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On the problem of finding small subdivision and homomorphism bases for classes of countable graphs

Reinhard Diestel
1985 Discrete Mathematics  
We consider the following problem: for which r is % well characterized by the simplicial decompositions of its members into prime graphs, that is for which 3 is it possible to find a small subset D of  ...  Let % be a class of countable graphs given by a set r of forbidden configurations.  ...  (TX)(%(HX)) has an uncountable base then so does %(TX') (%(HX')), for every X' 1 X with V(X') = V(X).  ... 
doi:10.1016/s0012-365x(85)80018-2 fatcat:2y3rywagozaddibcxc4xjlll5u

Simplicial decompositions of graphs: a survey of applications

Reinhard Diestel
1989 Discrete Mathematics  
the prism", all wheels" K" K"" the prism", all wheels"  ...  The third remark concerns small elements of the homomorphism base B(HQ.  ...  The applications are based on two theorems: one, due to Diestel, Halin and Vogler, which relates homomorphism and subdivision bases (see Section 1) to universal graphs, and another, due to Halin, which  ... 
doi:10.1016/0012-365x(89)90084-8 fatcat:n7g4dykpxnahzogrtopyf2nitq

Page 4981 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
{For the entire collection see MR 85k:05005. } Diestel, Reinhard (4-CAMB) 86k:05109 On the problem of finding small subdivision and homomorphism bases for classes of countable graphs.  ...  A class of prime graphs is called the subdivision (homomorphism) base of § if it is the class of prime graphs occurring in the simplicial decompositions of the  ... 

Towards a Characterization of Universal Categories [article]

J. Nesetril, P. Ossona de Mendez
2016 arXiv   pre-print
The proof of the characterization is based on the sparse--dense dichotomy and its model theoretic equivalent.  ...  In this note we characterize, within the framework of the theory of finite set, those categories of graphs that are algebraic universal in the sense that every concrete category embeds in them.  ...  On sees easily that the class C 1 of all such graphs G i j * ( C 2Ni , a, b) has bounded expansion: for any fixed integer d and any graph H, if the d-th subdivision of H is a subgraph of a graph in C 1  ... 
arXiv:1608.01112v1 fatcat:2rq6yp3aizfm7lc6kizxsus7hm

A surprising permanence of old motivations (a not-so-rigid story)

Jaroslav Nešetřil
2009 Discrete Mathematics  
Rather it is a personal statement written for a lifelong friend and collaborator. Still it is an ambition of this article to trace some of the key moments of our development in the past 40 years.  ...  In doing so perhaps some evidence has arisen which otherwise seems to be obscured by the hectic day-to-day academic life. Thus the title. * *  ...  Is then one of the graphs finite? This is formulated in [127] where it is proved that K 1 ,K 2 and K ω are the only maximal antichains of size 1 for the homomorphism order of countable graphs.  ... 
doi:10.1016/j.disc.2008.04.055 fatcat:pmcsmwfxovctleuckjaztb2vgq

Sparse Combinatorial Structures: Classification and Applications

Jaroslav Nešetřil, Patrice Ossona de Mendez
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
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.  ...  Finally we characterize the existence of all restricted dualities in terms of limit objects defined on the homomorphism order of graphs.  ...  (Universality of the homomorphism order) For every countable partial order P there is an embedding of P into (Graph, ≤).  ... 
doi:10.1142/9789814324359_0156 fatcat:io4dnuj4wzarvest64wnvvm4ge

Decomposing infinite graphs

Reinhard Diestel
1991 Discrete Mathematics  
It is intended for the non-specialist, and particular prominence is given to the presentation of open problems.  ...  ., Decomposing infinite graphs, Discrete Mathematics 95 (1991) 69-89. This paper gives an introduction to the theory of simplicial and related decompositions of graphs as developed in [I].  ...  countable, like the homomorphism base of K,) and uncountable bases (like that of Kg, which contains all the-uncountably many-countable maximally planar graphs).  ... 
doi:10.1016/0012-365x(91)90330-5 fatcat:kek3uw2ovbe6vkhjuxggjjr3yy

On Low Tree-Depth Decompositions [article]

Jaroslav Nesetril
2014 arXiv   pre-print
In more technical terms we survey some of the properties and applications of low tree depth decomposition of graphs.  ...  The theory of sparse structures usually uses tree like structures as building blocks. In the context of sparse/dense dichotomy this role is played by graphs with bounded tree depth.  ...  More generally, we are interested in the following problem: given a class of graphs C and a connected graph F , find a graph D C (F ) for C (which we shall refer to as a dual of F for C), such that F D  ... 
arXiv:1412.1581v1 fatcat:pqx6dojaxbevlbl5pr7v3tl55u

ASPECTS OF STRUCTURAL COMBINATORICS (Graph Homomorphisms and Their Use)

Jaroslav Nesetril
1999 Taiwanese journal of mathematics  
Due to space limitations we concentrate on a sample of areas only: representation of algebraic structures by combinatorial ones (graphs), the poset of colour classes and corresponding algorithmic questions  ...  This paper is based on a course delivered by the author at NCTS, National Chiao Tung University, Taiwan in Febuary 1999. We survey results related to structural aspects of graph homomorphism.  ...  A The Density Problem for a class K is the problem of describing all gaps of the class K. This is a challenging problem even in the simplest case of all undirected graphs.  ... 
doi:10.11650/twjm/1500407157 fatcat:5y566g6furgbdgbsx2lhmxueia

Groups of type FP via graphical small cancellation [article]

Thomas Brown, Ian J Leary
2022 arXiv   pre-print
In contrast to every previous construction of non-finitely presented groups of type FP we do not use Morse theory on cubical complexes; instead we use Gromov's graphical small cancellation theory.  ...  We construct an uncountable family of groups of type FP.  ...  Let Y = Y P be the presentation 2-complex for the small cancellation presentation for H P , and let f : Y → Y be the based cellular map that induces φ : H P → H P on fundamental groups.  ... 
arXiv:2004.04550v3 fatcat:jnjocv7fkbfjrgaoo4zyvvnruq

Structural Properties of Sparse Graphs [chapter]

Jaroslav NeŠetřil, Patrice Ossona De Mendez
2008 Bolyai Society Mathematical Studies  
The closest to our topic covered in this paper is the recent development which is based on the study of homomorphisms of graphs (and structures).  ...  For instance, consider a surface Σ and let C Σ be the class of the graphs which embed on Σ.  ... 
doi:10.1007/978-3-540-85221-6_13 fatcat:j7ifp6snovdybopviit5zaya2e

Structural Properties of Sparse Graphs

Jaroslav Nešetřil
2008 Electronic Notes in Discrete Mathematics  
The closest to our topic covered in this paper is the recent development which is based on the study of homomorphisms of graphs (and structures).  ...  For instance, consider a surface Σ and let C Σ be the class of the graphs which embed on Σ.  ... 
doi:10.1016/j.endm.2008.06.050 fatcat:hk5istkcqzftjnfm7kqgsia7sm

Exact and Approximate Pattern Counting in Degenerate Graphs: New Algorithms, Hardness Results, and Complexity Dichotomies [article]

Marco Bressan, Marc Roth
2021 arXiv   pre-print
We study the problems of counting the homomorphisms, counting the copies, and counting the induced copies of a k-vertex graph H in a d-degenerate n-vertex graph G.  ...  Those lower bounds for exact counting are complemented with new algorithms for approximate counting of subgraphs and induced subgraphs in degenerate graphs.  ...  Roth is grateful to Lior Gishboliner for introducing him to pattern counting in degenerate graphs.  ... 
arXiv:2103.05588v2 fatcat:pqq53dd72rdjzpc4unymf3igki

Finite dualities and map-critical graphs on a fixed surface

Jaroslav Nešetřil, Yared Nigussie
2012 Journal of combinatorial theory. Series B (Print)  
of Thomassen's theorem [15] for 5-colorable graphs on the torus.  ...  We prove that the class of planar graphs has no finite duality except for two trivial cases.  ...  Acknowledgment We thank Carsten Thomassen for personal communication.  ... 
doi:10.1016/j.jctb.2011.06.001 fatcat:qxftndmv5jclthwoccyeaaoyv4

First order properties on nowhere dense structures

Jaroslav Nešetřil, Patrice Ossona de Mendez
2010 Journal of Symbolic Logic (JSL)  
It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for  ...  This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs.  ...  We find it useful to study wide, almost wide and quasi-wide classes by means of the following functions Φ C and Φ C defined for classes of graphs.  ... 
doi:10.2178/jsl/1278682204 fatcat:fairowkzp5hu5iichwc374j2j4
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