12,019 research outputs found
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
A transfer principle and applications to eigenvalue estimates for graphs
In this paper, we prove a variant of the Burger-Brooks transfer principle
which, combined with recent eigenvalue bounds for surfaces, allows to obtain
upper bounds on the eigenvalues of graphs as a function of their genus. More
precisely, we show the existence of a universal constants such that the
-th eigenvalue of the normalized Laplacian of a graph
of (geometric) genus on vertices satisfies where denotes the maximum valence of
vertices of the graph. This result is tight up to a change in the value of the
constant , and improves recent results of Kelner, Lee, Price and Teng on
bounded genus graphs.
To show that the transfer theorem might be of independent interest, we relate
eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple
graph models, and discuss an application to the mesh partitioning problem,
extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang
to arbitrary meshes.Comment: Major revision, 16 page
Modular Graph Functions
In earlier work we studied features of non-holomorphic modular functions
associated with Feynman graphs for a conformal scalar field theory on a
two-dimensional torus with zero external momenta at all vertices. Such
functions, which we will refer to as modular graph functions, arise, for
example, in the low energy expansion of genus-one Type II superstring
amplitudes. We here introduce a class of single-valued elliptic multiple
polylogarithms, which are defined as elliptic functions associated with Feynman
graphs with vanishing external momenta at all but two vertices. These functions
depend on a coordinate, , on the elliptic curve and reduce to modular
graph functions when is set equal to . We demonstrate that these
single-valued elliptic multiple polylogarithms are linear combinations of
multiple polylogarithms, and that modular graph functions are sums of
single-valued elliptic multiple polylogarithms evaluated at the identity of the
elliptic curve, in both cases with rational coefficients. This insight suggests
the many interrelations between modular graph functions (a few of which were
established in earlier papers) may be obtained as a consequence of identities
involving multiple polylogarithms, and explains an earlier observation that the
coefficients of the Laurent polynomial at the cusp are given by rational
numbers times single-valued multiple zeta values.Comment: 42 pages, significant clarifications added in section 5, minor typos
corrected, and references added in version
Rank-width and Tree-width of H-minor-free Graphs
We prove that for any fixed r>=2, the tree-width of graphs not containing K_r
as a topological minor (resp. as a subgraph) is bounded by a linear (resp.
polynomial) function of their rank-width. We also present refinements of our
bounds for other graph classes such as K_r-minor free graphs and graphs of
bounded genus.Comment: 17 page
Automorphisms of free groups with boundaries
The automorphisms of free groups with boundaries form a family of groups
A_{n,k} closely related to mapping class groups, with the standard
automorphisms of free groups as A_{n,0} and (essentially) the symmetric
automorphisms of free groups as A_{0,k}. We construct a contractible space
L_{n,k} on which A_{n,k} acts with finite stabilizers and finite quotient space
and deduce a range for the virtual cohomological dimension of A_{n,k}. We also
give a presentation of the groups and calculate their first homology group.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-25.abs.htm
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