3,906 research outputs found
Obstructions to within a few vertices or edges of acyclic
Finite obstruction sets for lower ideals in the minor order are guaranteed to
exist by the Graph Minor Theorem. It has been known for several years that, in
principle, obstruction sets can be mechanically computed for most natural lower
ideals. In this paper, we describe a general-purpose method for finding
obstructions by using a bounded treewidth (or pathwidth) search. We illustrate
this approach by characterizing certain families of cycle-cover graphs based on
the two well-known problems: -{\sc Feedback Vertex Set} and -{\sc
Feedback Edge Set}. Our search is based on a number of algorithmic strategies
by which large constants can be mitigated, including a randomized strategy for
obtaining proofs of minimality.Comment: 16 page
Hyperplane Neural Codes and the Polar Complex
Hyperplane codes are a class of convex codes that arise as the output of a
one layer feed-forward neural network. Here we establish several natural
properties of stable hyperplane codes in terms of the {\it polar complex} of
the code, a simplicial complex associated to any combinatorial code. We prove
that the polar complex of a stable hyperplane code is shellable and show that
most currently known properties of the hyperplane codes follow from the
shellability of the appropriate polar complex.Comment: 23 pages, 5 figures. To appear in Proceedings of the Abel Symposiu
Obstructions to nonpositive curvature for open manifolds
We study algebraic conditions on a group G under which every properly
discontinuous, isometric G-action on a Hadamard manifold has a G-invariant
Busemann function. For such G we prove the following structure theorem: every
open complete nonpositively curved Riemannian K(G,1) manifold that is homotopy
equivalent to a finite complex of codimension >2 is an open regular
neighborhood of a subcomplex of the same codimension. In this setting we show
that each tangential homotopy type contains infinitely many open K(G,1)
manifolds that admit no complete nonpositively curved metric even though their
universal cover is the Euclidean space. A sample application is that an open
contractible manifold W is homeomorphic to a Euclidean space if and only if the
product of W and a circle admits a complete Riemannian metric of nonpositive
curvature.Comment: 29 page
Minor-Obstructions for Apex-Pseudoforests
A graph is called a pseudoforest if none of its connected components contains
more than one cycle. A graph is an apex-pseudoforest if it can become a
pseudoforest by removing one of its vertices. We identify 33 graphs that form
the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of
all minor-minimal graphs that are not apex-pseudoforests
Minimal Obstructions for Partial Representations of Interval Graphs
Interval graphs are intersection graphs of closed intervals. A generalization
of recognition called partial representation extension was introduced recently.
The input gives an interval graph with a partial representation specifying some
pre-drawn intervals. We ask whether the remaining intervals can be added to
create an extending representation. Two linear-time algorithms are known for
solving this problem.
In this paper, we characterize the minimal obstructions which make partial
representations non-extendible. This generalizes Lekkerkerker and Boland's
characterization of the minimal forbidden induced subgraphs of interval graphs.
Each minimal obstruction consists of a forbidden induced subgraph together with
at most four pre-drawn intervals. A Helly-type result follows: A partial
representation is extendible if and only if every quadruple of pre-drawn
intervals is extendible by itself. Our characterization leads to a linear-time
certifying algorithm for partial representation extension
Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links
The present paper studies the structure of characteristic varieties of
fundamental groups of graph manifolds. As a consequence, a simple proof of
Papadima's question is provided on the characterization of algebraic links that
have quasi-projective fundamental groups. The type of quasi-projective
obstructions used here are in the spirit of Papadima's original work.Comment: 22 pages, 6 figures, to appear in European Journal of Mathematic
FPT is Characterized by Useful Obstruction Sets
Many graph problems were first shown to be fixed-parameter tractable using
the results of Robertson and Seymour on graph minors. We show that the
combination of finite, computable, obstruction sets and efficient order tests
is not just one way of obtaining strongly uniform FPT algorithms, but that all
of FPT may be captured in this way. Our new characterization of FPT has a
strong connection to the theory of kernelization, as we prove that problems
with polynomial kernels can be characterized by obstruction sets whose elements
have polynomial size. Consequently we investigate the interplay between the
sizes of problem kernels and the sizes of the elements of such obstruction
sets, obtaining several examples of how results in one area yield new insights
in the other. We show how exponential-size minor-minimal obstructions for
pathwidth k form the crucial ingredient in a novel OR-cross-composition for
k-Pathwidth, complementing the trivial AND-composition that is known for this
problem. In the other direction, we show that OR-cross-compositions into a
parameterized problem can be used to rule out the existence of efficiently
generated quasi-orders on its instances that characterize the NO-instances by
polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201
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