A Borel graph is a simplicial graph on a standard Borel space X such that the edge set is a Borel subset of X^2. This has been studied in the context of countable Borel equivalence relations, and recently there are many attempts to apply the ideas of geometric group theory. Stallings-Swan's theorem states that groups of cohomological dimension 1 are free groups. We will talk about an analog of this theorem for Borel graphs: A Borel graph on X with uniformly bounded degrees of cohomological dimension 1 is Lipschitz equivalent to a Borel acyclic graph on X. This is proved by establishing a criterion for certain decomposition of Borel graphs, which is inspired by Dunwoody's work on accessibility of groups.