2,287 research outputs found
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
Heuristic algorithms for the min-max edge 2-coloring problem
In multi-channel Wireless Mesh Networks (WMN), each node is able to use
multiple non-overlapping frequency channels. Raniwala et al. (MC2R 2004,
INFOCOM 2005) propose and study several such architectures in which a computer
can have multiple network interface cards. These architectures are modeled as a
graph problem named \emph{maximum edge -coloring} and studied in several
papers by Feng et. al (TAMC 2007), Adamaszek and Popa (ISAAC 2010, JDA 2016).
Later on Larjomaa and Popa (IWOCA 2014, JGAA 2015) define and study an
alternative variant, named the \emph{min-max edge -coloring}.
The above mentioned graph problems, namely the maximum edge -coloring and
the min-max edge -coloring are studied mainly from the theoretical
perspective. In this paper, we study the min-max edge 2-coloring problem from a
practical perspective. More precisely, we introduce, implement and test four
heuristic approximation algorithms for the min-max edge -coloring problem.
These algorithms are based on a \emph{Breadth First Search} (BFS)-based
heuristic and on \emph{local search} methods like basic \emph{hill climbing},
\emph{simulated annealing} and \emph{tabu search} techniques, respectively.
Although several algorithms for particular graph classes were proposed by
Larjomaa and Popa (e.g., trees, planar graphs, cliques, bi-cliques,
hypergraphs), we design the first algorithms for general graphs.
We study and compare the running data for all algorithms on Unit Disk Graphs,
as well as some graphs from the DIMACS vertex coloring benchmark dataset.Comment: This is a post-peer-review, pre-copyedit version of an article
published in International Computing and Combinatorics Conference
(COCOON'18). The final authenticated version is available online at:
http://www.doi.org/10.1007/978-3-319-94776-1_5
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
Constructing Adjacency Arrays from Incidence Arrays
Graph construction, a fundamental operation in a data processing pipeline, is
typically done by multiplying the incidence array representations of a graph,
and , to produce an adjacency
array of the graph, , that can be processed with a variety of
algorithms. This paper provides the mathematical criteria to determine if the
product
will have the required structure of the adjacency array of the graph. The
values in the resulting adjacency array are determined by the corresponding
addition and multiplication operations used to perform the
array multiplication. Illustrations of the various results possible from
different and operations are provided using a small
collection of popular music metadata.Comment: 8 pages, 5 figures, accepted to IEEE IPDPS 2017 Workshop on Graph
Algorithm Building Block
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
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