Graph $G$ is 2-connected. It means that for each two edges there are exists at least to disjont (in terms of edges) paths.
Graph $G$ is not directed.
Our task is to find spanning subgraph $H$ of graph $G=(V,E)$ such that $H = (V, F)$ such that $F=O(|V|)$ and $H$ is 2-connected.
So far I have came up with some idea, but I have not proved it.
Let execute DFS in order to get DFS search tree. Now we will get every edge from this tree - then $H$ is spanning of $G$.
Additionally for each node in this tree we add back edges $e$ such that $e$ jump possibly high.
What about this idea ? Could you give me hint in case of I am wrong ?