Graph operations characterizing rank-width

Graph complexity measures like tree-width, clique-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. These algorithms are based on hierarchical decompositions of the considered graphs and on boundedness conditions on the graph operations that, more or less explicitly, recombine the components of decompositions into larger graphs. Rank-width is de ned in a combinatorial way, based on a tree, and not in terms of graph operations. We de ne operations on graphs that characterize rank-width and help for the construction of Fixed Parameter Tractable algorithms, especially for problems speci ed in monadic second-order logic.