Graph complexity measures such as 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 defined in a combinatorial way, based on a tree, and not in terms of graph operations. We define operations on graphs that characterize rank-width and help for the construction of Fixed Parameter Tractable algorithms, especially for problems specified in monadic second-order logic. © 2008 Elsevier B.V. All rights reserved.
Courcelle, B., & Kanté, M. M. (2009). Graph operations characterizing rank-width. Discrete Applied Mathematics, 157(4), 627–640. https://doi.org/10.1016/j.dam.2008.08.026