411 research outputs found
Switching Reconstruction of Digraphs
Switching about a vertex in a digraph means to reverse the direction of every
edge incident with that vertex. Bondy and Mercier introduced the problem of
whether a digraph can be reconstructed up to isomorphism from the multiset of
isomorphism types of digraphs obtained by switching about each vertex. Since
the largest known non-reconstructible oriented graphs have 8 vertices, it is
natural to ask whether there are any larger non-reconstructible graphs. In this
paper we continue the investigation of this question. We find that there are
exactly 44 non-reconstructible oriented graphs whose underlying undirected
graphs have maximum degree at most 2. We also determine the full set of
switching-stable oriented graphs, which are those graphs for which all
switchings return a digraph isomorphic to the original
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
On the structure of quadrilateral brane tilings
Brane tilings provide the most general framework in string and M-theory for
matching toric Calabi-Yau singularities probed by branes with superconformal
fixed points of quiver gauge theories. The brane tiling data consists of a
bipartite tiling of the torus which encodes both the classical superpotential
and gauge-matter couplings for the quiver gauge theory. We consider the class
of tilings which contain only tiles bounded by exactly four edges and present a
method for generating any tiling within this class by iterating combinations of
certain graph-theoretic moves. In the context of D3-branes in IIB string
theory, we consider the effect of these generating moves within the
corresponding class of supersymmetric quiver gauge theories in four dimensions.
Of particular interest are their effect on the superpotential, the vacuum
moduli space and the conditions necessary for the theory to reach a
superconformal fixed point in the infrared. We discuss the general structure of
physically admissible quadrilateral brane tilings and Seiberg duality in terms
of certain composite moves within this class.Comment: 57 pages, 22 figure
Reconstructing Gene Trees From Fitch's Xenology Relation
Two genes are xenologs in the sense of Fitch if they are separated by at
least one horizontal gene transfer event. Horizonal gene transfer is asymmetric
in the sense that the transferred copy is distinguished from the one that
remains within the ancestral lineage. Hence xenology is more precisely thought
of as a non-symmetric relation: is xenologous to if has been
horizontally transferred at least once since it diverged from the least common
ancestor of and . We show that xenology relations are characterized by a
small set of forbidden induced subgraphs on three vertices. Furthermore, each
xenology relation can be derived from a unique least-resolved edge-labeled
phylogenetic tree. We provide a linear-time algorithm for the recognition of
xenology relations and for the construction of its least-resolved edge-labeled
phylogenetic tree. The fact that being a xenology relation is a heritable graph
property, finally has far-reaching consequences on approximation problems
associated with xenology relations
Seurat games on Stockmeyer graphs
We define a family of vertex colouring games played over a pair of
graphs or digraphs (G, H) by players ∀ and ∃. These games arise from work on
a longstanding open problem in algebraic logic. It is conjectured that there is a
natural number n such that ∀ always has a winning strategy in the game with
n colours whenever G 6∼= H. This is related to the reconstruction conjecture
for graphs and the degree-associated reconstruction conjecture for digraphs.
We show that the reconstruction conjecture implies our game conjecture with
n = 3 for graphs, and the same is true for the degree-associated reconstruction
conjecture and our conjecture for digraphs. We show (for any k < ω) that
the 2-colour game can distinguish certain non-isomorphic pairs of graphs that
cannot be distinguished by the k-dimensional Weisfeiler-Leman algorithm. We
also show that the 2-colour game can distinguish the non-isomorphic pairs of
graphs in the families defined by Stockmeyer as counterexamples to the original
digraph reconstruction conjecture
Partial Homology Relations - Satisfiability in terms of Di-Cographs
Directed cographs (di-cographs) play a crucial role in the reconstruction of
evolutionary histories of genes based on homology relations which are binary
relations between genes. A variety of methods based on pairwise sequence
comparisons can be used to infer such homology relations (e.g.\ orthology,
paralogy, xenology). They are \emph{satisfiable} if the relations can be
explained by an event-labeled gene tree, i.e., they can simultaneously co-exist
in an evolutionary history of the underlying genes. Every gene tree is
equivalently interpreted as a so-called cotree that entirely encodes the
structure of a di-cograph. Thus, satisfiable homology relations must
necessarily form a di-cograph. The inferred homology relations might not cover
each pair of genes and thus, provide only partial knowledge on the full set of
homology relations. Moreover, for particular pairs of genes, it might be known
with a high degree of certainty that they are not orthologs (resp.\ paralogs,
xenologs) which yields forbidden pairs of genes. Motivated by this observation,
we characterize (partial) satisfiable homology relations with or without
forbidden gene pairs, provide a quadratic-time algorithm for their recognition
and for the computation of a cotree that explains the given relations
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