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 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 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 A New Perspective on FO Model Checking of Dense Graph Classes

2020 ◽  
Vol 21 (4) ◽  
pp. 1-23
Author(s):  
Jakub Gajarský ◽  
Petr Hliněný ◽  
Jan Obdržálek ◽  
Daniel Lokshtanov ◽  
M. S. Ramanujan
Keyword(s):  
Model Checking ◽  
Dense Graph ◽  
Graph Classes ◽  
New Perspective

scholarly journals A counterexample to the Bollobás–Riordan conjectures on sparse graph limits

2021 ◽  
pp. 1-4
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Jonathan Tidor ◽  
Yufei Zhao
Keyword(s):  
Sparse Graph ◽  
Sparse Graphs ◽  
Graph Limits

Abstract Bollobás and Riordan, in their paper ‘Metrics for sparse graphs’, proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom C4-counts), but with no limit expressible as a kernel.

scholarly journals Clustering Powers of Sparse Graphs

10.37236/9417 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Jaroslav Nešetřil ◽  
Patrice Ossona de Mendez ◽  
Michał Pilipczuk ◽  
Xuding Zhu
Keyword(s):  
Approximation Algorithms ◽  
Chromatic Number ◽  
Clique Number ◽  
Sparse Graph ◽  
Sparse Graphs ◽  
Single Vertex ◽  
Graph Theoretic ◽  
Bounded Expansion ◽  
Efficient Approximation

We prove that if $G$ is a sparse graph — it belongs to a fixed class of bounded expansion $\mathcal{C}$ — and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these clique to a single vertex again yields a sparse graph. This result has several graph-theoretic and algorithmic consequences for powers of sparse graphs, including bounds on their subchromatic number and efficient approximation algorithms for the chromatic number and the clique number.

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
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