1,349 research outputs found
On Visibility Representations of Non-planar Graphs
A rectangle visibility representation (RVR) of a graph consists of an
assignment of axis-aligned rectangles to vertices such that for every edge
there exists a horizontal or vertical line of sight between the rectangles
assigned to its endpoints. Testing whether a graph has an RVR is known to be
NP-hard. In this paper, we study the problem of finding an RVR under the
assumption that an embedding in the plane of the input graph is fixed and we
are looking for an RVR that reflects this embedding. We show that in this case
the problem can be solved in polynomial time for general embedded graphs and in
linear time for 1-plane graphs (i.e., embedded graphs having at most one
crossing per edge). The linear time algorithm uses a precise list of forbidden
configurations, which extends the set known for straight-line drawings of
1-plane graphs. These forbidden configurations can be tested for in linear
time, and so in linear time we can test whether a 1-plane graph has an RVR and
either compute such a representation or report a negative witness. Finally, we
discuss some extensions of our study to the case when the embedding is not
fixed but the RVR can have at most one crossing per edge
A Note on Flips in Diagonal Rectangulations
Rectangulations are partitions of a square into axis-aligned rectangles. A
number of results provide bijections between combinatorial equivalence classes
of rectangulations and families of pattern-avoiding permutations. Other results
deal with local changes involving a single edge of a rectangulation, referred
to as flips, edge rotations, or edge pivoting. Such operations induce a graph
on equivalence classes of rectangulations, related to so-called flip graphs on
triangulations and other families of geometric partitions. In this note, we
consider a family of flip operations on the equivalence classes of diagonal
rectangulations, and their interpretation as transpositions in the associated
Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This
complements results from Law and Reading (JCTA, 2012) and provides a complete
characterization of flip operations on diagonal rectangulations, in both
geometric and combinatorial terms
Lattice Topological Field Theory on Non-Orientable Surfaces
The lattice definition of the two-dimensional topological quantum field
theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is
generalized to arbitrary (not necessarily orientable) compact surfaces. It is
shown that there is a one-to-one correspondence between real associative
-algebras and the topological state sum invariants defined on such surfaces.
The partition and -point functions on all two-dimensional surfaces
(connected sums of the Klein bottle or projective plane and -tori) are
defined and computed for arbitrary -algebras in general, and for the the
group ring of discrete groups , in particular.Comment: Corrected Latex file, 39 pages, 28 figures available upon reques
Spacetime and Euclidean Geometry
Using only the principle of relativity and Euclidean geometry we show in this
pedagogical article that the square of proper time or length in a
two-dimensional spacetime diagram is proportional to the Euclidean area of the
corresponding causal domain. We use this relation to derive the Minkowski line
element by two geometric proofs of the "spacetime Pythagoras theorem".Comment: 11 pages, 9 figures; for a festschrift honoring Michael P. Ryan; v.2:
References to related work adde
Area-Universal Rectangular Layouts
A rectangular layout is a partition of a rectangle into a finite set of
interior-disjoint rectangles. Rectangular layouts appear in various
applications: as rectangular cartograms in cartography, as floorplans in
building architecture and VLSI design, and as graph drawings. Often areas are
associated with the rectangles of a rectangular layout and it might hence be
desirable if one rectangular layout can represent several area assignments. A
layout is area-universal if any assignment of areas to rectangles can be
realized by a combinatorially equivalent rectangular layout. We identify a
simple necessary and sufficient condition for a rectangular layout to be
area-universal: a rectangular layout is area-universal if and only if it is
one-sided. More generally, given any rectangular layout L and any assignment of
areas to its regions, we show that there can be at most one layout (up to
horizontal and vertical scaling) which is combinatorially equivalent to L and
achieves a given area assignment. We also investigate similar questions for
perimeter assignments. The adjacency requirements for the rectangles of a
rectangular layout can be specified in various ways, most commonly via the dual
graph of the layout. We show how to find an area-universal layout for a given
set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 figure
Singular link Floer homology
We define a grid presentation for singular links i.e. links with a finite
number of rigid transverse double points. Then we use it to generalize link
Floer homology to singular links. Besides the consistency of its definition, we
prove that this homology is acyclic under some conditions which naturally make
its Euler characteristic vanish.Comment: 29 pages, many figure
Metastability in the two-dimensional Ising model with free boundary conditions
We investigate metastability in the two dimensional Ising model in a square
with free boundary conditions at low temperatures. Starting with all spins down
in a small positive magnetic field, we show that the exit from this metastable
phase occurs via the nucleation of a critical droplet in one of the four
corners of the system. We compute the lifetime of the metastable phase
analytically in the limit , and via Monte Carlo simulations at
fixed values of and and find good agreement. This system models the
effects of boundary domains in magnetic storage systems exiting from a
metastable phase when a small external field is applied.Comment: 24 pages, TeX fil
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