Algorithmic Meta Theorems for Sparse Graph Classes

2014 ◽  
pp. 16-22 ◽  
Author(s):  
Martin Grohe
Keyword(s):  
Sparse Graph ◽  
Graph Classes

scholarly journals Kernelization using structural parameters on sparse graph classes

2017 ◽  
Vol 84 ◽  
pp. 219-242 ◽  
Author(s):  
Jakub Gajarský ◽  
Petr Hliněný ◽  
Jan Obdržálek ◽  
Sebastian Ordyniak ◽  
Felix Reidl ◽  
...  
Keyword(s):  
Structural Parameters ◽  
Sparse Graph ◽  
Graph Classes

scholarly journals Fixed-Parameter Tractable Distances to Sparse Graph Classes

Algorithmica ◽  
2016 ◽  
Vol 79 (1) ◽  
pp. 139-158 ◽  
Author(s):  
Jannis Bulian ◽  
Anuj Dawar
Keyword(s):  
Sparse Graph ◽  
Fixed Parameter Tractable ◽  
Graph Classes ◽  
Fixed Parameter

scholarly journals The Degree-Diameter Problem for Sparse Graph Classes

10.37236/4313 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Guillermo Pineda-Villavicencio ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Sparse Graphs ◽  
Graph Classes ◽  
Minor Free Graphs

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.

scholarly journals On Ultralimits of Sparse Graph Classes

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.

scholarly journals Anti-Path Cover on Sparse Graph Classes

2016 ◽  
Vol 233 ◽  
pp. 82-86 ◽  
Author(s):  
Pavel Dvořák ◽  
Dušan Knop ◽  
Tomáš Masařík
Keyword(s):  
Sparse Graph ◽  
Path Cover ◽  
Graph Classes

scholarly journals Tree Densities in Sparse Graph Classes

2021 ◽  
pp. 1-20
Author(s):  
Tony Huynh ◽  
David R. Wood
Keyword(s):  
Sparse Graph ◽  
Graph Classes

scholarly journals Kernelization Using Structural Parameters on Sparse Graph Classes

2013 ◽  
pp. 529-540 ◽  
Author(s):  
Jakub Gajarský ◽  
Petr Hliněný ◽  
Jan Obdržálek ◽  
Sebastian Ordyniak ◽  
Felix Reidl ◽  
...  
Keyword(s):  
Structural Parameters ◽  
Sparse Graph ◽  
Graph Classes

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

Algorithms ◽  
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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