An edge e of G is called a bridge if G-e has more components than G.

NB: This definition implies that if G is connected, then G-e is disconnected.

Lemma:

Let G be a connected graph. For some $x, y \in V(G)$, if $e=\{x, y\}$ is a bridge, then G-e has precisely 2 components; Furthermore, x and y are in different components.

Theorem:

An edge e is a bridge of a graph G iff it is not connected in any cycle of G.

Corollary:

If there are 2 distinct paths from vertex u to vertex v in G, then G contains a cycle.