# Prove that if $G$ connected and $G$ becomes disconnected after removing any edge, then $G$ is a tree .

Given a graph $G = (V,E)$, prove that if $G$ connected and $G$ becomes disconnected after removing any edge, then $G$ is a tree .

For this proof I would only like to ask on the contrapositive proof given by my Professor. So basically I am proving if $G$ is not a tree, then $G$ is either not connected or $G$ is still connected after removing any edge.

So if $G$ is not a tree, then he said if $G$ has a cycle , removing an edge in the cycle still makes the graph connected. This i completely understand, but the questions says any edge, so what if $G$ is connected but we remove an edge not from the cycle? Does it not make the $G$ disconnected still...? Am I missing something?

If we rewrite the original assertion as 'if for all edges, removing them from $G$ gives a connected graph, then $G$ is a tree', from here it is clear that the contrapositive is: 'if $G$ is not a tree, there exists an edge for which $G$ will remain connected when removing it'. So you can choose which edge to remove, as long as it makes $G$ to remain connected.