I saw stated in a paper the following result but without a reference or a proof.

Let $G$ be an Erdos-Renyi random graph with $n$ nodes and probability of connection $c/n$ with $c>1$. Let $H$ be its giant component (which exists and has $\alpha(c)n$ nodes almost surely where $\alpha$ is a well known function). Then the graph $H$ has paths with only one connection to the rest of the graph of length $O(\log n)$ asymptotically almost surely.

Can somebody show me why is this true or give me a reference? Thanks a lot!