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?
My solution:
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?
Best Answer
If the vertex $v$ has degree $1$, then $v$ is incident to only one edge, namely $e$. This means that $v$ is connected to the rest of $G$ by way of only one edge, and $v$ will only have one other vertex connected to it. So removing $v$ and its only incident edge $e$ from connected graph $G$ means what?