Tree Densities in Sparse Graph Classes

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\lfloor k/2\rfloor})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.

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On Ultralimits of Sparse Graph Classes

The Electronic Journal of Combinatorics ◽

10.37236/5519 ◽

2016 ◽

Vol 23 (2) ◽

Author(s):

Michał Pilipczuk ◽

Szymon Toruńczyk

Keyword(s):

Sparse Graph ◽

Sparse Graphs ◽

Dense Graph ◽

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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Anti-Path Cover on Sparse Graph Classes

Electronic Proceedings in Theoretical Computer Science ◽

10.4204/eptcs.233.8 ◽

2016 ◽

Vol 233 ◽

pp. 82-86 ◽

Cited By ~ 1

Author(s):

Pavel Dvořák ◽

Dušan Knop ◽

Tomáš Masařík

Keyword(s):

Sparse Graph ◽

Path Cover ◽

Graph Classes

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Algorithmic Meta Theorems for Sparse Graph Classes

Computer Science - Theory and Applications - Lecture Notes in Computer Science ◽

10.1007/978-3-319-06686-8_2 ◽

2014 ◽

pp. 16-22 ◽

Cited By ~ 2

Author(s):

Martin Grohe

Keyword(s):

Sparse Graph ◽

Graph Classes

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Efficient Union-Find for planar graphs and other sparse graph classes

Graph-Theoretic Concepts in Computer Science - Lecture Notes in Computer Science ◽

10.1007/3-540-62559-3_16 ◽

1997 ◽

pp. 181-195 ◽

Cited By ~ 1

Author(s):

Jens Gustedt

Keyword(s):

Planar Graphs ◽

Sparse Graph ◽

Graph Classes

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Kernelization Using Structural Parameters on Sparse Graph Classes

Lecture Notes in Computer Science - Algorithms – ESA 2013 ◽

10.1007/978-3-642-40450-4_45 ◽

2013 ◽

pp. 529-540 ◽

Cited By ~ 10

Author(s):

Jakub Gajarský ◽

Petr Hliněný ◽

Jan Obdržálek ◽

Sebastian Ordyniak ◽

Felix Reidl ◽

...

Keyword(s):

Structural Parameters ◽

Sparse Graph ◽

Graph Classes

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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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