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

1985
*
Discrete Mathematics
*

We consider

doi:10.1016/s0012-365x(85)80018-2
fatcat:2y3rywagozaddibcxc4xjlll5u
*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). ...##
###
Simplicial decompositions of graphs: a survey of applications

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 ...

##
###
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]

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 ...

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

2009
*
Discrete Mathematics
*

Rather it is a personal statement written

doi:10.1016/j.disc.2008.04.055
fatcat:pmcsmwfxovctleuckjaztb2vgq
*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*. ...##
###
Sparse Combinatorial Structures: Classification and Applications

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*, ≤). ...

##
###
Decomposing infinite graphs

1991
*
Discrete Mathematics
*

It is intended

doi:10.1016/0012-365x(91)90330-5
fatcat:kek3uw2ovbe6vkhjuxggjjr3yy
*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*). ...##
###
On Low Tree-Depth Decompositions
[article]

2014
*
arXiv
*
pre-print

In more technical terms we survey some

arXiv:1412.1581v1
fatcat:pqx6dojaxbevlbl5pr7v3tl55u
*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 ...##
###
ASPECTS OF STRUCTURAL COMBINATORICS (Graph Homomorphisms and Their Use)

1999
*
Taiwanese journal of mathematics
*

Due to space limitations we concentrate

doi:10.11650/twjm/1500407157
fatcat:5y566g6furgbdgbsx2lhmxueia
*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*. ...##
###
Groups of type FP via graphical small cancellation
[article]

2022
*
arXiv
*
pre-print

In contrast to every previous construction

arXiv:2004.04550v3
fatcat:jnjocv7fkbfjrgaoo4zyvvnruq
*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. ...##
###
Structural Properties of Sparse Graphs
[chapter]

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. ...

##
###
Structural Properties of Sparse Graphs

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. ...

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

2021
*
arXiv
*
pre-print

We study

arXiv:2103.05588v2
fatcat:pqq53dd72rdjzpc4unymf3igki
*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*. ...##
###
Finite dualities and map-critical graphs on a fixed surface

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. ...

##
###
First order properties on nowhere dense structures

2010
*
Journal of Symbolic Logic (JSL)
*

It has been proved previously that minor closed

doi:10.2178/jsl/1278682204
fatcat:fairowkzp5hu5iichwc374j2j4
*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*. ...
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