1,029 research outputs found
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense
Recent works have shown that random triangulations decorated by critical
() Bernoulli site percolation converge in the scaling limit to a
-Liouville quantum gravity (LQG) surface (equivalently, a Brownian
surface) decorated by SLE in two different ways:
1. The triangulation, viewed as a curve-decorated metric measure space
equipped with its graph distance, the counting measure on vertices, and a
single percolation interface converges with respect to a version of the
Gromov-Hausdorff topology.
2. There is a bijective encoding of the site-percolated triangulation by
means of a two-dimensional random walk, and this walk converges to the
correlated two-dimensional Brownian motion which encodes SLE-decorated
-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield
(2014); this is sometimes called .
We prove that one in fact has convergence in both of these
two senses simultaneously. We also improve the metric convergence result by
showing that the map decorated by the full collection of percolation interfaces
(rather than just a single interface) converges to -LQG decorated
by CLE in the metric space sense.
This is the first work to prove simultaneous convergence of any random planar
map model in the metric and peanosphere senses. Moreover, this work is an
important step in an ongoing program to prove that random triangulations
embedded into via the so-called converge
to -LQG.Comment: 55 pages; 13 Figures. Minor revision according to a referee report.
Accepted for publication at EJ
Geometric auxetics
We formulate a mathematical theory of auxetic behavior based on one-parameter
deformations of periodic frameworks. Our approach is purely geometric, relies
on the evolution of the periodicity lattice and works in any dimension. We
demonstrate its usefulness by predicting or recognizing, without experiment,
computer simulations or numerical approximations, the auxetic capabilities of
several well-known structures available in the literature. We propose new
principles of auxetic design and rely on the stronger notion of expansive
behavior to provide an infinite supply of planar auxetic mechanisms and several
new three-dimensional structures
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
- …