1,985 research outputs found
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
Isomorphism test for digraphs with weighted edges
Colour refinement is at the heart of all the most efficient graph isomorphism software packages. In this paper we present a method for extending the applicability of refinement algorithms to directed graphs with weighted edges. We use Traces as a reference software, but the proposed solution is easily transferrable to any other refinement-based graph isomorphism tool in the literature. We substantiate the claim that the performances of the original algorithm remain substantially unchanged by showing experiments for some classes of benchmark graphs
Directed path graphs
The concept of a line digraph is generalized to that of a directed path graph. The directed path graph of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. Several properties of are studied, in particular with respect to isomorphism and traversability
From expanded digraphs to lifts of voltage digraphs and line digraphs
In this note we present a general approach to construct large digraphs
from small ones. These are called expanded digraphs, and, as particular
cases, we show the close relationship between lifted digraphs of voltage
digraphs and line digraphs, which are two known ways to obtain dense
digraphs. In the same context, we show the equivalence between the
vertex-splitting and partial line digraph techniques. Then, we give a
sufficient condition for a lifted digraph of a base line digraph to be again
a line digraph. Some of the results are illustrated with two well-known
families of digraphs, namely, the De Bruijn and Kautz digraphs, where
it is shown that both families can be seen as lifts of smaller De Bruijn
digraphs with appropriate voltage assignments.Research of the first two authors is supported by MINECO under project MTM2014-60127-P, and by AGAUR under project 2014SGR1147. The first author has also received funding from the European Union’s Horizon 2020 res
Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)
Given an embedded planar acyclic digraph G, we define the problem of "acyclic
hamiltonian path completion with crossing minimization (Acyclic-HPCCM)" to be
the problem of determining an hamiltonian path completion set of edges such
that, when these edges are embedded on G, they create the smallest possible
number of edge crossings and turn G to a hamiltonian digraph. Our results
include:
--We provide a characterization under which a triangulated st-digraph G is
hamiltonian.
--For an outerplanar triangulated st-digraph G, we define the st-polygon
decomposition of G and, based on its properties, we develop a linear-time
algorithm that solves the Acyclic-HPCCM problem with at most one crossing per
edge of G.
--For the class of st-planar digraphs, we establish an equivalence between
the Acyclic-HPCCM problem and the problem of determining an upward 2-page
topological book embedding with minimum number of spine crossings. We infer
(based on this equivalence) for the class of outerplanar triangulated
st-digraphs an upward topological 2-page book embedding with minimum number of
spine crossings and at most one spine crossing per edge.
To the best of our knowledge, it is the first time that edge-crossing
minimization is studied in conjunction with the acyclic hamiltonian completion
problem and the first time that an optimal algorithm with respect to spine
crossing minimization is presented for upward topological book embeddings
Partition MCMC for inference on acyclic digraphs
Acyclic digraphs are the underlying representation of Bayesian networks, a
widely used class of probabilistic graphical models. Learning the underlying
graph from data is a way of gaining insights about the structural properties of
a domain. Structure learning forms one of the inference challenges of
statistical graphical models.
MCMC methods, notably structure MCMC, to sample graphs from the posterior
distribution given the data are probably the only viable option for Bayesian
model averaging. Score modularity and restrictions on the number of parents of
each node allow the graphs to be grouped into larger collections, which can be
scored as a whole to improve the chain's convergence. Current examples of
algorithms taking advantage of grouping are the biased order MCMC, which acts
on the alternative space of permuted triangular matrices, and non ergodic edge
reversal moves.
Here we propose a novel algorithm, which employs the underlying combinatorial
structure of DAGs to define a new grouping. As a result convergence is improved
compared to structure MCMC, while still retaining the property of producing an
unbiased sample. Finally the method can be combined with edge reversal moves to
improve the sampler further.Comment: Revised version. 34 pages, 16 figures. R code available at
https://github.com/annlia/partitionMCM
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