O(N)

In a bridge tree, every vertex corresponds to a 2-connected component, and edges correspond to bridges. If edge $$$(a, b)$$$ is on the path $$$u, v$$$ in the bridge tree, then the edge $$$(a, b)$$$ is a bridge that separates $$$u$$$ and $$$v$$$ in the original graph.

Block-cut trees are kind of similar, with cut vertices instead of bridges. It has the property that if vertex $$$x$$$ is on the path $$$u, v$$$ in the block-cut tree, then it is a cut vertex that separates $$$u$$$ and $$$v$$$ in the original graph.