Monadic second-order logic and hypergraph orientation

It is proved that in every undirected graph or, more generally, in every undirected hypergraph of bounded rank, one can specify an orientation of the edges or hyperedges by monadic second-order formulas using quantifications on sets of edges or hyperedges. The proof uses an extension to hypergraphs of the classical notion of a depth-first search spanning tree. Applications are given to the partially open problem of characterizing the classes of graphs (or hypergraphs) having decidable monadic theories, with and without quantifications on sets of edges (or hyperedges)