4,524 research outputs found
Total domination stable graphs upon edge addition
AbstractA set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge addition stable if the addition of an arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge addition stable graphs. We determine a sharp upper bound on the total domination number of total domination edge addition stable graphs, and we determine which combinations of order and total domination number are attainable. We finish this work with an investigation of claw-free total domination edge addition stable graphs
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
From rubber bands to rational maps: A research report
This research report outlines work, partially joint with Jeremy Kahn and
Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal
surfaces with boundary. One one hand, this lets us tell when one rubber band
network is looser than another, and on the other hand tell when one conformal
surface embeds in another.
We apply this to give a new characterization of hyperbolic critically finite
rational maps among branched self-coverings of the sphere, by a positive
criterion: a branched covering is equivalent to a hyperbolic rational map if
and only if there is an elastic graph with a particular "self-embedding"
property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example
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]
Oka manifolds: From Oka to Stein and back
Oka theory has its roots in the classical Oka-Grauert principle whose main
result is Grauert's classification of principal holomorphic fiber bundles over
Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds
and Stein spaces to Oka manifolds. It has emerged as a subfield of complex
geometry in its own right since the appearance of a seminal paper of M. Gromov
in 1989.
In this expository paper we discuss Oka manifolds and Oka maps. We describe
equivalent characterizations of Oka manifolds, the functorial properties of
this class, and geometric sufficient conditions for being Oka, the most
important of which is Gromov's ellipticity. We survey the current status of the
theory in terms of known examples of Oka manifolds, mention open problems and
outline the proofs of the main results.
In the appendix by F. Larusson it is explained how Oka manifolds and Oka
maps, along with Stein manifolds, fit into an abstract homotopy-theoretic
framework.
The article is an expanded version of the lectures given by the author at the
Winter School KAWA-4 in Toulouse, France, in January 2013. A more comprehensive
exposition of Oka theory is available in the monograph F. Forstneric, Stein
Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex
Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56,
Springer-Verlag, Berlin-Heidelberg (2011).Comment: With an appendix by Finnur Larusson. To appear in Ann. Fac. Sci.
Toulouse Math. (6), vol. 22, no. 4. This version is identical with the
published tex
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