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We consider the class US k of uniformly k-sparse simple graphs, i.e., the class of ÿnite or countable simple graphs, every ÿnite subgraph of which has a number of edges bounded by k times the number of vertices. We prove that for each k, every monadic second-order formula (intended to express a graph property) that uses variables denoting sets of edges can be e ectively translated into a monadic second-order formula where all set variables denote sets of vertices and that expresses the samedoi:10.1016/s0304-3975(02)00578-9 fatcat:4aozazrqarbvpmald6ifup3mxm