Let $G=(V,E)$ be a (simple) finite graph such that every vertex has degree at least 1. Then it is easy to see that there is a subset $E'$ of $E$ such every vertex in $G'=(V,E')$ still has degree at least 1 and all paths (with no repeating edges) in $G'$ are of length at most 2. (I just keep removing middle edges of paths of length 3 until I'm done.) My question is, does this hold for infinite graphs ?