61 research outputs found
On Universal Point Sets for Planar Graphs
A set P of points in R^2 is n-universal, if every planar graph on n vertices
admits a plane straight-line embedding on P. Answering a question by Kobourov,
we show that there is no n-universal point set of size n, for any n>=15.
Conversely, we use a computer program to show that there exist universal point
sets for all n<=10 and to enumerate all corresponding order types. Finally, we
describe a collection G of 7'393 planar graphs on 35 vertices that do not admit
a simultaneous geometric embedding without mapping, that is, no set of 35
points in the plane supports a plane straight-line embedding of all graphs in
G.Comment: Fixed incorrect numbers of universal point sets in the last par
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
Superpatterns and Universal Point Sets
An old open problem in graph drawing asks for the size of a universal point
set, a set of points that can be used as vertices for straight-line drawings of
all n-vertex planar graphs. We connect this problem to the theory of
permutation patterns, where another open problem concerns the size of
superpatterns, permutations that contain all patterns of a given size. We
generalize superpatterns to classes of permutations determined by forbidden
patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the
213-avoiding permutations, half the size of known superpatterns for
unconstrained permutations. We use our superpatterns to construct universal
point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16
factor. We prove that every proper subclass of the 213-avoiding permutations
has superpatterns of size O(n log^O(1) n), which we use to prove that the
planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA
An exponential bound for simultaneous embeddings of planar graphs
We show that there are planar graphs on vertices
which do not admit a simultaneous straight-line embedding on any -point set
in the plane. In particular, this improves the best known bound
significantly.Comment: 4 figures, 8 page
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