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The structure of almost all graphs in a hereditary property
[article]
2009
arXiv
pre-print
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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
2011
Journal of combinatorial theory. Series B (Print)
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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]
2020
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2004
arXiv
pre-print
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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
2010
Journal of Symbolic Logic (JSL)
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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
2002
Discrete Mathematics
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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
2009
Journal of Graph Theory
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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
2008
Journal of computer and system sciences (Print)
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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]
2020
arXiv
pre-print
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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]
2015
arXiv
pre-print
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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
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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
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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]
2008
Bolyai Society Mathematical Studies
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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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