• Corpus ID: 246652451

VC-density and abstract cell decomposition for edge relation in graphs of bounded twin-width

@article{Przybyszewski2022VCdensityAA,
  title={VC-density and abstract cell decomposition for edge relation in graphs of bounded twin-width},
  author={Wojciech Przybyszewski},
  journal={ArXiv},
  year={2022},
  volume={abs/2202.04006}
}
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… 

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References

SHOWING 1-10 OF 25 REFERENCES

Twin-width I: tractable FO model checking

TLDR
It is proved that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph) and unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets.

Regularity lemma for distal structures

It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into

Regularity Partitions and The Topology of Graphons

TLDR
It is proved that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension, which implies that such graphons have regularity partitions of polynomial size.

Discrepancy and approximations for bounded VC-dimension

TLDR
It is shown that if for anym-point subset $$Y \subseteq X$$ the number of distinct subsets induced by ℛ onY is bounded byO(md) for a fixed integerd, then there are improved upper bounds on the size of ε-approximations for (X,ℛ).

Cutting lemma and Zarankiewicz’s problem in distal structures

TLDR
A cutting lemma is established for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definite families of set on the plane in o-minimal expansions of fields, which generalizes the results in Fox et al.

Szemeredi''s Regularity Lemma and its applications in graph theory

Szemer\''edi''s Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps in

On the Density of Families of Sets

  • N. Sauer
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1972