We consider the class US k 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 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 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. © 2002 Elsevier Science B.V. All rights reserved.
Courcelle, B. (2003). The monadic second-order logic of graphs XIV: Uniformly sparse graphs and edge set quantifications. Theoretical Computer Science, 299(1–3), 1–36. https://doi.org/10.1016/S0304-3975(02)00578-9