49,804 research outputs found
The Directed Grid Theorem
The grid theorem, originally proved by Robertson and Seymour in Graph Minors
V in 1986, is one of the most central results in the study of graph minors. It
has found numerous applications in algorithmic graph structure theory, for
instance in bidimensionality theory, and it is the basis for several other
structure theorems developed in the graph minors project.
In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [Reed
97, Johnson, Robertson, Seymour, Thomas 01]), independently, conjectured an
analogous theorem for directed graphs, i.e. the existence of a function f : N
-> N such that every digraph of directed tree-width at least f(k) contains a
directed grid of order k. In an unpublished manuscript from 2001, Johnson,
Robertson, Seymour and Thomas give a proof of this conjecture for planar
digraphs. But for over a decade, this was the most general case proved for the
Reed, Johnson, Robertson, Seymour and Thomas conjecture.
Only very recently, this result has been extended to all classes of digraphs
excluding a fixed undirected graph as a minor (see [Kawarabayashi, Kreutzer
14]). In this paper, nearly two decades after the conjecture was made, we are
finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas
conjecture in full generality and to prove the directed grid theorem.
As consequence of our results we are able to improve results in Reed et al.
in 1996 [Reed, Robertson, Seymour, Thomas 96] (see also [Open Problem Garden])
on disjoint cycles of length at least l and in [Kawarabayashi, Kobayashi,
Kreutzer 14] on quarter-integral disjoint paths. We expect many more
algorithmic results to follow from the grid theorem.Comment: 43 pages, 21 figure
Search for the end of a path in the d-dimensional grid and in other graphs
We consider the worst-case query complexity of some variants of certain
\cl{PPAD}-complete search problems. Suppose we are given a graph and a
vertex . We denote the directed graph obtained from by
directing all edges in both directions by . is a directed subgraph of
which is unknown to us, except that it consists of vertex-disjoint
directed paths and cycles and one of the paths originates in . Our goal is
to find an endvertex of a path by using as few queries as possible. A query
specifies a vertex , and the answer is the set of the edges of
incident to , together with their directions. We also show lower bounds for
the special case when consists of a single path. Our proofs use the theory
of graph separators. Finally, we consider the case when the graph is a grid
graph. In this case, using the connection with separators, we give
asymptotically tight bounds as a function of the size of the grid, if the
dimension of the grid is considered as fixed. In order to do this, we prove a
separator theorem about grid graphs, which is interesting on its own right
Trees and Matchings
In this article, Temperley's bijection between spanning trees of the square
grid on the one hand, and perfect matchings (also known as dimer coverings) of
the square grid on the other, is extended to the setting of general planar
directed (and undirected) graphs, where edges carry nonnegative weights that
induce a weighting on the set of spanning trees. We show that the weighted,
directed spanning trees (often called arborescences) of any planar graph G can
be put into a one-to-one weight-preserving correspondence with the perfect
matchings of a related planar graph H.
One special case of this result is a bijection between perfect matchings of
the hexagonal honeycomb lattice and directed spanning trees of a triangular
lattice. Another special case gives a correspondence between perfect matchings
of the ``square-octagon'' lattice and directed weighted spanning trees on a
directed weighted version of the cartesian lattice.
In conjunction with results of Kenyon, our main theorem allows us to compute
the measures of all cylinder events for random spanning trees on any (directed,
weighted) planar graph. Conversely, in cases where the perfect matching model
arises from a tree model, Wilson's algorithm allows us to quickly generate
random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1
Adapting the Directed Grid Theorem into an FPT Algorithm
The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most
important tools in the field of structural graph theory, finding numerous
applications in the design of algorithms for undirected graphs. An analogous
version of the Grid Theorem in digraphs was conjectured by Johnson et al.
[JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely,
they showed that there is a function such that every digraph of directed
tree-width at least contains a cylindrical grid of size as a
butterfly minor and stated that their proof can be turned into an XP algorithm,
with parameter , that either constructs a decomposition of the appropriate
width, or finds the claimed large cylindrical grid as a butterfly minor. In
this paper, we adapt some of the steps of the proof of Kawarabayashi and
Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our
main technical contributions are two FPT algorithms with parameter . The
first one either produces an arboreal decomposition of width or finds a
haven of order in a digraph , improving on the original result for
arboreal decompositions by Johnson et al. The second algorithm finds a
well-linked set of order in a digraph of large directed tree-width. As
tools to prove these results, we show how to solve a generalized version of the
problem of finding balanced separators for a given set of vertices in FPT
time with parameter , a result that we consider to be of its own interest.Comment: 31 pages, 9 figure
Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs
A connected graph G is called matching covered if every edge of G is
contained in a perfect matching. Perfect matching width is a width parameter
for matching covered graphs based on a branch decomposition. It was introduced
by Norine and intended as a tool for the structural study of matching covered
graphs, especially in the context of Pfaffian orientations. Norine conjectured
that graphs of high perfect matching width would contain a large grid as a
matching minor, similar to the result on treewidth by Robertson and Seymour. In
this paper we obtain the first results on perfect matching width since its
introduction. For the restricted case of bipartite graphs, we show that perfect
matching width is equivalent to directed treewidth and thus the Directed Grid
Theorem by Kawarabayashi and Kreutzer for directed \treewidth implies Norine's
conjecture.Comment: Manuscrip
1-Safe Petri nets and special cube complexes: equivalence and applications
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net
unfolds into an event structure . By a result of Thiagarajan
(1996 and 2002), these unfoldings are exactly the trace regular event
structures. Thiagarajan (1996 and 2002) conjectured that regular event
structures correspond exactly to trace regular event structures. In a recent
paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on
the striking bijection between domains of event structures, median graphs, and
CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we
proved that Thiagarajan's conjecture is true for regular event structures whose
domains are principal filters of universal covers of (virtually) finite special
cube complexes.
In the current paper, we prove the converse: to any finite 1-safe Petri net
one can associate a finite special cube complex such that the
domain of the event structure (obtained as the unfolding of
) is a principal filter of the universal cover of .
This establishes a bijection between 1-safe Petri nets and finite special cube
complexes and provides a combinatorial characterization of trace regular event
structures.
Using this bijection and techniques from graph theory and geometry (MSO
theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet
another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that
the monadic second order logic of a 1-safe Petri net is decidable if and only
if its unfolding is grid-free.
Our counterexample is the trace regular event structure
which arises from a virtually special square complex . The domain of
is grid-free (because it is hyperbolic), but the MSO
theory of the event structure is undecidable
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