Bridge

Definition

Let $G$ be a graph and let $e$ be an edge of $G$. Let $G - e$ be the graph that $G$ gives when $e$ is deleted and all vertices are kept. The edge $e$ is a bridge of $G$ if $G - e$ has more connected components than $G$.

Remarks

Let $e = uv$. Then $e$ is a bridge if and only if $u$ and $v$ are not reachable from each other in $G - e$. In this case, deleting $e$ increases the number of components by exactly one. In a tree every edge is a bridge. In a cycle graph no edge is a bridge. Bridges are the edges on no cycle.

Explanations

L1 human

A bridge is an edge that holds the graph together: delete it, and one piece breaks into two. Think of the only road between two towns.