9,859 research outputs found
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
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
Simultaneous Orthogonal Planarity
We introduce and study the problem: Given planar
graphs each with maximum degree 4 and the same vertex set, do they admit an
OrthoSEFE, that is, is there an assignment of the vertices to grid points and
of the edges to paths on the grid such that the same edges in distinct graphs
are assigned the same path and such that the assignment induces a planar
orthogonal drawing of each of the graphs?
We show that the problem is NP-complete for even if the shared
graph is a Hamiltonian cycle and has sunflower intersection and for
even if the shared graph consists of a cycle and of isolated vertices. Whereas
the problem is polynomial-time solvable for when the union graph has
maximum degree five and the shared graph is biconnected. Further, when the
shared graph is biconnected and has sunflower intersection, we show that every
positive instance has an OrthoSEFE with at most three bends per edge.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
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