Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Necessary Condition

Theorem

Let $G = \struct {V, E}$ be a connected simple graph such that:

for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected.

Then $G$ is a tree.

Proof

Let $G$ be a connected simple graph such that for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected.

Hence, by definition, every edge of $G$ must be a bridge.

So by Condition for Edge to be Bridge, $G$ has no circuits.

Hence $G$ is a tree by definition.

$\blacksquare$