# 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…

## 5 Citations

### Bounds on the Twin-Width of Product Graphs

- MathematicsArXiv
- 2022

Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomassé & Watrigant. Given two graphs G and H and a graph product ⋆ , we address the question: is the twin-width of G ⋆ H…

### Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)

- MathematicsArXiv
- 2022

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying u and v, each edge incident to exactly one of u and v is…

### Twin-width and types

- MathematicsICALP
- 2022

We study problems connected to first-order logic in graphs of bounded twin-width. Inspired by the approach of Bonnet et al. [FOCS 2020], we introduce a robust methodology of local types and describe…

### Twin-width can be exponential in treewidth

- Computer ScienceArXiv
- 2022

This work builds a graph with a vertex deletion set of size t to a tree, and twin-width greater than 2 (1 − ε ) t, and shows that the twin- width is sometimes exponential in the treewidth, in the so-called oriented twin-Width and grid number.

### A Brief Tour in Twin-Width

- 2022

This is an introduction to the notion of twin-width, with emphasis on how it interacts with first-order model checking and enumerative combinatorics. Even though approximating twin-width remains a…

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