52,325 research outputs found
A Boundary Property for Upper Domination
An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard, but can be solved in polynomial time in some restricted graph classes, such as P4-free graphs or 2K2-free graphs. For classes defined by finitely many forbidden induced subgraphs, the boundary separating difficult instances of the problem from polynomially solvable ones consists of the so called boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. In the present paper, we discover the first boundary class for this problem
A lower bound for disconnection by random interlacements
We consider the vacant set of random interlacements on Z^d, with d bigger or
equal to 3, in the percolative regime. Motivated by the large deviation
principles obtained in our recent work arXiv:1304.7477, we investigate the
asymptotic behavior of the probability that a large body gets disconnected from
infinity by the random interlacements. We derive an asymptotic lower bound,
which brings into play tilted interlacements, and relates the problem to some
of the large deviations of the occupation-time profile considered in
arXiv:1304.7477.Comment: 28 pages, appeared in the Electronic Journal of Probabilit
Finiteness of outer automorphism groups of random right-angled Artin groups
We consider the outer automorphism group Out(A_Gamma) of the right-angled
Artin group A_Gamma of a random graph Gamma on n vertices in the Erdos--Renyi
model. We show that the functions (log(n)+log(log(n)))/n and
1-(log(n)+log(log(n)))/n bound the range of edge probability functions for
which Out(A_Gamma) is finite: if the probability of an edge in Gamma is
strictly between these functions as n grows, then asymptotically Out(A_Gamma)
is almost surely finite, and if the edge probability is strictly outside of
both of these functions, then asymptotically Out(A_Gamma) is almost surely
infinite. This sharpens results of Ruth Charney and Michael Farber from their
preprint "Random groups arising as graph products", arXiv:1006.3378v1.Comment: 29 pages. Mostly rewritten, results tightened, statements corrected,
gaps fille
Half domination arrangements in regular and semi-regular tessellation type graphs
We study the problem of half-domination sets of vertices in vertex transitive
infinite graphs generated by regular or semi-regular tessellations of the
plane. In some cases, the results obtained are sharp and in the rest, we show
upper bounds for the average densities of vertices in half-domination sets.Comment: 14 pages, 12 figure
Stochastic domination for the Ising and fuzzy Potts models
We discuss various aspects concerning stochastic domination for the Ising
model and the fuzzy Potts model. We begin by considering the Ising model on the
homogeneous tree of degree , \Td. For given interaction parameters ,
and external field h_1\in\RR, we compute the smallest external field
such that the plus measure with parameters and dominates
the plus measure with parameters and for all .
Moreover, we discuss continuity of with respect to the three
parameters , , and also how the plus measures are stochastically
ordered in the interaction parameter for a fixed external field. Next, we
consider the fuzzy Potts model and prove that on \Zd the fuzzy Potts measures
dominate the same set of product measures while on \Td, for certain parameter
values, the free and minus fuzzy Potts measures dominate different product
measures. For the Ising model, Liggett and Steif proved that on \Zd the plus
measures dominate the same set of product measures while on \T^2 that
statement fails completely except when there is a unique phase.Comment: 22 pages, 5 figure
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