98,087 research outputs found
On Topological Minors in Random Simplicial Complexes
For random graphs, the containment problem considers the probability that a
binomial random graph contains a given graph as a substructure. When
asking for the graph as a topological minor, i.e., for a copy of a subdivision
of the given graph, it is well-known that the (sharp) threshold is at .
We consider a natural analogue of this question for higher-dimensional random
complexes , first studied by Cohen, Costa, Farber and Kappeler for
.
Improving previous results, we show that is the
(coarse) threshold for containing a subdivision of any fixed complete
-complex. For higher dimensions , we get that is an
upper bound for the threshold probability of containing a subdivision of a
fixed -dimensional complex.Comment: 15 page
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
Semi-Transitive Orientations and Word-Representable Graphs
A graph is a \emph{word-representable graph} if there exists a word
over the alphabet such that letters and alternate in if and
only if for each .
In this paper we give an effective characterization of word-representable
graphs in terms of orientations. Namely, we show that a graph is
word-representable if and only if it admits a \emph{semi-transitive
orientation} defined in the paper. This allows us to prove a number of results
about word-representable graphs, in particular showing that the recognition
problem is in NP, and that word-representable graphs include all 3-colorable
graphs.
We also explore bounds on the size of the word representing the graph. The
representation number of is the minimum such that is a
representable by a word, where each letter occurs times; such a exists
for any word-representable graph. We show that the representation number of a
word-representable graph on vertices is at most , while there exist
graphs for which it is .Comment: arXiv admin note: text overlap with arXiv:0810.031
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