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Nowhere dense graph classes, stability, and the independence property
[article]
2010
arXiv
pre-print
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For subgraph-closed classes of graphs we prove equivalence to stability and to not having the independence property. ...
A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of cliques that occur as (topological) r-minors. ...
For subgraph-closed classes of graphs we show that nowhere density is equivalent to stability and to dependence (not having the independence property). ...
arXiv:1011.4016v1
fatcat:ugcfw2bh4rfuxkeabtvjcmpfee
Interpreting nowhere dense graph classes as a classical notion of model theory
2014
European journal of combinatorics (Print)
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For subgraph-closed classes of graphs we prove equivalence to stability and to not having the independence property. ...
Expressed in terms of PAC learning, the concept classes definable in first-order logic in a subgraph-closed graph class have bounded sample complexity, if and only if the class is nowhere dense. ...
All these graph classes are nowhere dense. Dawar and Kreutzer [4] gave efficient algorithms for domination problems on classes of nowhere dense graphs. ...
doi:10.1016/j.ejc.2013.06.048
fatcat:cmgyymeggvdxfkkhh5lhy5ugaa
Progressive Algorithms for Domination and Independence
2019
Symposium on Theoretical Aspects of Computer Science
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, such as powers of nowhere dense classes, map graphs, and (for r = 1) biclique-free graphs. ...
Similarly, for the Distance-r Independent Set problem the technique can be used to give a linear-time fixed-parameter algorithm on any nowhere dense class. ...
Acknowledgements We thank Ehud Hrushovski for pointing us to the nfcp property. ...
doi:10.4230/lipics.stacs.2019.27
dblp:conf/stacs/FabianskiPST19
fatcat:wutum5aw6bho3clozgv7nl4hxy
Combinatorial and Algorithmic Aspects of Monadic Stability
[article]
2022
arXiv
pre-print
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for the independent set and dominating set problems in powers of nowhere dense classes. ...
Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. ...
Therefore the first property follows trivially from the monadic stability of nowhere dense classes. The second property was proved by Fabiański et al. in [12, Theorem 11] . ...
arXiv:2206.14509v1
fatcat:rqolnlljxbgddh7yhuwauqdg3q
Properties of nowhere dense graph classes related to independent set problem
[article]
2019
arXiv
pre-print
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In this work we present an algorithm for this problem, which applied to a graph from any fixed nowhere dense class, works in time bounded by f(k, r)*G, for some function f. ...
In the r-independent set problem with parameter k, we ask whether in a given graph G there exists an r-independent set of size k. ...
Classes of nowhere dense graphs were introduced by Nešetřil and Ossona de Mendez in [8] and generalize many known families of sparse graphs, for example the following classes are nowhere dense: graphs ...
arXiv:1912.00948v1
fatcat:eio6w3a2pjgrvc2lvbknw3pama
On the number of types in sparse graphs
[article]
2017
arXiv
pre-print
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Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. ...
This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. ...
We would like to thank Patrice Ossona de Mendez for pointing us to the question of studying VC-density of nowhere dense graph classes. ...
arXiv:1705.09336v2
fatcat:vhxfnivt25ed5gm2hajts7cgxe
Sparse Combinatorial Structures: Classification and Applications
2011
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
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All these problems can be studied under the umbrella of classes of structures which are Nowhere Dense and in the context of Nowhere Dense -Somewhere Dense dichotomy. ...
The topics include: complexity of subgraph-and homomorphism-problems; model checking problems for first order formulas in special classes; property testing in sparse classes of structures. ...
stabilizes to Graph. ...
doi:10.1142/9789814324359_0156
fatcat:io4dnuj4wzarvest64wnvvm4ge
Kernelization and approximation of distance-r independent sets on nowhere dense graphs
[article]
2020
arXiv
pre-print
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We study the duality between the maximum size of a distance-2r independent set and the minimum size of a distance-r dominating set in nowhere dense graph classes, as well as the kernelization complexity ...
Specifically, we prove that the distance-r independent set problem admits an almost linear kernel on every nowhere dense graph class. ...
Let C be a nowhere dense class of graphs. ...
arXiv:1809.05675v2
fatcat:whh5ad7d5fga5ao4rh27m6b62i
Progressive Algorithms for Domination and Independence
[article]
2018
arXiv
pre-print
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, such as powers of nowhere dense classes, map graphs, and (for r=1) biclique-free graphs. ...
Similarly, for the distance-r independent set problem the technique can be used to give a linear-time fixed-parameter algorithm on any nowhere dense class. ...
The cases when C is a power of a nowhere dense class and when C is the class of map graphs follow as simple corollaries from the result for nowhere dense classes. ...
arXiv:1811.06799v1
fatcat:xj4ob4ihlrg6nepv3dhmce4tui
Polynomial Kernels and Wideness Properties of Nowhere Dense Graph Classes
[article]
2018
arXiv
pre-print
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Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. ...
in the parameter only (independently of the input graph size). ...
nowhere dense classes were recently established in [11] . ...
arXiv:1608.05637v4
fatcat:bodc5jmb6jdr7ijtizt7gma2xu
On nowhere dense graphs
2011
European journal of combinatorics (Print)
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In this paper, we define and analyze the nowhere dense classes of graphs. ...
In this paper, we show that this concept leads to a classification of general classes of graphs and to the dichotomy between nowhere dense and somewhere dense classes. ...
Acknowledgements This work was supported by grant 1M0021620808 of the Czech Ministry of Education and AEOLUS. ...
doi:10.1016/j.ejc.2011.01.006
fatcat:m4xrtlurfrev7divuibohk4fsu
Neighborhood complexity and kernelization for nowhere dense classes of graphs
[article]
2016
arXiv
pre-print
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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 r-neighborhoods ...
nowhere denseness and somewhere denseness. ...
An important step towards proving Conjecture 1 was made recently by a subset of the authors [21] , who gave a polynomial kernel for Distance-r Dominating Set on any nowhere dense class C. ...
arXiv:1612.08197v1
fatcat:qv32qhkhgnhcnlq3sbobpm57jq
Stable graphs of bounded twin-width
2022
Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science
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We prove that every class of graphs C that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. ...
This generalizes analogous results for classes of bounded linear cliquewidth [Nešetřil et al. 2021b ] and of bounded cliquewidth [Nešetřil et al. 2021a ]. ...
ACKNOWLEDGMENTS This paper is a part of projects LIPA (JG) and BOBR (MP, SzT) that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation ...
doi:10.1145/3531130.3533356
fatcat:odelyxerdbd5vb7onkwtkar5wm
FO model checking on geometric graphs
2019
Computational geometry
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for FO model checking of nowhere dense classes of graphs. ...
We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). ...
This is not a random coincidence, as it is known [1] that on monotone graph classes stability coincides with nowhere dense (which is the most general characterization allowing for FPT FO model checking ...
doi:10.1016/j.comgeo.2018.10.001
fatcat:4reekwyaqfez7idoh34lgyksoe
Structural Properties of Sparse Graphs
[chapter]
2008
Bolyai Society Mathematical Studies
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The dense graphs display a remarkable stability (and many of their properties do not change by deletions and additions of a small proportion of all edges, see e.g. ...
In [87] we characterized these classes and showed how they relate to the classes of nowhere dense graphs. ...
doi:10.1007/978-3-540-85221-6_13
fatcat:j7ifp6snovdybopviit5zaya2e
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