Classes of graphs with low complexity: The case of classes with bounded linear rankwidth

Classes with bounded rankwidth are MSO-transductions of trees and classes linear paths – a result that shows strong link between the properties these graph considered from point view structural theory finite model theory. We take both views on prove theoretic classes. The results we obtain following. (1) number unlabeled graphs order n rank-width at most r is [(2r+1)(r+1)!2r23r+1]n (2) Graphs linearly χ-bounded. Actually, they have c-chromatic number, meaning can be colored f(r) colors, each color inducing cograph. (3) To contrary, based Ramsey-like argument, for every proper hereditary family ℱ (like cographs) there class does not property in it by subgraph ℱ. From theoretical side following results: A direct short proof first-order transductions orders. This could also derived Colcombet’s theorem transduction orders equivalence cliquewidth. For C conditions equivalent: (a) stable, (b) excludes some half-graph as semi-induced subgraph, (c) pathwidth. These open perspective to study admitting low covers.