scholarly journals Efficient union-find for planar graphs and other sparse graph classes

1998 ◽  
Vol 203 (1) ◽  
pp. 123-141 ◽  
Author(s):  
Jens Gustedt
Keyword(s):  
Planar Graphs ◽  
Sparse Graph ◽  
Graph Classes

scholarly journals Subcritical Graph Classes Containing All Planar Graphs

2018 ◽  
Vol 27 (5) ◽  
pp. 763-773
Author(s):  
AGELOS GEORGAKOPOULOS ◽  
STEPHAN WAGNER
Keyword(s):  
Planar Graphs ◽  
Graph Classes

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.

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 Adjacency Labelling for Planar Graphs (and Beyond)

2021 ◽  
Vol 68 (6) ◽  
pp. 1-33
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Cyril Gavoille ◽  
Gwenaël Joret ◽  
Piotr Micek ◽  
...  
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Order Term ◽  
Optimal Result ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Graph Classes ◽  
Bounded Degree Graphs ◽  
Minor Free Graphs ◽  
Labelling Scheme

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an n -vertex planar graph G is assigned a (1 + o(1)) log 2 n -bit label and the labels of two vertices u and v are sufficient to determine if uv is an edge of G . This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every positive integer n , there exists a graph U n with n 1+o(1) vertices such that every n -vertex planar graph is an induced subgraph of U n . These results generalize to a number of other graph classes, including bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and k -planar graphs.

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.

BIPARTITE PERMUTATION GRAPHS ARE RECONSTRUCTIBLE

2012 ◽  
Vol 04 (03) ◽  
pp. 1250039
Author(s):  
MASASHI KIYOMI ◽  
TOSHIKI SAITOH ◽  
RYUHEI UEHARA
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Unit Interval ◽  
Regular Graphs ◽  
Pendant Vertex ◽  
Permutation Graphs ◽  
Graph Classes ◽  
Graph Reconstruction ◽  
Bipartite Permutation Graphs ◽  
Disconnected Graphs

The graph reconstruction conjecture is a long-standing open problem in graph theory. The conjecture has been verified for all graphs with at most 11 vertices. Further, the conjecture has been verified for regular graphs, trees, disconnected graphs, unit interval graphs, separable graphs with no pendant vertex, outer-planar graphs, and unicyclic graphs. We extend the list of graph classes for which the conjecture holds. We give a proof that bipartite permutation graphs are reconstructible.

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 Improved Bounds for Centered Colorings

2021 ◽  
Author(s):  
Michał Dębski ◽  
Piotr Micek ◽  
Felix Schröder ◽  
Stefan Felsner
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Structure Theorem ◽  
Upper Bounds ◽  
Log P ◽  
Connected Subgraph ◽  
Bounded Degree ◽  
Graph Classes ◽  
Proof Techniques ◽  
Bounded Degree Graphs

A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $O(p^3\log p)$ colors where the previous bound was $O(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $O(p)$ colors while it was conjectured that they may require exponential number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring. This bound matches the upper bound; (5) there are planar graphs that require $\Omega(p^2\log p)$ colors in any $p$-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth $3$. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

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