# Characterising Bounded Expansion by Neighbourhood Complexity

@article{Reidl2019CharacterisingBE, title={Characterising Bounded Expansion by Neighbourhood Complexity}, author={Felix Reidl and Fernando S{\'a}nchez Villaamil and Konstantinos S. Stavropoulos}, journal={ArXiv}, year={2019}, volume={abs/1603.09532} }

## Figures from this paper

## 26 Citations

Lossy Kernels for Connected Dominating Set on Sparse Graphs

- Mathematics, Computer ScienceSTACS
- 2018

It is shown that even though the kernelization complexity ofDominating Set and Connected Dominating Set diverges on sparse graphs this divergence is not as extreme as kernelization lower bounds suggest.

Algorithmic Properties of Sparse Digraphs

- Mathematics, Computer ScienceSTACS
- 2019

It is shown that the directed Steiner tree problem is fixed-parameter tractable on any class of directed bounded expansion parameterized by the number of non-terminals plus the maximal diameter in the subgraph induced by the terminals, and thereby highlights a rich algorithmic structure theory ofdirected bounded expansion classes.

Lossy kernels for connected distance-$r$ domination on nowhere dense graph classes

- Mathematics, Computer ScienceArXiv
- 2017

It is proved that for every nowhere dense class of graphs, every $\alpha>1$ and $r\in\mathbb{N}$ there exists a polynomial $p$ such that the connected distance-$r$ dominating set problem with parameter $k$ admits an $\alpha$-approximate bi-kernel of size $p(k)$.

Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs

- MathematicsSTACS
- 2017

It is shown that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case and that linear neighbourhood complexity does not characterise directed classes of bounded expansion.

Neighborhood complexity and kernelization for nowhere dense classes of graphs

- Mathematics, Computer ScienceICALP
- 2017

We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with…

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

- Mathematics, Computer ScienceArXiv
- 2022

It is proved 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, and that this fact can apply to such classes combinatorial tools based on the Distal cutting lemma and thedistal regularity lemma.

On the number of types in sparse graphs

- Mathematics, Computer ScienceLICS
- 2018

It is proved that for every class of graphs ℒ which is nowhere dense, and for every first order formula φ(x, y), the number of subsets of A|y| which are of the form ū for some valuation ū of x in G is bounded by O(|A||x|ε), which provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes.

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…

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

- Mathematics, Computer ScienceSEA
- 2018

A new algorithm for uniform quasi-wideness with polynomial size guarantees in graph classes of bounded expansion and a lower bound indicating that the guarantees of this algorithm are close to optimal ingraph classes with fixed excluded minor is shown.

## References

SHOWING 1-10 OF 40 REFERENCES

Kernelization and Sparseness: the case of Dominating Set

- Mathematics, Computer ScienceSTACS
- 2016

The results fall short of proving a sharp dichotomy for the parameterized complexity of r-Dominating Set on subgraph-monotone graph classes, but it is conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.

Characterisations and examples of graph classes with bounded expansion

- MathematicsEur. J. Comb.
- 2012

Deciding first-order properties of nowhere dense graphs

- Mathematics, Computer ScienceSTOC
- 2014

It is shown that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes, and a "rank-preserving" version of Gaifman's locality theorem is proved.

Deciding First-Order Properties for Sparse Graphs

- Mathematics2010 IEEE 51st Annual Symposium on Foundations of Computer Science
- 2010

An almost linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion and a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion for testing existential properties or the existence of short paths between prescribed vertices.

Structure theorem and isomorphism test for graphs with excluded topological subgraphs

- MathematicsSTOC '12
- 2012

It is proved that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices.

Deciding first-order properties of locally tree-decomposable structures

- Mathematics, Computer ScienceJACM
- 2001

It is shown that for each property φ of structures that is definable in first-order logic and for each locally tree-decomposable class C of structures, there is a linear time algorithm deciding whether a given structure A ∈ C hasproperty φ.

Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions

- Mathematics, Computer ScienceICALP
- 2013

It is shown that any parameterized graph problem (with parameter k) that has a finite integer index and such that Yes-instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H.

Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs

- MathematicsSTACS
- 2013

The first linear kernels for the (Connected) Dominating Set problems on H-topological minor free graphs are given and the existence of polynomial time algorithms that produce an equivalent instance of the problem, with treewidth $O(\sqrt{k})$.

Grad and classes with bounded expansion I. Decompositions

- MathematicsEur. J. Comb.
- 2008