Structural sparsity

Nešetřil, Jaroslav; Mendez, Patrice Ossona de

doi:10.1070/rm9688

Cited by 9 publications

(5 citation statements)

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“…Monotone classes of graphs which have modeling limits (of which graphing limits are a particular case) were characterized in [26] and coincides with nowhere dense classes (of graphs). This also coincides (in the case of monotone classes of graphs) with the notion of NIP and stable classes, see [1] (see also [23]). For hereditary classes the structure theory and the existence of modeling limits is more complicated (see [22]) and local-global convergence seems to provide a useful framework.…”

Section: Introductionsupporting

confidence: 59%

Local-global convergence, an analytic and structural approach

Nešetřil¹,

Mendez²

2019

Commentationes Mathematicae Universitatis Carolinae

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Section: Introductionsupporting

confidence: 59%

Local-global convergence, an analytic and structural approach

Nešetřil¹,

Mendez²

2019

Commentationes Mathematicae Universitatis Carolinae

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“…We note that this above technique can be applied under slightly weaker hypothesis than the one we state in Theorem 3. For instance, Nešetřil and Ossona de Mendez proved that for all nowhere dense graph classes (i.e., a broad generalization of proper minor-closed graph classes and bounded-degree graphs), for any graph in the class and for any constant k, the VC-dimension of the k-neighbourhood hypergraph is constantly upper-bounded [48]. It allows us to derive the following weaker version of our Theorem 3:…”

Section: Our Contributionsmentioning

confidence: 94%

“…A closer look at the proof of Theorem 3 shows that it also holds if, instead of having bounded distance VC-dimension, there rather exists some constant d such that, for every 1 ≤ i ≤ k − 1, the VC-dimension of the i-neighbourhood hypergraph is at most d (the latter value is sometimes called the distance-i VC-dimension of the graph [48]). It has algorithmic implications for some special cases of sparse graphs.…”

Section: Application To Nowhere Dense Graph Classesmentioning

confidence: 98%

“…Namely, H is an r-shallow minor of a graph G if it can be obtained from some subgraph of G by the contraction of pairwise disjoint subgraphs of radius at most r [51]; a graph family G is termed nowhere dense if, for any r, there exists a graph H r which is not an r-shallow minor for any graph in G [47]. Of interest here is that, for any graph class G nowhere dense, and for any i, the distance-i VC-dimension of any graph in G is upper-bounded by some constant d i [48]. By choosing d := max 1≤i≤k−1 d i , we so obtain the following weaker version of Theorem 3 for nowhere dense graphs:…”

Section: Application To Nowhere Dense Graph Classesmentioning

confidence: 99%

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Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Ducoffe¹,

Habib²,

Viennot³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic-time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes -where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radius and center in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| − 1. number of edges on a shortest path. Beyond its many practical applications, this fundamental problem in Graph Theory has attracted a lot of attention in the fine-grained complexity study of polynomial-time solvable problems [1,4,8,15,18,22,25,32,52]. More precisely, for every n-vertex m-edge unweighted graph the textbook algorithm for computing its diameter runs in time O(nm). In a seminal paper [52] this roughly quadratic running-time was matched by a quadratic lower-bound, assuming the Strong Exponential-Time Hypothesis (SETH). We stress that for graphs with millions of nodes and edges, quadratic time is already too prohibitive.The conditional lower-bound of [52] also holds for sparse graphs i.e., with only m = O(n) edges [1]. However it does not hold for many well-structured graph classes [1,11,13,20,14,21,23,25,31,33,35,49]. Our work proposes some new advances on the characterization of graph families for which we can compute the diameter in truly subquadratic time. Related workBefore we detail our contributions, we wish to mention a few recent (and not so recent) results that are most related to our approach.

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“…The theory has led to the nowhere dense/somewhere dense dichotomy that can be observed in several areas of graph theory, theoretical computer science, model theory, analysis, category theory and probability theory. Motivated by the connection with model theory -nowhere dense classes are monadically stable [1] and even have low VC-density [37] and by a possible extension of first-order model-checking algorithms for bounded expansion classes [11,12] and for nowhere dense classes [17], these notions were extended to classes that are obtained as first-order transductions of sparse classes, the structurally sparse classes [34,13]. The central tool used in our approach is the transduction machinery, which establishes a fruitful bridge between graph theory and finite model theory.…”

Section: Introductionmentioning

confidence: 99%

Classes of graphs with low complexity: the case of classes with bounded linear rankwidth

Nešetřil¹,

Mendez²,

Rabinovich³

et al. 2019

Preprint

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Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths -a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes. The structural results we obtain are the following. 1) The number of unlabeled graphs of order with linear rank-width at most is at most ( ∕2)! 2 ( 2 ) 3 +2 . 2) Graphs with linear rankwidth at most are linearly -bounded. Actually, they have bounded -chromatic number, meaning that they can be colored with ( ) colors, each color inducing a cograph. 3) To the contrary, based on a Ramsey-like argument, we prove for every proper hereditary family F of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in F .From the model theoretical side we obtain the following results: 1) A direct short proof that graphs with linear rankwidth at most are first-order transductions of linear orders. This result could also be derived from Colcombet's theorem on first-order transduction of linear orders and the equivalence of linear rankwidth with linear cliquewidth. 2) For a class C with bounded linear rankwidth the following conditions are equivalent: a) C is stable, b) C excludes some half-graph as a semi-induced subgraph, c) C is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.On devient jeune à soixante ans. Malheureusement, c'est trop tard.You become young when you're sixty. Unfortunately, it's too late.

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Structural sparsity

Cited by 9 publications

References 55 publications

Local-global convergence, an analytic and structural approach

Local-global convergence, an analytic and structural approach

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Classes of graphs with low complexity: the case of classes with bounded linear rankwidth

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