2,424 research outputs found
Recognizing graphs of acyclic cubical complexes
AbstractAcyclic cubical complexes have first been introduced by Bandelt and Chepoi in analogy to acyclic simplicial complexes. They characterized them by cube contraction and elimination schemes and showed that the graphs of acyclic cubical complexes are retracts of cubes characterized by certain forbidden convex subgraphs. In this paper we present an algorithm of time complexity O(mlogn) which recognizes whether a given graph G on n vertices with m edges is the graph of an acyclic cubical complex. This is significantly better than the complexity O(mn) of the fastest currently known algorithm for recognizing retracts of cubes in general
Hierarchically hyperbolic spaces I: curve complexes for cubical groups
In the context of CAT(0) cubical groups, we develop an analogue of the theory
of curve complexes and subsurface projections. The role of the subsurfaces is
played by a collection of convex subcomplexes called a \emph{factor system},
and the role of the curve graph is played by the \emph{contact graph}. There
are a number of close parallels between the contact graph and the curve graph,
including hyperbolicity, acylindricity of the action, the existence of
hierarchy paths, and a Masur--Minsky-style distance formula.
We then define a \emph{hierarchically hyperbolic space}; the class of such
spaces includes a wide class of cubical groups (including all virtually compact
special groups) as well as mapping class groups and Teichm\"{u}ller space with
any of the standard metrics. We deduce a number of results about these spaces,
all of which are new for cubical or mapping class groups, and most of which are
new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent
Lie group into a hierarchically hyperbolic space lies close to a product of
hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic
spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi,
Hamenst\"{a}dt, and Kleiner. We finally prove that each hierarchically
hyperbolic group admits an acylindrical action on a hyperbolic space. This
acylindricity result is new for cubical groups, in which case the hyperbolic
space admitting the action is the contact graph; in the case of the mapping
class group, this provides a new proof of a theorem of Bowditch.Comment: To appear in "Geometry and Topology". This version incorporates the
referee's comment
Homological Region Adjacency Tree for a 3D Binary Digital Image via HSF Model
Given a 3D binary digital image I, we define and compute
an edge-weighted tree, called Homological Region Tree (or Hom-Tree,
for short). It coincides, as unweighted graph, with the classical Region
Adjacency Tree of black 6-connected components (CCs) and white 26-
connected components of I. In addition, we define the weight of an edge
(R, S) as the number of tunnels that the CCs R and S “share”. The
Hom-Tree structure is still an isotopic invariant of I. Thus, it provides
information about how the different homology groups interact between
them, while preserving the duality of black and white CCs.
An experimentation with a set of synthetic images showing different
shapes and different complexity of connected component nesting is performed
for numerically validating the method.Ministerio de Economía y Competitividad MTM2016-81030-
Acylindrical hyperbolicity of cubical small-cancellation groups
We provide an analogue of Strebel's classification of geodesic triangles in
classical groups for groups given by Wise's cubical presentations
satisfying sufficiently strong metric cubical small cancellation conditions.
Using our classification, we prove that, except in specific degenerate cases,
such groups are acylindrically hyperbolic.Comment: Added figures. Exposition improved in Section 3,
correction/simplification in Section 5, background added and citations
updated in Section
A Geometric Approach to the Problem of Unique Decomposition of Processes
This paper proposes a geometric solution to the problem of prime
decomposability of concurrent processes first explored by R. Milner and F.
Moller in [MM93]. Concurrent programs are given a geometric semantics using
cubical areas, for which a unique factorization theorem is proved. An effective
factorization method which is correct and complete with respect to the
geometric semantics is derived from the factorization theorem. This algorithm
is implemented in the static analyzer ALCOOL.Comment: 15 page
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