3,864 research outputs found
The boundary action of a sofic random subgroup of the free group
We prove that the boundary action of a sofic random subgroup of a finitely
generated free group is conservative. This addresses a question asked by
Grigorchuk, Kaimanovich, and Nagnibeda, who studied the boundary actions of
individual subgroups of the free group. Following their work, we also
investigate the cogrowth and various limit sets associated to sofic random
subgroups. We make heavy use of the correspondence between subgroups and their
Schreier graphs, and central to our approach is an investigation of the
asymptotic density of a given set inside of large neighborhoods of the root of
a sofic random Schreier graph.Comment: 21 pages, 2 figures, made minor corrections, to appear in Groups,
Geometry, and Dynamic
Thinness of product graphs
The thinness of a graph is a width parameter that generalizes some properties
of interval graphs, which are exactly the graphs of thinness one. Many
NP-complete problems can be solved in polynomial time for graphs with bounded
thinness, given a suitable representation of the graph. In this paper we study
the thinness and its variations of graph products. We show that the thinness
behaves "well" in general for products, in the sense that for most of the graph
products defined in the literature, the thinness of the product of two graphs
is bounded by a function (typically product or sum) of their thinness, or of
the thinness of one of them and the size of the other. We also show for some
cases the non-existence of such a function.Comment: 45 page
Definability equals recognizability for graphs of bounded treewidth
We prove a conjecture of Courcelle, which states that a graph property is
definable in MSO with modular counting predicates on graphs of constant
treewidth if, and only if it is recognizable in the following sense:
constant-width tree decompositions of graphs satisfying the property can be
recognized by tree automata. While the forward implication is a classic fact
known as Courcelle's theorem, the converse direction remained openComment: 21 pages, an extended abstract will appear in the proceedings of LICS
201
The ergodic theory of hyperbolic groups
These notes are a self-contained introduction to the use of dynamical and
probabilistic methods in the study of hyperbolic groups. Most of this material
is standard; however some of the proofs given are new, and some results are
proved in greater generality than have appeared in the literature. These notes
originated in a minicourse given at a workshop in Melbourne, July 11-15 2011.Comment: 37 pages, 5 figures; incorporates referee's comment
Relations Between Graphs
Given two graphs G and H, we ask under which conditions there is a relation R
that generates the edges of H given the structure of graph G. This construction
can be seen as a form of multihomomorphism. It generalizes surjective
homomorphisms of graphs and naturally leads to notions of R-retractions,
R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are
unique up to isomorphism and can be computed in polynomial time.Comment: accepted by Ars Mathematica Contemporane
Precedence thinness in graphs
Interval and proper interval graphs are very well-known graph classes, for
which there is a wide literature. As a consequence, some generalizations of
interval graphs have been proposed, in which graphs in general are expressed in
terms of interval graphs, by splitting the graph in some special way.
As a recent example of such an approach, the classes of -thin and proper
-thin graphs have been introduced generalizing interval and proper interval
graphs, respectively. The complexity of the recognition of each of these
classes is still open, even for fixed .
In this work, we introduce a subclass of -thin graphs (resp. proper
-thin graphs), called precedence -thin graphs (resp. precedence proper
-thin graphs). Concerning partitioned precedence -thin graphs, we present
a polynomial time recognition algorithm based on -trees. With respect to
partitioned precedence proper -thin graphs, we prove that the related
recognition problem is \NP-complete for an arbitrary and polynomial-time
solvable when is fixed. Moreover, we present a characterization for these
classes based on threshold graphs.Comment: 33 page
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