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##
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The structure of almost all graphs in a hereditary property
[article]

2009
*
arXiv
*
pre-print

However, their results tell us very little about

arXiv:0905.1942v1
fatcat:hsmy644pmfez3ljduq6r3hfz3u
*the**structure**of**a*typical*graph*G ∈.*In*this paper we describe*the**structure**of**almost*every*graph**in**a**hereditary**property**of**graphs*, . ...*A**hereditary**property**of**graphs*is*a*collection*of**graphs*which is closed under taking induced subgraphs.*The*speed*of*is*the*function n |_n|, where _n denotes*the**graphs**of*order n*in*. ... Introduction*In*this paper we shall describe*the**structure**of**almost*every*graph**in*an arbitrary*hereditary**property**of**graphs*, P. ...##
###
The structure of almost all graphs in a hereditary property

2011
*
Journal of combinatorial theory. Series B (Print)
*

However, their results tell us very little about

doi:10.1016/j.jctb.2010.10.001
fatcat:2raub3zzdbfbbb7xfcwzcbv3um
*the**structure**of**a*typical*graph*G ∈ P.*In*this paper we describe*the**structure**of**almost*every*graph**in**a**hereditary**property**of**graphs*, P. ...*A**hereditary**property**of**graphs*is*a*collection*of**graphs*which is closed under taking induced subgraphs.*The*speed*of*P is*the*function n → |P n |, where P n denotes*the**graphs**of*order n*in*P. ... Introduction*In*this paper we shall describe*the**structure**of**almost*every*graph**in*an arbitrary*hereditary**property**of**graphs*, P. ...##
###
A Characterization of Uniquely Representable Graphs
[article]

2020
*
arXiv
*
pre-print

*In*this paper, we give

*a*characterization

*of*uniquely representable

*graphs*by showing that they are exactly

*the*block

*graphs*. ... Further, we prove that two related classes

*of*

*graphs*coincide with

*the*class

*of*block

*graphs*and

*the*class

*of*distance-

*hereditary*

*graphs*, respectively. ... Acknowledgment

*The*author is grateful to Pierre Aboulker for sharing his thoughts on

*the*results presented here, as well as on other, related results. ...

##
###
On the Typical Structure of Graphs in a Monotone Property
[article]

2014
*
arXiv
*
pre-print

Given

arXiv:1404.2456v1
fatcat:zmcfxxm6wjhrng2yb4lug4kpqu
*a**graph**property*P, it is interesting to determine*the*typical*structure**of**graphs*that satisfy P. ... Using results from*the*theory*of**graph*limits, we show that if P is*a*monotone*property*and r is*the*largest integer for which every r-colorable*graph*satisfies P, then*almost*every*graph*with P is close ...*In*this note, we use results from*graph*limit theory to study*the**structure**of**a*typical*graph**in**a*general monotone*property*. ...##
###
Typical structure of hereditary graph families. I. Apex-free families
[article]

2020
*
arXiv
*
pre-print

Alon, Balogh, Bollobás and Morris [

arXiv:2007.00686v1
fatcat:6ubre4cddbdlxeuabvl5wwgivi
*The**structure**of**almost**all**graphs**in**a**hereditary**property*, JCTB 2011] gave*a*rough description*of*typical*graphs**in**a**hereditary*family and used it to show for every ...*The*main result*of*this paper gives*a*more precise description*of*typical*structure*for*a*restricted class*of**hereditary*families. ... We are indebted to Bruce Reed for introducing us to*the*questions considered and to many*of**the*techniques employed*in*this paper. We thank Zachary Feng for valuable comments. ...##
###
Generating All Maximal Induced Subgraphs for Hereditary, Connected-Hereditary and Rooted-Hereditary Properties
[article]

2004
*
arXiv
*
pre-print

*The*problem

*of*computing

*all*maximal induced subgraphs

*of*

*a*

*graph*G that have

*a*

*graph*

*property*P, also called

*the*maximal P-subgraphs problem, is considered. ... This problem is studied for

*hereditary*, connected-

*hereditary*and rooted-

*hereditary*

*graph*

*properties*. ... Let P cbip be

*the*connected-

*hereditary*

*property*that contains

*all*connected bipartite

*graphs*. Suppose that G is

*a*

*graph*that

*almost*satisfies P cbip . ...

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

2010
*
Journal of Symbolic Logic (JSL)
*

This

doi:10.2178/jsl/1278682204
fatcat:fairowkzp5hu5iichwc374j2j4
*in*turn led to*the*notions*of*wide,*almost*wide and quasi-wide classes*of**graphs*. ...*A*set*A**of*vertices*of**a**graph*G is called d-scattered*in*G if no two d-neighborhoods*of*(distinct) vertices*of**A*intersect. ... Although*almost**all*results*of*this paper can be formulated*in**the*"local" form (for*a*single*graph*with special*properties*) we find it useful to formulate our results by means*of**properties**of*classes ...##
###
Almost distance-hereditary graphs

2002
*
Discrete Mathematics
*

*In*this paper, we extend

*the*notion

*of*distance-

*hereditary*

*graphs*by introducing

*the*class

*of*

*almost*distance-

*hereditary*

*graphs*(

*a*very weak increase

*of*

*the*distance is allowed by induced subgraphs). ... We obtain

*a*characterization

*of*these

*graphs*

*in*terms

*of*forbidden-induced subgraphs and derive other both combinatorial and metric

*properties*. ... Charles Payan (laboratoire Leibniz/imag Grenoble, France) and would thank referees for many constructive suggestions on

*the*revision

*of*this paper. ...

##
###
Stability-type results for hereditary properties

2009
*
Journal of Graph Theory
*

For

doi:10.1002/jgt.20388
fatcat:3odqiv2wnrbzzos6cg6aywn2fy
*a**hereditary**graph**property*P, consider modifying*the*edges*of**a*random*graph*G = G(n, 1/2) to obtain*a**graph*G * that satisfies P*in*(essentially)*the*most economical way. ...*In*this paper we extend*the*above notion*of*stability to*hereditary**graph**properties*. It turns out that to do so*the*complete*graph*K n has to be replaced by*a*random*graph*. ...*The*basic question we address is*the*following: given*a**hereditary**graph**property*P, is there*a*unique*structure*for*all**the**graphs**in*P that have (essentially)*the*minimum distance to G(n, 1/2)? ...##
###
Generating all maximal induced subgraphs for hereditary and connected-hereditary graph properties

2008
*
Journal of computer and system sciences (Print)
*

*In*

*the*maximal P-subgraphs problem,

*the*goal is to produce

*all*(locally) maximal subgraphs

*of*

*a*

*graph*that have

*property*P, whereas

*in*

*the*node-deletion problem,

*the*goal is to find

*a*single (globally) ... This paper investigates

*a*

*graph*enumeration problem, called

*the*maximal P-subgraphs problem, where P is

*a*

*hereditary*or connected-

*hereditary*

*graph*

*property*. ... Acknowledgments

*The*authors thank

*the*anonymous referees for helpful suggestions and for pointing out important related work. Appendix

*A*. Auxiliary procedures

*A*.1. ExtendMax ...

##
###
Typical structure of hereditary graph families. II. Exotic examples
[article]

2020
*
arXiv
*
pre-print

Recently, Reed and Scott proposed

arXiv:2007.00688v1
fatcat:cphjxxbl3vbtjlc5kytk2vvo6e
*a*conjectural description*of**the*typical*structure**of*H-free*graphs*for*all**graphs*H, which extends*all*previously known results*in**the*area. ...*The*study*of**the*typical*structure**of*H-free*graphs*was initiated by Erdős, Kleitman and Rothschild, who have shown that*almost**all*C_3-free*graphs*are bipartite. ... For*a*family*of**graphs*F , let F n denote*the*set*of**graphs**in*F with vertex set [n] = {1, 2, . . . , n}. We say that*a**property*P holds for*almost**all**graphs**in*F , if lim n→∞ |F n ∩ P| |F n | = 1. ...##
###
Distance Preserving Graphs
[article]

2015
*
arXiv
*
pre-print

We also find

arXiv:1507.03615v1
fatcat:5cppf5yhsrcilatnikvcbiocwe
*a*condition on*the*girth*of*G which implies that it is not dp.*In*closing, we discuss other work and open problems concerning dp*graphs*. ... We say*a**graph*G is distance preserving (dp) if it has an isometric subgraph*of*every possible order up to*the*order*of*G. ... Prove or disprove that*almost**all**graphs*are dp. We note that*almost**all**graphs*have diameter two. ...##
###
Page 7840 of Mathematical Reviews Vol. , Issue 2002K
[page]

2002
*
Mathematical Reviews
*

*A*

*graph*G = (V, E) is

*almost*distance-

*hereditary*if for

*all*connected induced subgraphs H = (V*, E*)

*of*G and for each pair

*of*vertices u,v € V*

*the*distance

*of*u,v

*in*H is at most

*the*distance

*of*u,v ...

*A*

*structural*characterization

*of*this class is obtained

*in*terms

*of*

*the*set

*of*forbidden induced subgraphs.

*The*other

*structural*and metric

*properties*are investigated as well. ...

##
###
On generating sets of induced-hereditary properties

2002
*
Discussiones Mathematicae Graph Theory
*

*In*this paper we focus on

*the*

*structure*

*of*specific generating sets which provide

*the*base for

*the*proof

*of*

*The*Unique Factorization Theorem for induced-

*hereditary*

*properties*

*of*

*graphs*. ... These problems can be very well described

*in*

*the*language

*of*reducible (induced)

*hereditary*

*properties*

*of*

*graphs*. ... Four Different Types

*of*Generating Sets

*In*

*the*case

*of*

*hereditary*

*properties*it is natural to characterize

*a*

*property*P by

*the*set

*of*

*graphs*containing

*all*

*the*

*graphs*

*in*P as subgraphs. ...

##
###
Structural Properties of Sparse Graphs
[chapter]

2008
*
Bolyai Society Mathematical Studies
*

For

doi:10.1007/978-3-540-85221-6_13
fatcat:j7ifp6snovdybopviit5zaya2e
*a*class*of**graphs*C, we denote by H(C)*the*class containing*all**the*induced subgraphs*of**graphs**in*C, that is*the*inclusion-minimal*hereditary*class*of**graphs*containing C. 3.1.1. Limits. ... Let C be*a**hereditary*class*of**graphs*. Then*the*following are equivalent: • C is*almost*wide; • C is uniformly*almost*wide; • There are s ∈ N and t : N → N such that K s,t(r) / ∈ C r (for*all*r ∈ N). ...
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