869 research outputs found
On total domination vertex critical graphs of high connectivity
AbstractA graph is called Îł-critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is Îłt-critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A Îłt-critical graph that has total domination number k, is called k-Îłt-critical. In this paper, we introduce a class of k-Îłt-critical graphs of high connectivity for each integer kâ„3. In particular, we provide a partial answer to the question âWhich graphs are Îł-critical and Îłt-critical or one but not the other?â posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255â261]
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
Ge
The Total Acquisition Number of Random Graphs
Let be a graph in which each vertex initially has weight 1. In each step,
the weight from a vertex can be moved to a neighbouring vertex ,
provided that the weight on is at least as large as the weight on . The
total acquisition number of , denoted by , is the minimum possible
size of the set of vertices with positive weight at the end of the process.
LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of
such that with high probability, where
is a binomial random graph. We show that is a sharp threshold for this
property. We also show that almost all trees satisfy ,
confirming a conjecture of West.Comment: 18 pages, 1 figur
The random geometry of equilibrium phases
This is a (long) survey about applications of percolation theory in
equilibrium statistical mechanics. The chapters are as follows:
1. Introduction
2. Equilibrium phases
3. Some models
4. Coupling and stochastic domination
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation
8. Phase transition and percolation
9. Random interactions
10. Continuum modelsComment: 118 pages. Addresses: [email protected]
http://www.mathematik.uni-muenchen.de/~georgii.html [email protected]
http://www.math.chalmers.se/~olleh [email protected]
Uniqueness and non-uniqueness in percolation theory
This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on and,
more generally, on transitive graphs. For iid percolation on ,
uniqueness of the infinite cluster is a classical result, while on certain
other transitive graphs uniqueness may fail. Key properties of the graphs in
this context turn out to be amenability and nonamenability. The same problem is
considered for certain dependent percolation models -- most prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for others the
proofs are merely sketched, and for yet others they are omitted. Several open
problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A statistical network analysis of the HIV/AIDS epidemics in Cuba
The Cuban contact-tracing detection system set up in 1986 allowed the
reconstruction and analysis of the sexual network underlying the epidemic
(5,389 vertices and 4,073 edges, giant component of 2,386 nodes and 3,168
edges), shedding light onto the spread of HIV and the role of contact-tracing.
Clustering based on modularity optimization provides a better visualization and
understanding of the network, in combination with the study of covariates. The
graph has a globally low but heterogeneous density, with clusters of high
intraconnectivity but low interconnectivity. Though descriptive, our results
pave the way for incorporating structure when studying stochastic SIR epidemics
spreading on social networks
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