Rainbow independent sets on dense graph classes

The notion of nowhere denseness is one of the central concepts of the recently developed theory of sparse graphs. We study the properties of nowhere dense graph classes by investigating appropriate limit objects defined using the ultraproduct construction. It appears that different equivalent definitions of nowhere denseness, for example via quasi-wideness or the splitter game, correspond to natural notions for the limit objects that are conceptually simpler and allow for less technically involved reasonings.

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Nowhere Dense Graph Classes and Dimension

COMBINATORICA ◽

10.1007/s00493-019-3892-8 ◽

2019 ◽

Vol 39 (5) ◽

pp. 1055-1079 ◽

Cited By ~ 1

Author(s):

Gwenaël Joret ◽

Piotr Micek ◽

Patrice Ossona de Mendez ◽

Veit Wiechert

Keyword(s):

Dense Graph ◽

Graph Classes

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A New Perspective on FO Model Checking of Dense Graph Classes

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16 ◽

10.1145/2933575.2935314 ◽

2016 ◽

Cited By ~ 3

Author(s):

J. Gajarský ◽

P. Hliněný ◽

J. Obdržálek ◽

D. Lokshtanov ◽

M. S. Ramanujan

Keyword(s):

Model Checking ◽

Dense Graph ◽

Graph Classes ◽

New Perspective

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Polynomial Kernels and Wideness Properties of Nowhere Dense Graph Classes

ACM Transactions on Algorithms ◽

10.1145/3274652 ◽

2019 ◽

Vol 15 (2) ◽

pp. 1-19

Author(s):

Stephan Kreutzer ◽

Roman Rabinovich ◽

Sebastian Siebertz

Keyword(s):

Dense Graph ◽

Graph Classes ◽

Polynomial Kernels

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Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes

The Electronic Journal of Combinatorics ◽

10.37236/8683 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Michael Drmota ◽

Lander Ramos ◽

Clément Requilé ◽

Juanjo Rué

Keyword(s):

Functional Equations ◽

Independent Sets ◽

Singularity Analysis ◽

Graph Classes ◽

Several Variables ◽

In Series ◽

Quantitative Results ◽

Related Graph ◽

Implicit Systems ◽

Maximal Independent Sets

The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].

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Polynomial Kernels and Wideness Properties of Nowhere Dense Graph Classes

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms ◽

10.1137/1.9781611974782.100 ◽

2017 ◽

Cited By ~ 1

Author(s):

Stephan Kreutzer ◽

Roman Rabinovich ◽

Sebastian Siebertz

Keyword(s):

Dense Graph ◽

Graph Classes ◽

Polynomial Kernels

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[Extendable Functions with a Dense Graph]

Real Analysis Exchange ◽

10.2307/44153671 ◽

1990 ◽

Vol 16 (1) ◽

pp. 44

Author(s):

Gibson

Keyword(s):

Dense Graph

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A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽

10.3390/a14060164 ◽

2021 ◽

Vol 14 (6) ◽

pp. 164

Author(s):

Tobias Rupp ◽

Stefan Funke

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Real World ◽

Road Network ◽

Upper And Lower Bounds ◽

Bounded Degree ◽

Shortest Path Routing ◽

Graph Classes ◽

Speed Up ◽

To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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