15,375 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
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
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
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
Universal Communication, Universal Graphs, and Graph Labeling
We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family ?, and inputs x and y. Alice and Bob use shared randomness to send a message to a third party who cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ?(x), ?(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ? k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ? 2 in modular lattices (a superset of distributive lattices) has super-constant ?(n^{1/4}) communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an O(k) protocol for deciding dist(x,y) ? k and planar graphs have an O(1) protocol for dist(x,y) ? 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs
Dynamic and Multi-functional Labeling Schemes
We investigate labeling schemes supporting adjacency, ancestry, sibling, and
connectivity queries in forests. In the course of more than 20 years, the
existence of labeling schemes supporting each of these
functions was proven, with the most recent being ancestry [Fraigniaud and
Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower
or upper bounds of or
respectively. Notably an upper bound of for
adjacency+siblings and a lower bound of for each of the
functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We
improve the constants hidden in the -notation. In particular we show a lower bound for connectivity+ancestry and
connectivity+siblings, as well as an upper bound of for connectivity+adjacency+siblings by altering existing
methods.
In the context of dynamic labeling schemes it is known that ancestry requires
bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower
bounds on the label size for adjacency, siblings, and connectivity of
bits, and to support all three functions. There exist efficient
adjacency labeling schemes for planar, bounded treewidth, bounded arboricity
and interval graphs. In a dynamic setting, we show a lower bound of
for each of those families.Comment: 17 pages, 5 figure
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