VC-density and abstract cell decomposition for edge relation in graphs of bounded twin-width
share
We study set systems formed by neighborhoods in graphs of bounded twin-width. In particular, we prove that such classes of graphs admit linear neighborhood complexity, in analogy to previous results concerning classes with bounded expansion and classes of bounded clique-width. Additionally, we show how, for a given graph from a class of graphs of bounded twin-width, to efficiently encode the neighborhood of a vertex in a given set of vertices A of the graph. For the encoding we use only a constant number of vertices from A. The obtained encoding can be decoded using FO formulas. This proves that the edge relation in graphs of bounded twin-width, seen as first-order structures, admits a definable distal cell decomposition. From this fact we derive that we can apply to such classes combinatorial tools based on the Distal cutting lemma and the Distal regularity lemma.
READ FULL TEXT
