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

Abstract

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

Harary polynomials

“…In the light of Theorem 5 biclique-free is renamed to weakly sparse in [36]. However, clique-free graphs can be rather dense, with c(t) • n 2 edges rather than n 2−ǫ(t) edges.…”

Obstructions for bounded shrub-depth and rank-depth

“…(Please see [5] for the definition of an pa, kq-shrubbery.) Lemma 2.16 of Nešetřil, Ossona de Mendez, Rabinovich, and Siebertz [17] states that every class of bounded shrub-depth can be partitioned into bounded number of vertex-disjoint induced subgraphs, each of which is a cograph. Its (short and easy) proof shows that a graph with an pa, kq-shrubbery can be partitioned into at most a vertex-disjoint induced subgraphs, each of which is a cograph.…”