794 research outputs found
Reconfiguring Graph Homomorphisms on the Sphere
Given a loop-free graph , the reconfiguration problem for homomorphisms to
(also called -colourings) asks: given two -colourings of of a
graph , is it possible to transform into by a sequence of
single-vertex colour changes such that every intermediate mapping is an
-colouring? This problem is known to be polynomial-time solvable for a wide
variety of graphs (e.g. all -free graphs) but only a handful of hard
cases are known. We prove that this problem is PSPACE-complete whenever is
a -free quadrangulation of the -sphere (equivalently, the plane)
which is not a -cycle. From this result, we deduce an analogous statement
for non-bipartite -free quadrangulations of the projective plane. This
include several interesting classes of graphs, such as odd wheels, for which
the complexity was known, and -chromatic generalized Mycielski graphs, for
which it was not.
If we instead consider graphs and with loops on every vertex (i.e.
reflexive graphs), then the reconfiguration problem is defined in a similar way
except that a vertex can only change its colour to a neighbour of its current
colour. In this setting, we use similar ideas to show that the reconfiguration
problem for -colourings is PSPACE-complete whenever is a reflexive
-free triangulation of the -sphere which is not a reflexive triangle.
This proof applies more generally to reflexive graphs which, roughly speaking,
resemble a triangulation locally around a particular vertex. This provides the
first graphs for which -Recolouring is known to be PSPACE-complete for
reflexive instances.Comment: 22 pages, 9 figure
A Brightwell-Winkler type characterisation of NU graphs
In 2000, Brightwell and Winkler characterised dismantlable graphs as the
graphs for which the Hom-graph , defined on the set of
homomorphisms from to , is connected for all graphs . This shows that
the reconfiguration version of the -colouring
problem, in which one must decide for a given whether is
connected, is trivial if and only if is dismantlable.
We prove a similar starting point for the reconfiguration version of the
-extension problem. Where is the subgraph of the
Hom-graph induced by the -colourings extending the
-precolouring of , the reconfiguration version
of the -extension problem asks, for a given -precolouring of a graph
, if is connected. We show that the graphs for which
is connected for every choice of are exactly the
graphs. This gives a new characterisation of graphs, a
nice class of graphs that is important in the algebraic approach to the -dichotomy.
We further give bounds on the diameter of for
graphs , and show that shortest path between two vertices of can be found in parameterised polynomial time. We apply our
results to the problem of shortest path reconfiguration, significantly
extending recent results.Comment: 17 pages, 1 figur
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Isolation and analysis of genes involved in mouse gonad development
Sex determination in the mammalian embryo is initiated when the bipotential gonad becomes committed to either the testicular or ovarian fate, depending on the presence or absence of the sex determining gene Sry. In the presence of Sry the gonad forms a testis. Sry is expressed in the supporting cells of the gonad and results in the differentiation of Sertoli cells. This process instigates a cascade of gene expression leading to testis cord development and masculinisation of the embryo. Many steps are required for this process. Little is known about these steps or how they are co-ordinated after the expression of Sry. In the absence of Sry the gonad develops into an ovary. Again little is known about ovarian development as the majority of research in this field has focused on the development of the testis. The aim of this project was therefore to find new genes, in new functional classes involved in the development of male and female gonads. The starting point of this project was the use of wholemount in situ hybridisation analyses to validate the expression of candidate sexual development genes. These had been shown to be expressed in a sexually dimorphic manner in the developing gonad using microarrays. Two genes, Gata2 and Vanin-1, were chosen for further study. Gata2 is expressed in the germ cells of the developing ovary and is currently the only gene whose expression is found to be female-specific during gonadogenesis. Vanin-1 is a GPI-anchored cell surface molecule thought to be involved in the migration of bone marrow cells to the thymus. Cell migration is an important feature of testis cord development, therefore Vanin-1 could play similar role in this process. It is expressed in the developing male gonad prior to overt testis cord formation and in the Sertoli cells once they form. Organ culture studies with an anti-Vanin-1 monoclonal antibody showed that cord formation in XY gonads could be blocked in vitro. However, studies of the Vanin-1-/- mouse revealed no overt abnormalities in testis development
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