Characterising bounded expansion by neighbourhood complexity

Reidl, Felix; Villaamil, Fernando Sánchez; Stavropoulos, Konstantinos S.

doi:10.1016/j.ejc.2018.08.001

Cited by 22 publications

(26 citation statements)

References 28 publications

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“…Lemma 23 (Theorem 4 of [32]). Let G be a graph, r be a half-integer, and let H t r G. Then |E(H)| |V (H)| (2r + 1) · max ν 1 (G) 4 · log 2 ν 1 (G), ν 2 (G), .…”

Section: Lower Bounds For Non-sparse Classesmentioning

confidence: 99%

On the number of types in sparse graphs

Pilipczuk

Siebertz

Toruńczyk

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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We prove that for every class of graphs C which is nowhere dense, as defined by Nešetřil and Ossona de Mendez [27,28], and for every first order formula ϕ(x,ȳ), whenever one draws a graph G ∈ C and a subset of its nodes A, the number of subsets of A |ȳ| which are of the form {v ∈ A |ȳ| : G |= ϕ(ū,v)} for some valuationū ofx in G is bounded by O(|A| |x|+ε ), for every ε > 0. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes.We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler [1] stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices). * The work of M. Pilipczuk and S.

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“…Lemma 23 (Theorem 4 of [32]). Let G be a graph, r be a half-integer, and let H t r G. Then |E(H)| |V (H)| (2r + 1) · max ν 1 (G) 4 · log 2 ν 1 (G), ν 2 (G), .…”

Section: Lower Bounds For Non-sparse Classesmentioning

confidence: 99%

On the number of types in sparse graphs

Pilipczuk

Siebertz

Toruńczyk

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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“…For A Ď V pGq and v P V pGqzA, the distance-r profile of v on A is π r G pv, Aq :" tpw, dist r G pv, wqq : w P Au, and the distance-r diversity on A is π r G pAq :" |tπ r G pv, Aq : v P V pGqzAu|. This definition is different to similar definitions in [19,41] in that we only consider v P V pGqzA. We use different notation to avoid confusion.…”

Section: Neighbourhood Diversitymentioning

confidence: 95%

“…The following concept will help to optimise our bounds on reduced bandwidth and twin-width, and is of independent interest because of connections to VC-dimension [22,37,41] and sparsity theory [19,41].…”

Section: Neighbourhood Diversitymentioning

confidence: 99%

Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)

Bonnet¹,

Kwon²,

Wood³

2022

Preprint

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In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying u and v, each edge incident to exactly one of u and v is coloured red. Bonnet, Kim, Thomassé and Watrigant [J. ACM 2022] defined the twin-width of a graph G to be the minimum integer k such that there is a reduction sequence of G in which every red graph has maximum degree at most k. For any graph parameter f , we define the reduced f of a graph G to be the minimum integer k such that there is a reduction sequence of G in which every red graph has f at most k. Our focus is on graph classes with bounded reduced bandwidth, which implies and is stronger than bounded twin-width (reduced maximum degree). We show that every proper minor-closed class has bounded reduced bandwidth, which is qualitatively stronger than an analogous result of Bonnet et al. for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least 2 1000 . We show that planar graphs have reduced bandwidth at most 466 and twin-width at most 583. Our bounds for graphs of Euler genus γ are Opγq. Lastly, we show that fixed powers of graphs in a proper minor-closed class have bounded reduced bandwidth (irrespective of the degree of the vertices). In particular, we show that map graphs of Euler genus γ have reduced bandwidth Opγ 4 q. Lastly, we separate twin-width and reduced bandwidth by showing that any infinite class of expanders excluding a fixed complete bipartite subgraph has unbounded reduced bandwidth, while there are bounded-degree expanders with twin-width at most 6.

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“…Zhu's connection of these parameters to bounded expansion in 2009 [16] renewed interest in analyzing and utilizing them. Reidl et al used the weak coloring numbers to prove that neighborhood complexity characterizes bounded expansion [14], while Nadara et al initiated an empirical study of algorithmic techniques used to compute the generalized coloring numbers in practice [9]. The characterization also led to new algorithms for bounded expansion; Dvořák gave a constant factor approximation for Dominating Set [3], and Reidl and Sullivan gave an exact FPT algorithm for Subgraph Counting [13].…”

Section: Introductionmentioning

confidence: 99%

Hardness of the Generalized Coloring Numbers

Breen-McKay¹,

Lavallee²,

Sullivan³

2021

Preprint

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The generalized coloring numbers of Kierstead and Yang [7] offer an algorithmically useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we show that it is NP-hard to compute the weak 2-coloring number (answering an open question of Grohe et al. [5]). We then complete the picture by proving that the r-coloring number is also NP-hard to compute for all r ≥ 2. Finally, we give an approximation algorithm for the r-coloring number which improves both the runtime and approximation factor of the existing approach of Dvořák [3]. Our algorithm greedily orders vertices with small enough i-reach for every i ≤ r and achieves an O(C r−1 k r−1 )-approximation, where C j is the jth Catalan number.

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Characterising bounded expansion by neighbourhood complexity

Cited by 22 publications

References 28 publications

On the number of types in sparse graphs

On the number of types in sparse graphs

Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)

Hardness of the Generalized Coloring Numbers

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