DOCSLIB.ORG

Isomorphism and Embedding Problems for Infinite Limits of Scale

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Isomorphism and Embedding Problems for Infinite Limits of Scale Isomorphism and Embedding Problems for Infinite Limits of Scale-Free Graphs Robert D. Kleinberg ∗ Jon M. Kleinberg y Abstract structure of finite PA graphs; in particular, we give a The study of random graphs has traditionally been characterization of the graphs H for which the expected dominated by the closely-related models (n; m), in number of subgraph embeddings of H in an n-node PA which a graph is sampled from the uniform distributionG graph remains bounded as n goes to infinity. on graphs with n vertices and m edges, and (n; p), in n G 1 Introduction which each of the 2 edges is sampled independently with probability p. Recen tly, however, there has been For decades, the study of random graphs has been dom- considerable interest in alternate random graph models inated by the closely-related models (n; m), in which designed to more closely approximate the properties of a graph is sampled from the uniformG distribution on complex real-world networks such as the Web graph, graphs with n vertices and m edges, and (n; p), in n G the Internet, and large social networks. Two of the most which each of the 2 edges is sampled independently well-studied of these are the closely related \preferential with probability p.The first was introduced by Erd}os attachment" and \copying" models, in which vertices and R´enyi in [16], the second by Gilbert in [19]. While arrive one-by-one in sequence and attach at random in these random graphs have remained a central object \rich-get-richer" fashion to d earlier vertices. of study and continue to have many important applica- Here we study the infinite limits of the preferential tions in combinatorics and theoretical computer science, attachment process | namely, the asymptotic behavior recently there has also been a great deal of interest in al- of finite graphs produced by preferential attachment ternative random graph models whose properties more (briefly, PA graphs), as well as the infinite graphs closely resemble those of complex real-world networks obtained by continuing the process indefinitely. We are such as the Web graph, the Internet, and large social guided in part by a striking result of Erd}os and R´enyi networks. Two of the most well-studied of these are the on countable graphs produced by the infinite analogue closely related \preferential attachment" and \copying" of the (n; p) model, showing that any two graphs models; the former was introduced by Barab´asi and Al- producedG by this model are isomorphic with probability bert in [3] and subsequently formalized by Bollob´as and 1; it is natural to ask whether a comparable result holds Riordan in [8], while the latter was introduced by Ku- for the preferential attachment process. mar et al. in [22]. We find, somewhat surprisingly, that the answer de- A random graph in the preferential attachment pends critically on the out-degree d of the model. For model (henceforth, the PA model) is built up one 1 d = 1 and d = 2, there exist infinite graphs Rd such vertex at a time, with each new vertex v linking to the that a random graph generated according to the in- preceding ones by d new edges, where the out-degree d finite preferential attachment process is isomorphic to is a parameter of the model. Roughly, the head of each 1 Rd with probability 1. For d 3, on the other hand, edge emanating from v is chosen by sampling from the two different samples generated≥from the infinite prefer- preceding vertices with probabilities weighted according ential attachment process are non-isomorphic with pos- to their total degree (in-degree plus out-degree); this itive probability. The main technical ingredients under- is the preferential, or \rich-get-richer," aspect of the lying this result have fundamental implications for the model, since nodes of higher in-degree attract new in- coming edges more readily. (We will sometimes use ∗Department of Mathematics, MIT, Cambridge MA 02139. the term \PA graph" as an informal shorthand to Email: [email protected]. Supported by a Fannie and John refer to a random graph drawn from the distribution Hertz Foundation Fellowship. defined by the PA model.) As we discuss further yDepartment of Computer Science, Cornell University, Ithaca NY 14853. Email: [email protected]. Supported in part below, there has been considerable work aimed at by a David and Lucile Packard Foundation Fellowship and NSF determining fundamental graph-theoretic properties in grants 0081334 and 0311333. the PA model, exposing both similarities and contrasts with the classical (n; p) model. the random process will almost surely be a tree with In the presenGt paper, we seek to understand the countably many nodes, in which each node has infinite infinite limits of the PA model | namely, the asymp- degree. For the case of out-degree d = 2, the resulting 1 totic behavior of graphs produced by this model as the graph R2 is much more complicated. Its structure can number of nodes goes to infinity, and the distribution be characterized axiomatically, but it is also possible to 1 1 d on random graphs with countably many vertices give explicit constructions of graphs isomorphic to R2 . obtainedPA by continuing the PA process indefinitely. We For example, it is isomorphic to the graph whose vertices were inspired by the following classical theorem about consist of all finite rooted binary trees with integer the “infinite version" of the (n; p) model [17]. labels, where the vertex corresponding to a labeled tree G T has edges to its left sub-tree and to its right sub-tree. Theorem 1.1. Let ( ; p) denote the probability dis- G 1 The global structure of the proof for the case d = 2 tribution on graphs with vertex set N, in which each edge is a standard \back-and-forth" argument, which will (i; j) is included independently with probability p. (Here be familiar to readers acquainted with Theorem 1.1. p is any constant in (0; 1).) There exists an infinite The key step, however | establishing that there is an graph R, such that a random sample from ( ; p) is G 1 adequate supply of vertices to sustain the back-and- isomorphic to R with probability 1. forth construction of the isomorphism | is much more When one first encounters this theorem, it can complicated than in the classical case, since the PA seem quite startling: infinite random graphs are not process introduces difficult conditioning problems. \random" at all; they are almost surely isomorphic to One might imagine that for the cases of out-degrees a single fixed graph R. A rich theory has developed d = 3; 4; 5; : : : one could establish isomorphisms with 1 1 around the infinite model ( ; p), with connections probability 1 to increasingly complex graphs R3 , R4 , reaching into mathematical logic,G 1 algebra, and a number and so on. But in fact, we have the following result. of other areas (see e.g. [13]). Theorem 1.3. For each out-degree d 3, it is not the On the other hand, essentially nothing is known case that two independent random samples≥ from 1 about the the infinite version of the PA model. Does d are isomorphic with probability 1. PA something analogous to Theorem 1.1 hold here as well, or is the situation fundamentally different? At a more This contrast between the cases of d = 2 and d 3 fine-grained level, we are also interested in understand- comes to us as something of a surprise, since it does ≥not ing what can be said about the local structure of finite have an obvious analogue in the prior work on graphs graphs produced by the PA model as the number of generated according to the PA process. There, typically, nodes goes to infinity. As we discuss further below, the the out-degree d has a clear quantitative effect on the only prior work addressing the infinite graphs generated underlying graph parameters, but not a qualitative by such processes, as far as we are aware, are some in- effect of this sort. teresting recent papers by Bonato and Janssen [11, 12], This contrasting pair of results is a particularly suc- which proposed the notion of studying infinite limits of cinct consequence of one of the main technical com- random graph evolution processes related to the copy- ponents of the paper, which addresses a fundamental ing model of [23]. These papers consider the relationship structural issue for both the finite and infinite versions between such infinite random graphs and certain deter- of the PA model | a characterization of the graphs H ministic adjacency axioms. Some of these axioms have for which the expected number of subgraph embeddings a unique infinite model up to isomorphism, while others of H in an n-node PA graph remains bounded as n goes are satisfied with probability 1 by the infinite limits of to infinity. Phrased equivalently as a statement about the random graph processes considered in these papers. 1 the infinite model d , we show that if a finite graph However, none of their theorems resolve the question H is equal to its 3-corePA (i.e. the union of all subgraphs of whether an analogue of Theorem 1.1 holds for such of H of minimum degree 3), then the number of sub- infinite random graphs. 1 graph embeddings of H in a random sample from d Our first result is the following, where again 1 PA PAd has a positive finite expectation, while if H is not equal denotes the distribution associated with the infinite PA to its 3-core, then the number of embeddings is almost model.
Load more

Recommended publications
  • Planar Embeddings of Minc's Continuum and Generalizations
    PLANAR EMBEDDINGS OF MINC’S CONTINUUM AND GENERALIZATIONS ANA ANUSIˇ C´ Abstract. We show that if f : I → I is piecewise monotone, post-critically finite, x X I,f and locally eventually onto, then for every point ∈ =←− lim( ) there exists a planar embedding of X such that x is accessible. In particular, every point x in Minc’s continuum XM from [11, Question 19 p. 335] can be embedded accessibly. All constructed embeddings are thin, i.e., can be covered by an arbitrary small chain of open sets which are connected in the plane. 1. Introduction The main motivation for this study is the following long-standing open problem: Problem (Nadler and Quinn 1972 [20, p. 229] and [21]). Let X be a chainable contin- uum, and x ∈ X. Is there a planar embedding of X such that x is accessible? The importance of this problem is illustrated by the fact that it appears at three independent places in the collection of open problems in Continuum Theory published in 2018 [10, see Question 1, Question 49, and Question 51]. We will give a positive answer to the Nadler-Quinn problem for every point in a wide class of chainable continua, which includes←− lim(I, f) for a simplicial locally eventually onto map f, and in particular continuum XM introduced by Piotr Minc in [11, Question 19 p. 335]. Continuum XM was suspected to have a point which is inaccessible in every planar embedding of XM . A continuum is a non-empty, compact, connected, metric space, and it is chainable if arXiv:2010.02969v1 [math.GN] 6 Oct 2020 it can be represented as an inverse limit with bonding maps fi : I → I, i ∈ N, which can be assumed to be onto and piecewise linear.
    [Show full text]
  • On the Tree–Depth of Random Graphs Arxiv:1104.2132V2 [Math.CO] 15 Feb 2012
    On the tree–depth of random graphs ∗ G. Perarnau and O.Serra November 11, 2018 Abstract The tree–depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree–depth of random graphs. For dense graphs, p n−1, the tree–depth of a random graph G is a.a.s. td(G) = n − O(pn=p). Random graphs with p = c=n, have a.a.s. linear tree–depth when c > 1, the tree–depth is Θ(log n) when c = 1 and Θ(log log n) for c < 1. The result for c > 1 is derived from the computation of tree–width and provides a more direct proof of a conjecture by Gao on the linearity of tree–width recently proved by Lee, Lee and Oum [?]. We also show that, for c = 1, every width parameter is a.a.s. constant, and that random regular graphs have linear tree–depth. 1 Introduction An elimination tree of a graph G is a rooted tree on the set of vertices such that there are no edges in G between vertices in different branches of the tree. The natural elimination scheme provided by this tree is used in many graph algorithmic problems where two non adjacent subsets of vertices can be managed independently. One good example is the Cholesky decomposition of symmetric matrices (see [?, ?, ?, ?]). Given an elimination tree, a distributed algorithm can be designed which takes care of disjoint subsets of vertices in different parallel processors. Starting arXiv:1104.2132v2 [math.CO] 15 Feb 2012 by the furthest level from the root, it proceeds by exposing at each step the vertices at a given depth.
    [Show full text]
  • Neural Subgraph Matching
    Neural Subgraph Matching NEURAL SUBGRAPH MATCHING Rex Ying, Andrew Wang, Jiaxuan You, Chengtao Wen, Arquimedes Canedo, Jure Leskovec Stanford University and Siemens Corporate Technology ABSTRACT Subgraph matching is the problem of determining the presence of a given query graph in a large target graph. Despite being an NP-complete problem, the subgraph matching problem is crucial in domains ranging from network science and database systems to biochemistry and cognitive science. However, existing techniques based on combinatorial matching and integer programming cannot handle matching problems with both large target and query graphs. Here we propose NeuroMatch, an accurate, efficient, and robust neural approach to subgraph matching. NeuroMatch decomposes query and target graphs into small subgraphs and embeds them using graph neural networks. Trained to capture geometric constraints corresponding to subgraph relations, NeuroMatch then efficiently performs subgraph matching directly in the embedding space. Experiments demonstrate that NeuroMatch is 100x faster than existing combinatorial approaches and 18% more accurate than existing approximate subgraph matching methods. 1.I NTRODUCTION Given a query graph, the problem of subgraph isomorphism matching is to determine if a query graph is isomorphic to a subgraph of a large target graph. If the graphs include node and edge features, both the topology as well as the features should be matched. Subgraph matching is a crucial problem in many biology, social network and knowledge graph applications (Gentner, 1983; Raymond et al., 2002; Yang & Sze, 2007; Dai et al., 2019). For example, in social networks and biomedical network science, researchers investigate important subgraphs by counting them in a given network (Alon et al., 2008).
    [Show full text]
  • Testing Isomorphism in the Bounded-Degree Graph Model (Preliminary Version)
    Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 102 (2019) Testing Isomorphism in the Bounded-Degree Graph Model (preliminary version) Oded Goldreich∗ August 11, 2019 Abstract We consider two versions of the problem of testing graph isomorphism in the bounded-degree graph model: A version in which one graph is fixed, and a version in which the input consists of two graphs. We essentially determine the query complexity of these testing problems in the special case of n-vertex graphs with connected components of size at most poly(log n). This is done by showing that these problems are computationally equivalent (up to polylogarithmic factors) to corresponding problems regarding isomorphism between sequences (over a large alphabet). Ignoring the dependence on the proximity parameter, our main results are: 1. The query complexity of testing isomorphism to a fixed object (i.e., an n-vertex graph or an n-long sequence) is Θ(e n1=2). 2. The query complexity of testing isomorphism between two input objects is Θ(e n2=3). Testing isomorphism between two sequences is shown to be related to testing that two distributions are equivalent, and this relation yields reductions in three of the four relevant cases. Failing to reduce the problem of testing the equivalence of two distribution to the problem of testing isomorphism between two sequences, we adapt the proof of the lower bound on the complexity of the first problem to the second problem. This adaptation constitutes the main technical contribution of the current work. Determining the complexity of testing graph isomorphism (in the bounded-degree graph model), in the general case (i.e., for arbitrary bounded-degree graphs), is left open.
    [Show full text]
  • Practical Verification of MSO Properties of Graphs of Bounded Clique-Width
    Practical verification of MSO properties of graphs of bounded clique-width Ir`ene Durand (joint work with Bruno Courcelle) LaBRI, Universit´ede Bordeaux 20 Octobre 2010 D´ecompositions de graphes, th´eorie, algorithmes et logiques, 2010 Objectives Verify properties of graphs of bounded clique-width Properties ◮ connectedness, ◮ k-colorability, ◮ existence of cycles ◮ existence of paths ◮ bounds (cardinality, degree, . ) ◮ . How : using term automata Note that we consider finite graphs only 2/33 Graphs as relational structures For simplicity, we consider simple,loop-free undirected graphs Extensions are easy Every graph G can be identified with the relational structure (VG , edgG ) where VG is the set of vertices and edgG ⊆VG ×VG the binary symmetric relation that defines edges. v7 v6 v2 v8 v1 v3 v5 v4 VG = {v1, v2, v3, v4, v5, v6, v7, v8} edgG = {(v1, v2), (v1, v4), (v1, v5), (v1, v7), (v2, v3), (v2, v6), (v3, v4), (v4, v5), (v5, v8), (v6, v7), (v7, v8)} 3/33 Expression of graph properties First order logic (FO) : ◮ quantification on single vertices x, y . only ◮ too weak ; can only express ”local” properties ◮ k-colorability (k > 1) cannot be expressed Second order logic (SO) ◮ quantifications on relations of arbitrary arity ◮ SO can express most properties of interest in Graph Theory ◮ too complex (many problems are undecidable or do not have a polynomial solution). 4/33 Monadic second order logic (MSO) ◮ SO formulas that only use quantifications on unary relations (i.e., on sets). ◮ can express many useful graph properties like connectedness, k-colorability, planarity... Example : k-colorability Stable(X ) : ∀u, v(u ∈ X ∧ v ∈ X ⇒¬edg(u, v)) Partition(X1,..., Xm) : ∀x(x ∈ X1 ∨ .
    [Show full text]
  • Harary Polynomials
    numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 1:2 (2021) Article #S2R13 ecajournal.haifa.ac.il Harary Polynomials Orli Herscovici∗, Johann A. Makowskyz and Vsevolod Rakitay ∗Department of Mathematics, University of Haifa, Israel Email: [email protected] zDepartment of Computer Science, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] yDepartment of Mathematics, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] Received: November 13, 2020, Accepted: February 7, 2021, Published: February 19, 2021 The authors: Released under the CC BY-ND license (International 4.0) Abstract: Given a graph property P, F. Harary introduced in 1985 P-colorings, graph colorings where each color class induces a graph in P. Let χP (G; k) counts the number of P-colorings of G with at most k colors. It turns out that χP (G; k) is a polynomial in Z[k] for each graph G. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial χ(G; k). We show that the characteristic and the Laplacian polynomial, the matching, the independence and the domination polynomials are not Harary polynomials. We show that for various notions of sparse, non-trivial properties P, the polynomial χP (G; k) is, in contrast to χ(G; k), not a chromatic, and even not an edge elimination invariant. Finally, we study whether the Harary polynomials are definable in monadic second-order Logic. Keywords: Generalized colorings; Graph polynomials; Courcelle's Theorem 2020 Mathematics Subject Classification: 05; 05C30; 05C31 1.
    [Show full text]
  • Some Planar Embeddings of Chainable Continua Can Be
    Some planar embeddings of chainable continua can be expressed as inverse limit spaces by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Montana State University © Copyright by Susan Pamela Schwartz (1992) Abstract: It is well known that chainable continua can be expressed as inverse limit spaces and that chainable continua are embeddable in the plane. We give necessary and sufficient conditions for the planar embeddings of chainable continua to be realized as inverse limit spaces. As an example, we consider the Knaster continuum. It has been shown that this continuum can be embedded in the plane in such a manner that any given composant is accessible. We give inverse limit expressions for embeddings of the Knaster continuum in which the accessible composant is specified. We then show that there are uncountably many non-equivalent inverse limit embeddings of this continuum. SOME PLANAR EMBEDDINGS OF CHAIN ABLE OONTINUA CAN BE EXPRESSED AS INVERSE LIMIT SPACES by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of . Doctor of Philosophy in Mathematics MONTANA STATE UNIVERSITY Bozeman, Montana February 1992 D 3 l% ii APPROVAL of a thesis submitted by Susan Pamela Schwartz This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. g / / f / f z Date Chairperson, Graduate committee Approved for the Major Department ___ 2 -J2 0 / 9 Date Head, Major Department Approved for the College of Graduate Studies Date Graduate Dean iii STATEMENT OF PERMISSION TO USE .
    [Show full text]
  • Cauchy Graph Embedding
    Cauchy Graph Embedding Dijun Luo [email protected] Chris Ding [email protected] Feiping Nie [email protected] Heng Huang [email protected] The University of Texas at Arlington, 701 S. Nedderman Drive, Arlington, TX 76019 Abstract classify unsupervised embedding approaches into two cat- Laplacian embedding provides a low- egories. Approaches in the first category are to embed data dimensional representation for the nodes of into a linear space via linear transformations, such as prin- a graph where the edge weights denote pair- ciple component analysis (PCA) (Jolliffe, 2002) and mul- wise similarity among the node objects. It is tidimensional scaling (MDS) (Cox & Cox, 2001). Both commonly assumed that the Laplacian embed- PCA and MDS are eigenvector methods and can model lin- ding results preserve the local topology of the ear variabilities in high-dimensional data. They have been original data on the low-dimensional projected long known and widely used in many machine learning ap- subspaces, i.e., for any pair of graph nodes plications. with large similarity, they should be embedded However, the underlying structure of real data is often closely in the embedded space. However, in highly nonlinear and hence cannot be accurately approx- this paper, we will show that the Laplacian imated by linear manifolds. The second category ap- embedding often cannot preserve local topology proaches embed data in a nonlinear manner based on differ- well as we expected. To enhance the local topol- ent purposes. Recently several promising nonlinear meth- ogy preserving property in graph embedding, ods have been proposed, including IsoMAP (Tenenbaum we propose a novel Cauchy graph embedding et al., 2000), Local Linear Embedding (LLE) (Roweis & which preserves the similarity relationships of Saul, 2000), Local Tangent Space Alignment (Zhang & the original data in the embedded space via a Zha, 2004), Laplacian Embedding/Eigenmap (Hall, 1971; new objective.
    [Show full text]
  • Strong Inverse Limit Reflection
    Strong Inverse Limit Reflection Scott Cramer March 4, 2016 Abstract We show that the axiom Strong Inverse Limit Reflection holds in L(Vλ+1) assuming the large cardinal axiom I0. This reflection theorem both extends results of [4], [5], and [3], and has structural implications for L(Vλ+1), as described in [3]. Furthermore, these results together highlight an analogy between Strong Inverse Limit Reflection and the Axiom of Determinacy insofar as both act as fundamental regularity properties. The study of L(Vλ+1) was initiated by H. Woodin in order to prove properties of L(R) under large cardinal assumptions. In particular he showed that L(R) satisfies the Axiom of Determinacy (AD) if there exists a non-trivial elementary embedding j : L(Vλ+1) ! L(Vλ+1) with crit (j) < λ (an axiom called I0). We investigate an axiom called Strong Inverse Limit Reflection for L(Vλ+1) which is in some sense analogous to AD for L(R). Our main result is to show that if I0 holds at λ then Strong Inverse Limit Reflection holds in L(Vλ+1). Strong Inverse Limit Reflection is a strong form of a reflection property for inverse limits. Axioms of this form generally assert the existence of a collection of embeddings reflecting a certain amount of L(Vλ+1), together with a largeness assumption on the collection. There are potentially many different types of axioms of this form which could be considered, but we concentrate on a particular form which, by results in [3], has certain structural consequences for L(Vλ+1), such as a version of the perfect set property.
    [Show full text]
  • Every Monotone Graph Property Is Testable∗
    Every Monotone Graph Property is Testable∗ Noga Alon † Asaf Shapira ‡ Abstract A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on . This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemer´edi’s Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P, any graph that is -far from satisfying P, contains a subgraph of size depending on only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is -far from satisfying a (possibly infinite) set of monotone graph properties P, then it is at least δP ()-far from satisfying one of the properties.
    [Show full text]
  • Graph Properties and Hypergraph Colourings
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 98 (1991) 81-93 81 North-Holland Graph properties and hypergraph colourings Jason I. Brown Department of Mathematics, York University, 4700 Keele Street, Toronto, Ont., Canada M3l lP3 Derek G. Corneil Department of Computer Science, University of Toronto, Toronto, Ont., Canada MSS IA4 Received 20 December 1988 Revised 13 June 1990 Abstract Brown, J.I., D.G. Corneil, Graph properties and hypergraph colourings, Discrete Mathematics 98 (1991) 81-93. Given a graph property P, graph G and integer k 20, a P k-colouring of G is a function Jr:V(G)+ (1,. ) k} such that the subgraph induced by each colour class has property P. When P is closed under induced subgraphs, we can construct a hypergraph HG such that G is P k-colourable iff Hg is k-colourable. This correlation enables us to derive interesting new results in hypergraph chromatic theory from a ‘graphic’ approach. In particular, we build vertex critical hypergraphs that are not edge critical, construct uniquely colourable hypergraphs with few edges and find graph-to-hypergraph transformations that preserve chromatic numbers. 1. Introduction A property P is a collection of graphs (closed under isomorphism) containing K, and K, (all graphs considered here are finite); P is hereditary iff P is closed under induced subgraphs and nontrivial iff P does not contain every graph. Any graph G E P is called a P-graph. Throughout, we shall only be interested in hereditary and nontrivial properties, and P will always denote such a property.
    [Show full text]
  • Lecture 9: the Whitney Embedding Theorem
    LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically, the word \manifold" (Mannigfaltigkeit in German) first appeared in Riemann's doctoral thesis in 1851. At the early times, manifolds are defined extrinsi- cally: they are the set of all possible values of some variables with certain constraints. Translated into modern language,\smooth manifolds" are objects that are (locally) de- fined by smooth equations and, according to last lecture, are embedded submanifolds in Euclidean spaces. In 1912 Weyl gave an intrinsic definition for smooth manifolds. A natural question is: what is the difference between the extrinsic definition and the intrinsic definition? Is there any \abstract" manifold that cannot be embedded into any Euclidian space? In 1930s, Whitney and others settled this foundational problem: the two ways of defining smooth manifolds are in fact the same. In fact, Whitney's result is much more stronger than this. He showed that not only one can embed any smooth manifold into some Euclidian space, but that the dimension of the Euclidian space can be chosen to be (as low as) twice the dimension of the manifold itself! Theorem 0.1 (The Whitney embedding theorem). Any smooth manifold M of di- mension m can be embedded into R2m+1. Remark. In 1944, by using completely different techniques (now known as the \Whitney trick"), Whitney was able to prove Theorem 0.2 (The Strong Whitney Embedding Theorem). Any smooth man- ifold M of dimension m ≥ 2 can be embedded into R2m (and can be immersed into R2m−1). We will not prove this stronger version in this course, but just mention that the Whitney trick was further developed in h-cobordism theory by Smale, using which he proved the Poincare conjecture in dimension ≥ 5 in 1961! Remark.
    [Show full text]