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Nowhere dense graph classes, stability, and the independence property [article]

Hans Adler, Isolde Adler
2010 arXiv   pre-print
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

Hans Adler, Isolde Adler
2014 European journal of combinatorics (Print)  
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

Grzegorz Fabianski, Michal Pilipczuk, Sebastian Siebertz, Szymon Torunczyk, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science  
, 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]

Jan Dreier, Nikolas Mählmann, Amer E. Mouawad, Sebastian Siebertz, Alexandre Vigny
2022 arXiv   pre-print
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]

Grzegorz Fabiański
2019 arXiv   pre-print
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]

Michał Pilipczuk, Sebastian Siebertz, Szymon Toruńczyk
2017 arXiv   pre-print
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

Jaroslav Nešetřil, Patrice Ossona de Mendez
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
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]

Michał Pilipczuk, Sebastian Siebertz
2020 arXiv   pre-print
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]

Grzegorz Fabiański and Michał Pilipczuk and Sebastian Siebertz and Szymon Toruńczyk
2018 arXiv   pre-print
, 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]

Stephan Kreutzer, Roman Rabinovich, Sebastian Siebertz
2018 arXiv   pre-print
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

Jaroslav Nešetřil, Patrice Ossona de Mendez
2011 European journal of combinatorics (Print)  
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]

Kord Eickmeyer and Archontia C. Giannopoulou and Stephan Kreutzer and O-joung Kwon and Michał Pilipczuk and Roman Rabinovich and Sebastian Siebertz
2016 arXiv   pre-print
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

Jakub Gajarský, Michał Pilipczuk, Szymon Toruńczyk
2022 Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science  
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

Petr Hliněný, Filip Pokrývka, Bodhayan Roy
2019 Computational geometry  
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]

Jaroslav NeŠetřil, Patrice Ossona De Mendez
2008 Bolyai Society Mathematical Studies  
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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