5,093 research outputs found
Stochastic domination: the contact process, Ising models and FKG measures
We prove for the contact process on , and many other graphs, that the
upper invariant measure dominates a homogeneous product measure with large
density if the infection rate is sufficiently large. As a
consequence, this measure percolates if the corresponding product measure
percolates. We raise the question of whether domination holds in the symmetric
case for all infinite graphs of bounded degree. We study some asymmetric
examples which we feel shed some light on this question. We next obtain
necessary and sufficient conditions for domination of a product measure for
``downward'' FKG measures. As a consequence of this general result, we show
that the plus and minus states for the Ising model on dominate the same
set of product measures. We show that this latter fact fails completely on the
homogenous 3-ary tree. We also provide a different distinction between
and the homogenous 3-ary tree concerning stochastic domination and Ising
models; while it is known that the plus states for different temperatures on
are never stochastically ordered, on the homogenous 3-ary tree, almost
the complete opposite is the case. Next, we show that on , the set of
product measures which the plus state for the Ising model dominates is strictly
increasing in the temperature. Finally, we obtain a necessary and sufficient
condition for a finite number of variables, which are both FKG and
exchangeable, to dominate a given product measure.Comment: 27 page
Upper bounds on the k-forcing number of a graph
Given a simple undirected graph and a positive integer , the
-forcing number of , denoted , is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most non-colored neighbors, then
each of its non-colored neighbors becomes colored. When , this is
equivalent to the zero forcing number, usually denoted with , a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the -forcing number.
Notable among these, we show that if is a graph with order and
maximum degree , then . This simplifies to, for the zero forcing number case
of , . Moreover, when and the graph is -connected, we prove that , which is an improvement when , and
specializes to, for the zero forcing number case, . These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the -forcing number and the connected -domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure
Maximum Number of Minimum Dominating and Minimum Total Dominating Sets
Given a connected graph with domination (or total domination) number
\gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of
dominating and total dominating sets of size \gamma. An exact answer is
provided for \gamma=2and lower bounds are given for \gamma>=3.Comment: 6 page
Open k-monopolies in graphs: complexity and related concepts
Closed monopolies in graphs have a quite long range of applications in
several problems related to overcoming failures, since they frequently have
some common approaches around the notion of majorities, for instance to
consensus problems, diagnosis problems or voting systems. We introduce here
open -monopolies in graphs which are closely related to different parameters
in graphs. Given a graph and , if is the
number of neighbors has in , is an integer and is a positive
integer, then we establish in this article a connection between the following
three concepts:
- Given a nonempty set a vertex of is said to be
-controlled by if . The set
is called an open -monopoly for if it -controls every vertex of
.
- A function is called a signed total
-dominating function for if for all
.
- A nonempty set is a global (defensive and offensive)
-alliance in if holds for every .
In this article we prove that the problem of computing the minimum
cardinality of an open -monopoly in a graph is NP-complete even restricted
to bipartite or chordal graphs. In addition we present some general bounds for
the minimum cardinality of open -monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016
New bounds on the signed total domination number of graphs
In this paper, we study the signed total domination number in graphs and
present new sharp lower and upper bounds for this parameter. For example by
making use of the classic theorem of Turan, we present a sharp lower bound on
this parameter for graphs with no complete graph of order r+1 as a subgraph.
Also, we prove that n-2(s-s') is an upper bound on the signed total domination
number of any tree of order n with s support vertices and s' support vertives
of degree two. Moreover, we characterize all trees attainig this bound.Comment: This paper contains 11 pages and one figur
- β¦