understand correctly I think G′ will be disconnected. The reason is that for graph to be connected each pair of vertexes must be connected. By removing one vertex and edge the node will be disconnected.
Is this correct?