29,215 research outputs found
The Complexity of Simultaneous Geometric Graph Embedding
Given a collection of planar graphs on the same set of
vertices, the simultaneous geometric embedding (with mapping) problem, or
simply -SGE, is to find a set of points in the plane and a bijection
such that the induced straight-line drawings of
under are all plane.
This problem is polynomial-time equivalent to weak rectilinear realizability
of abstract topological graphs, which Kyn\v{c}l (doi:10.1007/s00454-010-9320-x)
proved to be complete for , the existential theory of the
reals. Hence the problem -SGE is polynomial-time equivalent to several other
problems in computational geometry, such as recognizing intersection graphs of
line segments or finding the rectilinear crossing number of a graph.
We give an elementary reduction from the pseudoline stretchability problem to
-SGE, with the property that both numbers and are linear in the
number of pseudolines. This implies not only the -hardness
result, but also a lower bound on the minimum size of a
grid on which any such simultaneous embedding can be drawn. This bound is
tight. Hence there exists such collections of graphs that can be simultaneously
embedded, but every simultaneous drawing requires an exponential number of bits
per coordinates. The best value that can be extracted from Kyn\v{c}l's proof is
only
Snapping Graph Drawings to the Grid Optimally
In geographic information systems and in the production of digital maps for
small devices with restricted computational resources one often wants to round
coordinates to a rougher grid. This removes unnecessary detail and reduces
space consumption as well as computation time. This process is called snapping
to the grid and has been investigated thoroughly from a computational-geometry
perspective. In this paper we investigate the same problem for given drawings
of planar graphs under the restriction that their combinatorial embedding must
be kept and edges are drawn straight-line. We show that the problem is NP-hard
for several objectives and provide an integer linear programming formulation.
Given a plane graph G and a positive integer w, our ILP can also be used to
draw G straight-line on a grid of width w and minimum height (if possible).Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
On a Tree and a Path with no Geometric Simultaneous Embedding
Two graphs and admit a geometric simultaneous
embedding if there exists a set of points P and a bijection M: P -> V that
induce planar straight-line embeddings both for and for . While it
is known that two caterpillars always admit a geometric simultaneous embedding
and that two trees not always admit one, the question about a tree and a path
is still open and is often regarded as the most prominent open problem in this
area. We answer this question in the negative by providing a counterexample.
Additionally, since the counterexample uses disjoint edge sets for the two
graphs, we also negatively answer another open question, that is, whether it is
possible to simultaneously embed two edge-disjoint trees. As a final result, we
study the same problem when some constraints on the tree are imposed. Namely,
we show that a tree of depth 2 and a path always admit a geometric simultaneous
embedding. In fact, such a strong constraint is not so far from closing the gap
with the instances not admitting any solution, as the tree used in our
counterexample has depth 4.Comment: 42 pages, 33 figure
Improved Algorithms for the Point-Set Embeddability problem for Plane 3-Trees
In the point set embeddability problem, we are given a plane graph with
vertices and a point set with points. Now the goal is to answer the
question whether there exists a straight-line drawing of such that each
vertex is represented as a distinct point of as well as to provide an
embedding if one does exist. Recently, in \cite{DBLP:conf/gd/NishatMR10}, a
complete characterization for this problem on a special class of graphs known
as the plane 3-trees was presented along with an efficient algorithm to solve
the problem. In this paper, we use the same characterization to devise an
improved algorithm for the same problem. Much of the efficiency we achieve
comes from clever uses of the triangular range search technique. We also study
a generalized version of the problem and present improved algorithms for this
version of the problem as well
A logarithmic bound for simultaneous embeddings of planar graphs
A set of planar graphs on the same number of vertices is
called simultaneously embeddable if there exists a set of points in the
plane such that every graph admits a (crossing-free)
straight-line embedding with vertices placed at points of . A well-known
open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu,
Kobourov, Lubiw and Mitchell, asks whether for some there exists a set
consisting of two planar graphs on vertices that are not
simultaneously embeddable. While this remains widely open, we give a short
proof that for every and sufficiently large there exists a
collection of at most planar graphs on vertices
which cannot be simultaneously embedded. This significantly improves the
previous exponential bound of for the same problem which
was recently established by Goenka, Semnani and Yip.Comment: note, 5 page
Planar embeddability of the vertices of a graph using a fixed point set is NP-hard
Let G = (V, E) be a graph with n vertices and let P be a set of n points in the plane. We show that deciding whether there is a planar straight-line embedding of G such that the vertices V are embedded onto the points P
is NP-complete, even when G is 2-connected and 2-outerplanar. This settles an open problem posed in [P. Bose. On embedding an outer-planar graph in a point set. Comput. Geom. Theory Appl., 23:303-312, November 2002. A preliminary version appeared in Graph Drawing (Proc. GD ’97), LNCS 1353, pg. 25-36, F. Brandenberg, D. Eppstein, M.T. Goodrich, S.G. Kobourov, G. Liotta, and P. Mutzel. Selected open problems in graph drawing. In Graph Drawing (Proc. GD’03), LNCS, 2003. To appear y M. Kaufmann and R. Wiese. Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications, 6(1):115–129, 2002. A preliminary version appeared in Graph Drawing (Proc. GD ’99), LNCS 1731, pg. 165–174].Cornelis Lely Stichtin
New Algorithm for Drawings of 3-Planar Graphs
Graphs arise in a natural way in many applications, together with the need to be drawn. Except for very small instances, drawing a graph by hand becomes a very complex task, which must be performed by automatic tools. The field of graph drawing is concerned with finding algorithms to draw graph in an aesthetically pleasant way, based upon a certain number of aesthetic criteria that define what a good drawing, (synonyms: diagrams, pictures, layouts), of a graph should be. This problem can be found in many such as in the computer networks, data networks, class inter-relationship diagrams in object oriented databases and object oriented programs, visual programming interfaces, database design systems, software engineering…etc. Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edge-intersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper we introduce a new algorithms that finds an embedding of 3-planar graph. Keywords: 3- Planar Graph; Graph Drawing; drawing on grid
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