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On Nowhere Dense Graphs View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 32 (2011) 600–617 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc On nowhere dense graphs Jaroslav Nešetřil a, Patrice Ossona de Mendez b a Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, Malostranské nám.25, 11800 Praha 1, Czech Republic b Centre d'Analyse et de Mathématiques Sociales, CNRS, UMR 8557, 54 Bd Raspail, 75006 Paris, France article info a b s t r a c t Article history: In this paper, we define and analyze the nowhere dense classes of Received 4 June 2009 graphs. This notion is a common generalization of proper minor Accepted 6 April 2010 closed classes, classes of graphs with bounded degree, locally Available online 17 February 2011 planar graphs, classes with bounded expansion, to name just a few classes which are studied extensively in combinatorial and computer science contexts. In this paper, we show that this concept leads to a classification of general classes of graphs and to the dichotomy between nowhere dense and somewhere dense classes. This classification is surprisingly stable as it can be expressed in terms of the most commonly used basic combinatorial parameters, such as the independence number α, the clique number !, and the chromatic number χ. The remarkable stability of this notion and its robustness has a number of applications to mathematical logic, complexity of algorithms, and combinatorics. We also express the nowhere dense versus somewhere dense dichotomy in terms of edge densities as a trichotomy theorem. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction For a (finite or infinite) family F of graphs, we denote by Forbm.F / the class of all finite graphs G not containing any F 2 F as a subgraph. By a subgraph we mean here not necessarily induced subgraph (and ``m'' in Forbm.F / stands for monomorphism). In this paper, we study mainly asymptotic properties of such classes which are defined by means of edge densities. In order to digest this we list some examples: • Forbm.K2/ is the class of all discrete graphs; • Forbm.Pk/ is the class of graphs with a bounded tree depth ([25]; see Section 3); E-mail addresses: [email protected] (J. Nešetřil), [email protected] (P. Ossona de Mendez). 0195-6698/$ – see front matter ' 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2011.01.006 J. Nešetřil, P. Ossona de Mendez / European Journal of Combinatorics 32 (2011) 600–617 601 • Forbm.S1;k/ is the class of all graphs with all degrees bounded by k; • Forbm.T / for any given tree T is a class of graphs with bounded degeneracy (or bounded maximal average degree). On the other side of our spectrum is the class Forbm.fC` V ` ≤ gg/ of all graphs with girth >g. This class may be regarded as a class of random-like graphs. In the next section, we shall define the notions of shallow minor and topological minor and this will lead to much more general classes of type Forbm.F / which are called bounded expansion classes, bounded local expansion classes and finally here newly defined classes of nowhere dense graphs. For all these classes we shall be able to prove characterizations. Classes of nowhere dense structures are defined in this paper (in Section 2.1) and they have several interesting (and we believe surprising) properties and equivalent formulations; see Theorem 4.1. This result, for example, alternatively defines nowhere dense classes as classes of structures where all shallow minors have edge density n1Co.1/. The interplay between dense classes (more precisely the classes of somewhere dense graphs) and classes of nowhere dense graphs is very interesting and it is expressed by the trichotomy Theorem 3.2. Despite the generality of these classes we can deduce from the results of [26,27] several algorithmic consequences. For example any first-order sentence preserved under homomorphisms on a class C of structures may be decided in time O.n/ if C has bounded expansion and in time n1Co.1/ if C is a class of nowhere dense structures. The nowhere dense classes strictly contain studied classes of structures such as classes with bounded local tree width, locally excluded minors, etc; see [8,20,4,3,5,10]; see Fig. 3 for the schema of inclusion of these classes. Yet we can prove for all these new classes that the relativized homomorphism preservation theorems hold even for them. Perhaps this also provides a proper setting for questions related to scattered sets in graphs. This leads to notions wide, semi-wide and quasi-wide classes and we obtain characterization theorems for these classes. This is mentioned in the last section devoted to applications. In Section 3, we prove Theorem 4.1 which should be seen as the main result of this paper. It not only characterizes classes of nowhere dense graphs (and structures) but it also explains the pleasing interplay of various invariants defined for bounded expansion classes. This has several consequences for first-order definable classes, for dualities and leads to efficient algorithms. Some of these are listed in Section 4 which is devoted to applications. 2. Definitions For graphs, and more generally, relational structures, we use standard notation and terminology. In this section we give the key definitions of this paper. 2.1. Distances, shallow minors and grads The distance in a graph G between two vertices x and y is the minimum length of a path linking x and y (or 1 if x and y do not belong to the same connected component of G) and is denoted by distG.x; y/. D G 2 Let G .V ; E/ be a graph and let d be an integer. The d-neighborhood Nd .u/ of a vertex u V is the G D f 2 V ≤ g subset of vertices of G at distance at most d from u in G: Nd .u/ v V distG.u; v/ d . For a graph G D .V ; E/, we denote by jGj the order of G (that is, jV j) and by kGk the size of G (that is, jEj). For any graphs H and G and any integer d, the graph H is said to be a shallow minor of G at depth d ([34] attribute this notion, then called low depth minor to Ch. Leiserson and S. Toledo) if there exists a subset fx1;:::; xpg of G and a collection of disjoint subsets V1;:::; Vp of vertices of G, each inducing a connected subgraph of G, such that xi 2 Vi, every vertex in Vi is at distance at most d from xi in the subgraph of G induced by Vi, and so that H is a subgraph of the graph obtained from G by contracting each Vi into xi and removing loops and multiple edges (see Fig. 1). The set of all shallow minors of G at depth d is denoted by G O d. In particular, G O 0 is the set of all subgraphs of G. 602 J. Nešetřil, P. Ossona de Mendez / European Journal of Combinatorics 32 (2011) 600–617 Fig. 1. A shallow minor of depth r of a graph G is a simple subgraph of a minor of G obtained by contracting vertex disjoint subgraphs with radius at most r. The greatest reduced average density (shortly grad) with rank r of a graph G [26] is defined by the formula kHk rr .G/ D max V H 2 G O r : (1) jHj By extension, for a class of graphs C, we denote by C O i the set of all shallow minors at depth i of graphs of C, that is, [ C O i D .G O i/: G2C Hence we have C ⊆ C O 0 ⊆ C O 1 ⊆ ::: ⊆ C O i ⊆ :::: Also, for a class C of graphs we define the expansion of the class C as ri.C/ D sup ri.G/: G2C Notice that ri.C/ D r0.C O i/. These definitions may be carried over for any graph invariant. Explicitly, let us define !i.C/ D sup !.G/ D !.C O i/: G2C O i We of course put !i.C/ D 1 if the supremum is infinite. The (non-decreasing) sequence !1.C/; !2.C/; : : : plays an important role in this paper and it leads to the central definition. Definition 2.1. A class C of graphs is somewhere dense if there exists an integer τ such that !τ .C/ D 1. Otherwise, if !i.C/ < 1 for each integer i, the class C is nowhere dense. This definition seems to be ad hoc and arbitrary. In this paper, we give an evidence of the contrary: we show many equivalent formulations of the nowhere dense versus somewhere dense dichotomy. We start with a more detailed analysis of shallow minors. 2.2. Shallow topological minors and top-grads For any (simple) graphs H and G and any integer d, the graph H is said to be a shallow topological minor of G at depth d if there exists a subset fx1;:::; xpg of G and a collection of internally vertex disjoint paths P1 ::: Pq each of length at most 2d C 1 of G with endpoints in fx1;:::; xpg whose contraction into single edges define on fx1;:::; xpg a graph isomorphic to H (see Fig. 2). Putting otherwise G contains a subdivision of H with at most 2d subdivision vertices on each edge.
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