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

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Theorem

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

Then $G$ is a tree if and only if:

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

Proof

Sufficient Condition

Let $G$ be a tree.

Then by definition $G$ has no circuits.

From Condition for Edge to be Bridge, every edge of $G$ is a bridge.

Thus by definition of bridge, removing any edge of $G$ will disconnect $G$.

$\Box$

Necessary Condition

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.

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.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 2.3.4.1$: Free Trees: Theorem $\mathrm A$