The monadic second-order logic of graphs XIV: uniformly sparse graphs and edge set quantifications

We consider the class USk of uniformly k-sparse simple graphs, i.e., the class of finite or countable simple graphs, every finite 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 effectively translated into a monadic second-order formula where all set variables denote sets of vertices and that expresses the same property of the graphs in USk. 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 USk consisting of finite 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.