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 same

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... erty of the graphs in US k . This result extends to the class of uniformly k-sparse simple hypergraphs of rank at most m (for any k and m). It follows that every subclass of US k consisting of ÿnite graphs of bounded clique-width has bounded tree-width. Clique-width is a graph complexity measure similar to tree-width and relevant to the construction of polynomial algorithms for NP-complete problems on special classes of graphs.