3,622 research outputs found
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]
Location-domination in line graphs
A set of vertices of a graph is locating if every two distinct
vertices outside have distinct neighbors in ; that is, for distinct
vertices and outside , , where
denotes the open neighborhood of . If is also a dominating set (total
dominating set), it is called a locating-dominating set (respectively,
locating-total dominating set) of . A graph is twin-free if every two
distinct vertices of have distinct open and closed neighborhoods. It is
conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the
metric dimension and the determining number of a graph. Applied Mathematics and
Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning.
Locating-total dominating sets in twin-free graphs: a conjecture. The
Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any
twin-free graph without isolated vertices has a locating-dominating set of
size at most one-half its order and a locating-total dominating set of size at
most two-thirds its order. In this paper, we prove these two conjectures for
the class of line graphs. Both bounds are tight for this class, in the sense
that there are infinitely many connected line graphs for which equality holds
in the bounds.Comment: 23 pages, 2 figure
A Survey on Alliances and Related Parameters in Graphs
In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, -domination, -independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global -alliances in graphs.We also propose a new framework, called (global) -alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations
Bounds and extremal graphs for total dominating identifying codes
An identifying code of a graph is a dominating set of such that
any two distinct vertices of have distinct closed neighbourhoods within
. The smallest size of an identifying code of is denoted
. When every vertex of also has a neighbour in ,
it is said to be a total dominating identifying code of , and the smallest
size of a total dominating identifying code of is denoted by
.
Extending similar characterizations for identifying codes from the
literature, we characterize those graphs of order with
(the only such connected graph is ) and
(such graphs either satisfy
or are built from certain such graphs by adding a
set of universal vertices, to each of which a private leaf is attached).
Then, using bounds from the literature, we remark that any (open and closed)
twin-free tree of order has a total dominating identifying code of size at
most . This bound is tight, and we characterize the trees
reaching it. Moreover, by a new proof, we show that this bound actually holds
for the larger class of all twin-free graphs of girth at least 5. The cycle
also attains this bound. We also provide a generalized bound for all
graphs of girth at least 5 (possibly with twins).
Finally, we relate to the related parameter
as well as the location-domination number of and
its variants, providing bounds that are either tight or almost tight
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
Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach
Chordal structure and bounded treewidth allow for efficient computation in
numerical linear algebra, graphical models, constraint satisfaction and many
other areas. In this paper, we begin the study of how to exploit chordal
structure in computational algebraic geometry, and in particular, for solving
polynomial systems. The structure of a system of polynomial equations can be
described in terms of a graph. By carefully exploiting the properties of this
graph (in particular, its chordal completions), more efficient algorithms can
be developed. To this end, we develop a new technique, which we refer to as
chordal elimination, that relies on elimination theory and Gr\"obner bases. By
maintaining graph structure throughout the process, chordal elimination can
outperform standard Gr\"obner basis algorithms in many cases. The reason is
that all computations are done on "smaller" rings, of size equal to the
treewidth of the graph. In particular, for a restricted class of ideals, the
computational complexity is linear in the number of variables. Chordal
structure arises in many relevant applications. We demonstrate the suitability
of our methods in examples from graph colorings, cryptography, sensor
localization and differential equations.Comment: 40 pages, 5 figure
Graph Algorithms and Applications
The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity
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