101 research outputs found
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
New Parameters for Beyond-Planar Graphs
Parameters for graphs appear frequently throughout the history of research in this field. They represent very important measures for the properties of graphs and graph drawings, and are often a main criterion for their classification and their aesthetic perception. In this direction, we provide new results for the following graph parameters:
– The segment complexity of trees;
– the membership of graphs of bounded vertex degree to certain graph classes;
– the maximal complete and complete bipartite graphs contained in certain graph classes beyond-planarity;
– the crossing number of graphs;
– edge densities for outer-gap-planar graphs and for bipartite gap-planar graphs with certain properties;
– edge densities and inclusion relationships for 2-layer graphs, as well as characterizations for complete bipartite graphs in the 2-layer setting
Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends
We study the following classes of beyond-planar graphs: 1-planar, IC-planar,
and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar,
and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every
edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs
of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two
pairs of crossing edges share two vertices. We study the relations of these
beyond-planar graph classes (beyond-planar graphs is a collective term for the
primary attempts to generalize the planar graphs) to right-angle crossing (RAC)
graphs that admit compact drawings on the grid with few bends. We present four
drawing algorithms that preserve the given embeddings. First, we show that
every -vertex NIC-planar graph admits a NIC-planar RAC drawing with at most
one bend per edge on a grid of size . Then, we show that
every -vertex 1-planar graph admits a 1-planar RAC drawing with at most two
bends per edge on a grid of size . Finally, we make two
known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at
most one bend per edge and for drawing IC-planar RAC graphs straight-line
Shallow Minors, Graph Products and Beyond Planar Graphs
The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek,
Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a
subgraph of the strong product of a graph with bounded treewidth and a path.
This result has been the key tool to resolve important open problems regarding
queue layouts, nonrepetitive colourings, centered colourings, and adjacency
labelling schemes. In this paper, we extend this line of research by utilizing
shallow minors to prove analogous product structure theorems for several beyond
planar graph classes. The key observation that drives our work is that many
beyond planar graphs can be described as a shallow minor of the strong product
of a planar graph with a small complete graph. In particular, we show that
powers of planar graphs, -planar, -cluster planar, fan-planar and
-fan-bundle planar graphs have such a shallow-minor structure. Using a
combination of old and new results, we deduce that these classes have bounded
queue-number, bounded nonrepetitive chromatic number, polynomial -centred
chromatic numbers, linear strong colouring numbers, and cubic weak colouring
numbers. In addition, we show that -gap planar graphs have at least
exponential local treewidth and, as a consequence, cannot be described as a
subgraph of the strong product of a graph with bounded treewidth and a path
Macroscopic network circulation for planar graphs
The analysis of networks, aimed at suitably defined functionality, often
focuses on partitions into subnetworks that capture desired features. Chief
among the relevant concepts is a 2-partition, that underlies the classical
Cheeger inequality, and highlights a constriction (bottleneck) that limits
accessibility between the respective parts of the network. In a similar spirit,
the purpose of the present work is to introduce a new concept of maximal global
circulation and to explore 3-partitions that expose this type of macroscopic
feature of networks. Herein, graph circulation is motivated by transportation
networks and probabilistic flows (Markov chains) on graphs. Our goal is to
quantify the large-scale imbalance of network flows and delineate key parts
that mediate such global features. While we introduce and propose these notions
in a general setting, in this paper, we only work out the case of planar
graphs. We explain that a scalar potential can be identified to encapsulate the
concept of circulation, quite similarly as in the case of the curl of planar
vector fields. Beyond planar graphs, in the general case, the problem to
determine global circulation remains at present a combinatorial problem
Drawing Graphs with Circular Arcs and Right-Angle Crossings
In a RAC drawing of a graph, vertices are represented by points in the plane,
adjacent vertices are connected by line segments, and crossings must form right
angles. Graphs that admit such drawings are RAC graphs. RAC graphs are
beyond-planar graphs and have been studied extensively. In particular, it is
known that a RAC graph with n vertices has at most 4n - 10 edges.
We introduce a superclass of RAC graphs, which we call arc-RAC graphs. A
graph is arc-RAC if it admits a drawing where edges are represented by circular
arcs and crossings form right angles. We provide a Tur\'an-type result showing
that an arc-RAC graph with n vertices has at most 14n - 12 edges and that there
are n-vertex arc-RAC graphs with 4.5n - o(n) edges
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