131 research outputs found
Stable set meeting every longest path
AbstractLaborde, Payan and Xuong conjectured that every digraph has a stable set meeting every longest path. We prove that this conjecture holds for digraphs with stability number at most 2
A contribution to the second neighborhood problem
Seymour's Second Neighborhood Conjecture asserts that every digraph (without
digons) has a vertex whose first out-neighborhood is at most as large as its
second out-neighborhood. It is proved for tournaments, tournaments missing a
matching and tournaments missing a generalized star. We prove this conjecture
for classes of digraphs whose missing graph is a comb, a complete graph minus 2
independent edges, or a complete graph minus the edges of a cycle of length 5
b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
A b-coloring of a graph is a proper coloring such that every color class
contains a vertex that is adjacent to all other color classes. The b-chromatic
number of a graph G, denoted by \chi_b(G), is the maximum number t such that G
admits a b-coloring with t colors. A graph G is called b-continuous if it
admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and
b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of
G, and every induced subgraph H_2 of H_1.
We investigate the b-chromatic number of graphs with stability number two.
These are exactly the complements of triangle-free graphs, thus including all
complements of bipartite graphs. The main results of this work are the
following:
- We characterize the b-colorings of a graph with stability number two in
terms of matchings with no augmenting paths of length one or three. We derive
that graphs with stability number two are b-continuous and b-monotonic.
- We prove that it is NP-complete to decide whether the b-chromatic number of
co-bipartite graph is at most a given threshold.
- We describe a polynomial time dynamic programming algorithm to compute the
b-chromatic number of co-trees.
- Extending several previous results, we show that there is a polynomial time
dynamic programming algorithm for computing the b-chromatic number of
tree-cographs. Moreover, we show that tree-cographs are b-continuous and
b-monotonic
Lower Bounds for the Graph Homomorphism Problem
The graph homomorphism problem (HOM) asks whether the vertices of a given
-vertex graph can be mapped to the vertices of a given -vertex graph
such that each edge of is mapped to an edge of . The problem
generalizes the graph coloring problem and at the same time can be viewed as a
special case of the -CSP problem. In this paper, we prove several lower
bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main
result is a lower bound .
This rules out the existence of a single-exponential algorithm and shows that
the trivial upper bound is almost asymptotically
tight.
We also investigate what properties of graphs and make it difficult
to solve HOM. An easy observation is that an upper
bound can be improved to where
is the minimum size of a vertex cover of . The second
lower bound shows that the upper bound is
asymptotically tight. As to the properties of the "right-hand side" graph ,
it is known that HOM can be solved in time and
where is the maximum degree of
and is the treewidth of . This gives
single-exponential algorithms for graphs of bounded maximum degree or bounded
treewidth. Since the chromatic number does not exceed
and , it is natural to ask whether similar
upper bounds with respect to can be obtained. We provide a negative
answer to this question by establishing a lower bound for any
function . We also observe that similar lower bounds can be obtained for
locally injective homomorphisms.Comment: 19 page
Contraception and screening for cervical and breast cancer in neuromuscular disease: A retrospective study of 50Â patients monitored at a clinical reference centre
AbstractObjectiveTo analyse contraceptive methods and the extent of screening for breast and cervical cancer in women with neuromuscular disease, compare these results with data and guidelines for the general population and determine the environmental and attitudinal barriers encountered.Patients and methodsA retrospective, descriptive study in a population of female neuromuscular disease patients (aged 20 to 74) monitored at a clinical reference centre.ResultsComplete datasets were available for 49Â patients. Seventy percent used contraception (hormonal contraception in most cases). Sixty-eight percent had undergone screening for cervical cancer at some time in the previous 3Â years and 100% of the patients over 50 had undergone a mammography. Architectural accessibility and practical problems were the most common barriers to care and were more frequently encountered by wheelchair-bound, ventilated patients.ConclusionsIn general, the patients had good access to contraceptive care and cervical and breast cancer screening. However, specific measures may be useful for the most severely disabled patients
Nonrepetitive Colouring via Entropy Compression
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively -choosable if given lists of at least colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
is -choosable, for some constant . We prove this result
with (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth is nonrepetitively -colourable.Comment: v4: Minor changes made following helpful comments by the referee
Prognostic significance of cortactin levels in head and neck squamous cell carcinoma: comparison with epidermal growth factor receptor status
Cortactin is an actin-binding Src substrate involved in cell motility and invasion. In this study, we sought to examine the prognostic importance of cortactin protein expression in head and neck squamous cell carcinoma (HNSCC). To do so, cortactin and EGF receptor (EGFR) expression was retrospectively evaluated by immunohistochemistry in a tissue microarray composed of 176 HNSCCs with a mean follow-up time of 5 years. Cortactin immunoreactivity was weak to absent in normal epithelial tissue. Overexpression of the protein in 77 out of 176 tumours (44%) was associated with more advanced tumour-node-metastasis stage and higher histologic grade. Cortactin overexpression was associated with significantly increased local recurrence rates (49 vs 28% for high and low expressing carcinomas, respectively), decreased disease-free survival (17 vs 61%), and decreased the 5-year overall survival of (21 vs 58%), independently of the EGFR status. In multivariate analysis, cortactin expression status remained an independent prognostic factor for local recurrence, disease-free survival, and overall survival. Importantly, we identified a subset of patients with cortactin-overexpressing tumours that displayed low EGFR levels and a survival rate that equalled that of patients with tumoral overexpression of both EGFR and cortactin. These findings identify cortactin as a relevant prognostic marker and may have implications for targeted therapies in patients with HNSCC
A Unified Approach to Distance-Two Colouring of Graphs on Surfaces
In this paper we introduce the notion of -colouring of a graph :
For given subsets of neighbours of , for every , this
is a proper colouring of the vertices of such that, in addition, vertices
that appear together in some receive different colours. This
concept generalises the notion of colouring the square of graphs and of cyclic
colouring of graphs embedded in a surface. We prove a general result for graphs
embeddable in a fixed surface, which implies asymptotic versions of Wegner's
and Borodin's Conjecture on the planar version of these two colourings. Using a
recent approach of Havet et al., we reduce the problem to edge-colouring of
multigraphs, and then use Kahn's result that the list chromatic index is close
to the fractional chromatic index.
Our results are based on a strong structural lemma for graphs embeddable in a
fixed surface, which also implies that the size of a clique in the square of a
graph of maximum degree embeddable in some fixed surface is at most
plus a constant.Comment: 36 page
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