Path-components of $G-v$ do not have neighbours between them i.e if $G_1$ and $G_2$ are any two path-components of $G-v$, there is no edge from a vertex in $G_1$ to a vertex in $G_2$ (this is by definition of path-component). So the only way the original graph would be connected is there is a path from $G_1$ to $G_2$ through $v$.

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GV Connect