293 research outputs found
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Convex-Arc Drawings of Pseudolines
A weak pseudoline arrangement is a topological generalization of a line
arrangement, consisting of curves topologically equivalent to lines that cross
each other at most once. We consider arrangements that are outerplanar---each
crossing is incident to an unbounded face---and simple---each crossing point is
the crossing of only two curves. We show that these arrangements can be
represented by chords of a circle, by convex polygonal chains with only two
bends, or by hyperbolic lines. Simple but non-outerplanar arrangements
(non-weak) can be represented by convex polygonal chains or convex smooth
curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was
made as a poster at the 21st International Symposium on Graph Drawing,
Bordeaux, France, September 2013, and published in Lecture Notes in Computer
Science 8242, Springer, 2013, pp. 522--52
Unit Grid Intersection Graphs: Recognition and Properties
It has been known since 1991 that the problem of recognizing grid
intersection graphs is NP-complete. Here we use a modified argument of the
above result to show that even if we restrict to the class of unit grid
intersection graphs (UGIGs), the recognition remains hard, as well as for all
graph classes contained inbetween. The result holds even when considering only
graphs with arbitrarily large girth. Furthermore, we ask the question of
representing UGIGs on grids of minimal size. We show that the UGIGs that can be
represented in a square of side length 1+epsilon, for a positive epsilon no
greater than 1, are exactly the orthogonal ray graphs, and that there exist
families of trees that need an arbitrarily large grid
Convex-Arc Drawings of Pseudolines ⋆
Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have distance-preserving embedding
Fixed parameter tractability of crossing minimization of almost-trees
We investigate exact crossing minimization for graphs that differ from trees
by a small number of additional edges, for several variants of the crossing
minimization problem. In particular, we provide fixed parameter tractable
algorithms for the 1-page book crossing number, the 2-page book crossing
number, and the minimum number of crossed edges in 1-page and 2-page book
drawings.Comment: Graph Drawing 201
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
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
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