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

Noga Alon, Jozsef Balogh, Bela Bollobas, Robert Morris
2009 arXiv   pre-print
However, their results tell us very little about 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.  ... 
arXiv:0905.1942v1 fatcat:hsmy644pmfez3ljduq6r3hfz3u

The structure of almost all graphs in a hereditary property

Noga Alon, József Balogh, Béla Bollobás, Robert Morris
2011 Journal of combinatorial theory. Series B (Print)  
However, their results tell us very little about 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.  ... 
doi:10.1016/j.jctb.2010.10.001 fatcat:2raub3zzdbfbbb7xfcwzcbv3um

A Characterization of Uniquely Representable Graphs [article]

Péter G. N. Szabó
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.  ... 
arXiv:1708.01272v4 fatcat:mgaox6lufjeahblyhoc3kqjrxa

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

Svante Janson, Andrew J. Uzzell
2014 arXiv   pre-print
Given 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.  ... 
arXiv:1404.2456v1 fatcat:zmcfxxm6wjhrng2yb4lug4kpqu

Typical structure of hereditary graph families. I. Apex-free families [article]

Sergey Norin, Yelena Yuditsky
2020 arXiv   pre-print
Alon, Balogh, Bollobás and Morris [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.  ... 
arXiv:2007.00686v1 fatcat:6ubre4cddbdlxeuabvl5wwgivi

Generating All Maximal Induced Subgraphs for Hereditary, Connected-Hereditary and Rooted-Hereditary Properties [article]

Sara Cohen, Yehoshua Sagiv
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 .  ... 
arXiv:cs/0410039v1 fatcat:xutjxuyoo5cb7i66lkthqz5dnm

First order properties on nowhere dense structures

Jaroslav Nešetřil, Patrice Ossona de Mendez
2010 Journal of Symbolic Logic (JSL)  
This 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  ... 
doi:10.2178/jsl/1278682204 fatcat:fairowkzp5hu5iichwc374j2j4

Almost distance-hereditary graphs

Méziane Aı̈der
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.  ... 
doi:10.1016/s0012-365x(00)00401-5 fatcat:4krevbt3grdxhnxfprc444qyd4

Stability-type results for hereditary properties

Noga Alon, Uri Stav
2009 Journal of Graph Theory  
For 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)?  ... 
doi:10.1002/jgt.20388 fatcat:3odqiv2wnrbzzos6cg6aywn2fy

Generating all maximal induced subgraphs for hereditary and connected-hereditary graph properties

Sara Cohen, Benny Kimelfeld, Yehoshua Sagiv
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  ... 
doi:10.1016/j.jcss.2008.04.003 fatcat:b4t2bwzupfaf7npwgjhbjfsuym

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

Sergey Norin, Yelena Yuditsky
2020 arXiv   pre-print
Recently, Reed and Scott proposed 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.  ... 
arXiv:2007.00688v1 fatcat:cphjxxbl3vbtjlc5kytk2vvo6e

Distance Preserving Graphs [article]

Emad Zahedi
2015 arXiv   pre-print
We also find 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.  ... 
arXiv:1507.03615v1 fatcat:5cppf5yhsrcilatnikvcbiocwe

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

Gabriel Semanišin
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.  ... 
doi:10.7151/dmgt.1167 fatcat:56ywq2kkfrahrikioy4jbesau4

Structural Properties of Sparse Graphs [chapter]

Jaroslav NeŠetřil, Patrice Ossona De Mendez
2008 Bolyai Society Mathematical Studies  
For 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).  ... 
doi:10.1007/978-3-540-85221-6_13 fatcat:j7ifp6snovdybopviit5zaya2e
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