5,017 research outputs found
Essential Constraints of Edge-Constrained Proximity Graphs
Given a plane forest of points, we find the minimum
set of edges such that the edge-constrained minimum spanning
tree over the set of vertices and the set of constraints contains .
We present an -time algorithm that solves this problem. We
generalize this to other proximity graphs in the constraint setting, such as
the relative neighbourhood graph, Gabriel graph, -skeleton and Delaunay
triangulation. We present an algorithm that identifies the minimum set
of edges of a given plane graph such that for , where is the
constraint -skeleton over the set of vertices and the set of
constraints. The running time of our algorithm is , provided that the
constrained Delaunay triangulation of is given.Comment: 24 pages, 22 figures. A preliminary version of this paper appeared in
the Proceedings of 27th International Workshop, IWOCA 2016, Helsinki,
Finland. It was published by Springer in the Lecture Notes in Computer
Science (LNCS) serie
Obstacle Numbers of Planar Graphs
Given finitely many connected polygonal obstacles in the
plane and a set of points in general position and not in any obstacle, the
{\em visibility graph} of with obstacles is the (geometric)
graph with vertex set , where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph
is the smallest integer such that is the visibility graph of a set of
points with obstacles. If is planar, we define the planar obstacle
number of by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of ). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order is
, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least is .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
L-Visibility Drawings of IC-planar Graphs
An IC-plane graph is a topological graph where every edge is crossed at most
once and no two crossed edges share a vertex. We show that every IC-plane graph
has a visibility drawing where every vertex is an L-shape, and every edge is
either a horizontal or vertical segment. As a byproduct of our drawing
technique, we prove that an IC-plane graph has a RAC drawing in quadratic area
with at most two bends per edge
Grid Representations and the Chromatic Number
A grid drawing of a graph maps vertices to grid points and edges to line
segments that avoid grid points representing other vertices. We show that there
is a number of grid points that some line segment of an arbitrary grid drawing
must intersect. This number is closely connected to the chromatic number.
Second, we study how many columns we need to draw a graph in the grid,
introducing some new \NP-complete problems. Finally, we show that any planar
graph has a planar grid drawing where every line segment contains exactly two
grid points. This result proves conjectures asked by David Flores-Pe\~naloza
and Francisco Javier Zaragoza Martinez.Comment: 22 pages, 8 figure
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
Eigenvector localization as a tool to study small communities in online social networks
We present and discuss a mathematical procedure for identification of small
"communities" or segments within large bipartite networks. The procedure is
based on spectral analysis of the matrix encoding network structure. The
principal tool here is localization of eigenvectors of the matrix, by means of
which the relevant network segments become visible. We exemplified our approach
by analyzing the data related to product reviewing on Amazon.com. We found
several segments, a kind of hybrid communities of densely interlinked reviewers
and products, which we were able to meaningfully interpret in terms of the type
and thematic categorization of reviewed items. The method provides a
complementary approach to other ways of community detection, typically aiming
at identification of large network modules
The Complexity of Drawing Graphs on Few Lines and Few Planes
It is well known that any graph admits a crossing-free straight-line drawing
in and that any planar graph admits the same even in
. For a graph and , let denote
the minimum number of lines in that together can cover all edges
of a drawing of . For , must be planar. We investigate the
complexity of computing these parameters and obtain the following hardness and
algorithmic results.
- For , we prove that deciding whether for a
given graph and integer is -complete.
- Since , deciding is NP-hard for . On the positive side, we show that the problem
is fixed-parameter tractable with respect to .
- Since , both and
are computable in polynomial space. On the negative side, we show
that drawings that are optimal with respect to or
sometimes require irrational coordinates.
- Let be the minimum number of planes in needed
to cover a straight-line drawing of a graph . We prove that deciding whether
is NP-hard for any fixed . Hence, the problem is
not fixed-parameter tractable with respect to unless
BIGMAC : breaking inaccurate genomes and merging assembled contigs for long read metagenomic assembly.
BackgroundThe problem of de-novo assembly for metagenomes using only long reads is gaining attention. We study whether post-processing metagenomic assemblies with the original input long reads can result in quality improvement. Previous approaches have focused on pre-processing reads and optimizing assemblers. BIGMAC takes an alternative perspective to focus on the post-processing step.ResultsUsing both the assembled contigs and original long reads as input, BIGMAC first breaks the contigs at potentially mis-assembled locations and subsequently scaffolds contigs. Our experiments on metagenomes assembled from long reads show that BIGMAC can improve assembly quality by reducing the number of mis-assemblies while maintaining or increasing N50 and N75. Moreover, BIGMAC shows the largest N75 to number of mis-assemblies ratio on all tested datasets when compared to other post-processing tools.ConclusionsBIGMAC demonstrates the effectiveness of the post-processing approach in improving the quality of metagenomic assemblies
Bounded-Angle Minimum Spanning Trees
Motivated by the connectivity problem in wireless networks with directional antennas, we study bounded-angle spanning trees. Let P be a set of points in the plane and let ? be an angle. An ?-ST of P is a spanning tree of the complete Euclidean graph on P with the property that all edges incident to each point p ? P lie in a wedge of angle ? centered at p. We study the following closely related problems for ? = 120 degrees (however, our approximation ratios hold for any ? ? 120 degrees).
1) The ?-minimum spanning tree problem asks for an ?-ST of minimum sum of edge lengths. Among many interesting results, Aschner and Katz (ICALP 2014) proved the NP-hardness of this problem and presented a 6-approximation algorithm. Their algorithm finds an ?-ST of length at most 6 times the length of the minimum spanning tree (MST). By adopting a somewhat similar approach and using different proof techniques we improve this ratio to 16/3.
2) To examine what is possible with non-uniform wedge angles, we define an ??-ST to be a spanning tree with the property that incident edges to all points lie in wedges of average angle ?. We present an algorithm to find an ??-ST whose largest edge-length and sum of edge lengths are at most 2 and 1.5 times (respectively) those of the MST. These ratios are better than any achievable when all wedges have angle ?. Our algorithm runs in linear time after computing the MST
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