On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width

Rank-width is a structural graph measure introduced by Oum and Seymour and aimed at better handling of graphs of bounded clique-width. We propose a formal framework and tools for easy design of dynamic algorithms running directly on a rank-decomposition of a graph (on contrary to the usual approach which translates a rankdecomposition into a clique-width expression, with a possible exponential jump in the parameter). Our new approach links to a previous work of Courcelle and Kanté [WG 2007] who first proposed algebraic expressions with a so-called bilinear graph product as a better way of handling rankdecompositions.