75,013 research outputs found
On the structure of (pan, even hole)-free graphs
A hole is a chordless cycle with at least four vertices. A pan is a graph
which consists of a hole and a single vertex with precisely one neighbor on the
hole. An even hole is a hole with an even number of vertices. We prove that a
(pan, even hole)-free graph can be decomposed by clique cutsets into
essentially unit circular-arc graphs. This structure theorem is the basis of
our -time certifying algorithm for recognizing (pan, even hole)-free
graphs and for our -time algorithm to optimally color them.
Using this structure theorem, we show that the tree-width of a (pan, even
hole)-free graph is at most 1.5 times the clique number minus 1, and thus the
chromatic number is at most 1.5 times the clique number.Comment: Accepted to appear in the Journal of Graph Theor
The complexity of detecting taut angle structures on triangulations
There are many fundamental algorithmic problems on triangulated 3-manifolds
whose complexities are unknown. Here we study the problem of finding a taut
angle structure on a 3-manifold triangulation, whose existence has implications
for both the geometry and combinatorics of the triangulation. We prove that
detecting taut angle structures is NP-complete, but also fixed-parameter
tractable in the treewidth of the face pairing graph of the triangulation.
These results have deeper implications: the core techniques can serve as a
launching point for approaching decision problems such as unknot recognition
and prime decomposition of 3-manifolds.Comment: 22 pages, 10 figures, 3 tables; v2: minor updates. To appear in SODA
2013: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete
Algorithm
A Note on Graphs of Linear Rank-Width 1
We prove that a connected graph has linear rank-width 1 if and only if it is
a distance-hereditary graph and its split decomposition tree is a path. An
immediate consequence is that one can decide in linear time whether a graph has
linear rank-width at most 1, and give an obstruction if not. Other immediate
consequences are several characterisations of graphs of linear rank-width 1. In
particular a connected graph has linear rank-width 1 if and only if it is
locally equivalent to a caterpillar if and only if it is a vertex-minor of a
path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors
of graphs of small tree-width, arxiv:1203.3606] if and only if it does not
contain the co-K_2 graph, the Net graph and the 5-cycle graph as vertex-minors
[Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for
linear rank-width at most 1, arxiv:1106.2533].Comment: 9 pages, 2 figures. Not to be publishe
A Convolutional Neural Network for Modelling Sentences
The ability to accurately represent sentences is central to language
understanding. We describe a convolutional architecture dubbed the Dynamic
Convolutional Neural Network (DCNN) that we adopt for the semantic modelling of
sentences. The network uses Dynamic k-Max Pooling, a global pooling operation
over linear sequences. The network handles input sentences of varying length
and induces a feature graph over the sentence that is capable of explicitly
capturing short and long-range relations. The network does not rely on a parse
tree and is easily applicable to any language. We test the DCNN in four
experiments: small scale binary and multi-class sentiment prediction, six-way
question classification and Twitter sentiment prediction by distant
supervision. The network achieves excellent performance in the first three
tasks and a greater than 25% error reduction in the last task with respect to
the strongest baseline
Letter graphs and geometric grid classes of permutations: characterization and recognition
In this paper, we reveal an intriguing relationship between two seemingly
unrelated notions: letter graphs and geometric grid classes of permutations. An
important property common for both of them is well-quasi-orderability,
implying, in a non-constructive way, a polynomial-time recognition of geometric
grid classes of permutations and -letter graphs for a fixed . However,
constructive algorithms are available only for . In this paper, we present
the first constructive polynomial-time algorithm for the recognition of
-letter graphs. It is based on a structural characterization of graphs in
this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author
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