2,838 research outputs found
Vertex decompositions of two-dimensional complexes and graphs
We investigate families of two-dimensional simplicial complexes defined in
terms of vertex decompositions. They include nonevasive complexes, strongly
collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary
and Palmer. We investigate the complexity of recognition problems for those
families and some of their combinatorial properties. Certain results follow
from analogous decomposition techniques for graphs. For example, we prove that
it is NP-complete to decide if a graph can be reduced to a discrete graph by a
sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug
The complex of pant decompositions of a surface
We exhibit a set of edges (moves) and 2-cells (relations) making the complex
of pant decompositions on a surface a simply connected complex. Our
construction, unlike the previous ones, keeps the arguments concerning the
structural transformations independent from those deriving from the action of
the mapping class group. The moves and the relations turn out to be supported
in subsurfaces with 3g-3+n=1,2 (where g is the genus and n is the number of
boundary components), illustrating in this way the so called Grothendieck
principle.Comment: Minor changes in the introductio
Subdivisional spaces and graph braid groups
We study the problem of computing the homology of the configuration spaces of
a finite cell complex . We proceed by viewing , together with its
subdivisions, as a subdivisional space--a kind of diagram object in a category
of cell complexes. After developing a version of Morse theory for subdivisional
spaces, we decompose and show that the homology of the configuration spaces
of is computed by the derived tensor product of the Morse complexes of the
pieces of the decomposition, an analogue of the monoidal excision property of
factorization homology.
Applying this theory to the configuration spaces of a graph, we recover a
cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this
model enhances it with various convenient functorialities, exact sequences, and
module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version
published in Documenta Mathematic
JSJ-decompositions of finitely presented groups and complexes of groups
A JSJ-splitting of a group over a certain class of subgroups is a graph
of groups decomposition of which describes all possible decompositions of
as an amalgamated product or an HNN extension over subgroups lying in the
given class. Such decompositions originated in 3-manifold topology. In this
paper we generalize the JSJ-splitting constructions of Sela, Rips-Sela and
Dunwoody-Sageev and we construct a JSJ-splitting for any finitely presented
group with respect to the class of all slender subgroups along which the group
splits. Our approach relies on Haefliger's theory of group actions on CAT
spaces
Vertex decomposable graphs and obstructions to shellability
Inspired by several recent papers on the edge ideal of a graph G, we study
the equivalent notion of the independence complex of G. Using the tool of
vertex decomposability from geometric combinatorics, we show that 5-chordal
graphs with no chordless 4-cycles are shellable and sequentially
Cohen-Macaulay. We use this result to characterize the obstructions to
shellability in flag complexes, extending work of Billera, Myers, and Wachs. We
also show how vertex decomposability may be used to show that certain graph
constructions preserve shellability.Comment: 13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and
additional references. v3: minor corrections for publicatio
Not all simplicial polytopes are weakly vertex-decomposable
In 1980 Provan and Billera defined the notion of weak -decomposability for
pure simplicial complexes. They showed the diameter of a weakly
-decomposable simplicial complex is bounded above by a polynomial
function of the number of -faces in and its dimension. For weakly
0-decomposable complexes, this bound is linear in the number of vertices and
the dimension. In this paper we exhibit the first examples of non-weakly
0-decomposable simplicial polytopes
On the one-endedness of graphs of groups
We give a technical result that implies a straightforward necessary and
sufficient conditions for a graph of groups with virtually cyclic edge groups
to be one ended. For arbitrary graphs of groups, we show that if their
fundamental group is not one-ended, then we can blow up vertex groups to graphs
of groups with simpler vertex and edge groups. As an application, we generalize
a theorem of Swarup to decompositions of virtually free groups.Comment: Improved exposition. 17 pages, 7 figures. To appear in Pacific J.
Mat
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