14,587 research outputs found
Parameterized Complexity of Graph Constraint Logic
Graph constraint logic is a framework introduced by Hearn and Demaine, which
provides several problems that are often a convenient starting point for
reductions. We study the parameterized complexity of Constraint Graph
Satisfiability and both bounded and unbounded versions of Nondeterministic
Constraint Logic (NCL) with respect to solution length, treewidth and maximum
degree of the underlying constraint graph as parameters. As a main result we
show that restricted NCL remains PSPACE-complete on graphs of bounded
bandwidth, strengthening Hearn and Demaine's framework. This allows us to
improve upon existing results obtained by reduction from NCL. We show that
reconfiguration versions of several classical graph problems (including
independent set, feedback vertex set and dominating set) are PSPACE-complete on
planar graphs of bounded bandwidth and that Rush Hour, generalized to boards, is PSPACE-complete even when is at most a constant
Monoids of O-type, subword reversing, and ordered groups
We describe a simple scheme for constructing finitely generated monoids in
which left-divisibility is a linear ordering and for practically investigating
these monoids. The approach is based on subword reversing, a general method of
combinatorial group theory, and connected with Garside theory, here in a
non-Noetherian context. As an application we describe several families of
ordered groups whose space of left-invariant orderings has an isolated point,
including torus knot groups and some of their amalgamated products.Comment: updated version with new result
The group of parenthesized braids
We investigate a group that includes Artin's braid group
and Thompson's group . The elements of are
represented by braids diagrams in which the distances between the strands are
not uniform and, besides the usual crossing generators, new rescaling operators
shrink or strech the distances between the strands. We prove that
is a group of fractions, that it is orderable, admits a non-trivial
self-distributive structure, i.e., one involving the law ,
embeds in the mapping class group of a sphere with a Cantor set of punctures,
and that Artin's representation of into the automorphisms of a free
group extends to
A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties
A conjugation-free geometric presentation of a fundamental group is a
presentation with the natural topological generators and the
cyclic relations: with no conjugations on the
generators.
We have already proved that if the graph of the arrangement is a disjoint
union of cycles, then its fundamental group has a conjugation-free geometric
presentation. In this paper, we extend this property to arrangements whose
graphs are a disjoint union of cycle-tree graphs.
Moreover, we study some properties of this type of presentations for a
fundamental group of a line arrangement's complement. We show that these
presentations satisfy a completeness property in the sense of Dehornoy, if the
corresponding graph of the arrangement has no edges. The completeness property
is a powerful property which leads to many nice properties concerning the
presentation (such as the left-cancellativity of the associated monoid and
yields some simple criterion for the solvability of the word problem in the
group).Comment: 17 pages, 9 figures; final version, which corrects a mistake in the
published versio
Periods, Lefschetz numbers and entropy for a class of maps on a bouquet of circles
We consider some smooth maps on a bouquet of circles. For these maps we can
compute the number of fixed points, the existence of periodic points and an
exact formula for topological entropy. We use Lefschetz fixed point theory and
actions of our maps on both the fundamental group and the first homology group.Comment: 19 pages, 2 figure
Tamari Lattices and the symmetric Thompson monoid
We investigate the connection between Tamari lattices and the Thompson group
F, summarized in the fact that F is a group of fractions for a certain monoid
F+sym whose Cayley graph includes all Tamari lattices. Under this
correspondence, the Tamari lattice operations are the counterparts of the least
common multiple and greatest common divisor operations in F+sym. As an
application, we show that, for every n, there exists a length l chain in the
nth Tamari lattice whose endpoints are at distance at most 12l/n.Comment: 35page
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