94,686 research outputs found
Recoloring bounded treewidth graphs
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Any graph is
-mixing, where is the treewidth of the graph (Cereceda 2006). We
prove that the shortest sequence between any two -colorings is at most
quadratic, a problem left open in Bonamy et al. (2012).
Jerrum proved that any graph is -mixing if is at least the maximum
degree plus two. We improve Jerrum's bound using the grundy number, which is
the worst number of colors in a greedy coloring.Comment: 11 pages, 5 figure
Recoloring graphs via tree decompositions
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Jerrum proved that any
graph is -mixing if is at least the maximum degree plus two. We first
improve Jerrum's bound using the grundy number, which is the worst number of
colors in a greedy coloring.
Any graph is -mixing, where is the treewidth of the graph
(Cereceda 2006). We prove that the shortest sequence between any two
-colorings is at most quadratic (which is optimal up to a constant
factor), a problem left open in Bonamy et al. (2012).
We also prove that given any two -colorings of a cograph (resp.
distance-hereditary graph) , we can find a linear (resp. quadratic) sequence
between them. In both cases, the bounds cannot be improved by more than a
constant factor for a fixed . The graph classes are also optimal in
some sense: one of the smallest interesting superclass of distance-hereditary
graphs corresponds to comparability graphs, for which no such property holds
(even when relaxing the constraint on the length of the sequence). As for
cographs, they are equivalently the graphs with no induced , and there
exist -free graphs that admit no sequence between two of their
-colorings.
All the proofs are constructivist and lead to polynomial-time recoloring
algorithmComment: 20 pages, 8 figures, partial results already presented in
http://arxiv.org/abs/1302.348
Random graphs from a block-stable class
A class of graphs is called block-stable when a graph is in the class if and
only if each of its blocks is. We show that, as for trees, for most -vertex
graphs in such a class, each vertex is in at most blocks, and each path passes through at most blocks.
These results extend to `weakly block-stable' classes of graphs
Accelerating Consensus by Spectral Clustering and Polynomial Filters
It is known that polynomial filtering can accelerate the convergence towards
average consensus on an undirected network. In this paper the gain of a
second-order filtering is investigated. A set of graphs is determined for which
consensus can be attained in finite time, and a preconditioner is proposed to
adapt the undirected weights of any given graph to achieve fastest convergence
with the polynomial filter. The corresponding cost function differs from the
traditional spectral gap, as it favors grouping the eigenvalues in two
clusters. A possible loss of robustness of the polynomial filter is also
highlighted
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde
Analysis of Performance of Dynamic Multicast Routing Algorithms
In this paper, three new dynamic multicast routing algorithms based on the
greedy tree technique are proposed; Source Optimised Tree, Topology Based Tree
and Minimum Diameter Tree. A simulation analysis is presented showing various
performance aspects of the algorithms, in which a comparison is made with the
greedy and core based tree techniques. The effects of the tree source location
on dynamic membership change are also examined. The simulations demonstrate
that the Source Optimised Tree algorithm achieves a significant improvement in
terms of delay and link usage when compared to the Core Based Tree, and greedy
algorithm
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