22,266 research outputs found
A New Perspective on Vertex Connectivity
Edge connectivity and vertex connectivity are two fundamental concepts in
graph theory. Although by now there is a good understanding of the structure of
graphs based on their edge connectivity, our knowledge in the case of vertex
connectivity is much more limited. An essential tool in capturing edge
connectivity are edge-disjoint spanning trees. The famous results of Tutte and
Nash-Williams show that a graph with edge connectivity contains
\floor{\lambda/2} edge-disjoint spanning trees.
We present connected dominating set (CDS) partition and packing as tools that
are analogous to edge-disjoint spanning trees and that help us to better grasp
the structure of graphs based on their vertex connectivity. The objective of
the CDS partition problem is to partition the nodes of a graph into as many
connected dominating sets as possible. The CDS packing problem is the
corresponding fractional relaxation, where CDSs are allowed to overlap as long
as this is compensated by assigning appropriate weights. CDS partition and CDS
packing can be viewed as the counterparts of the well-studied edge-disjoint
spanning trees, focusing on vertex disjointedness rather than edge
disjointness.
We constructively show that every -vertex-connected graph with nodes
has a CDS packing of size and a CDS partition of size
. We prove that the CDS packing bound is
existentially optimal.
Using CDS packing, we show that if vertices of a -vertex-connected graph
are independently sampled with probability , then the graph induced by the
sampled vertices has vertex connectivity . Moreover,
using our CDS packing, we get a store-and-forward broadcast
algorithm with optimal throughput in the networking model where in each round,
each node can send one bounded-size message to all its neighbors
A R\'enyi entropy perspective on topological order in classical toric code models
Concepts of information theory are increasingly used to characterize
collective phenomena in condensed matter systems, such as the use of
entanglement entropies to identify emergent topological order in interacting
quantum many-body systems. Here we employ classical variants of these concepts,
in particular R\'enyi entropies and their associated mutual information, to
identify topological order in classical systems. Like for their quantum
counterparts, the presence of topological order can be identified in such
classical systems via a universal, subleading contribution to the prevalent
volume and boundary laws of the classical R\'enyi entropies. We demonstrate
that an additional subleading contribution generically arises for all
R\'enyi entropies with when driving the system towards a
phase transition, e.g. into a conventionally ordered phase. This additional
subleading term, which we dub connectivity contribution, tracks back to partial
subsystem ordering and is proportional to the number of connected parts in a
given bipartition. Notably, the Levin-Wen summation scheme -- typically used to
extract the topological contribution to the R\'enyi entropies -- does not fully
eliminate this additional connectivity contribution in this classical context.
This indicates that the distillation of topological order from R\'enyi
entropies requires an additional level of scrutiny to distinguish topological
from non-topological contributions. This is also the case for quantum
systems, for which we discuss which entropies are sensitive to these
connectivity contributions. We showcase these findings by extensive numerical
simulations of a classical variant of the toric code model, for which we study
the stability of topological order in the presence of a magnetic field and at
finite temperatures from a R\'enyi entropy perspective.Comment: 17 pages, 19 figure
On the fixed-parameter tractability of the maximum connectivity improvement problem
In the Maximum Connectivity Improvement (MCI) problem, we are given a
directed graph and an integer and we are asked to find new
edges to be added to in order to maximize the number of connected pairs of
vertices in the resulting graph. The MCI problem has been studied from the
approximation point of view. In this paper, we approach it from the
parameterized complexity perspective in the case of directed acyclic graphs. We
show several hardness and algorithmic results with respect to different natural
parameters. Our main result is that the problem is -hard for parameter
and it is FPT for parameters and , the matching number of
. We further characterize the MCI problem with respect to other
complementary parameters.Comment: 15 pages, 1 figur
Edge-Orders
Canonical orderings and their relatives such as st-numberings have been used
as a key tool in algorithmic graph theory for the last decades. Recently, a
unifying concept behind all these orders has been shown: they can be described
by a graph decomposition into parts that have a prescribed vertex-connectivity.
Despite extensive interest in canonical orderings, no analogue of this
unifying concept is known for edge-connectivity. In this paper, we establish
such a concept named edge-orders and show how to compute (1,1)-edge-orders of
2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs
in linear time, respectively. While the former can be seen as the edge-variants
of st-numberings, the latter are the edge-variants of Mondshein sequences and
non-separating ear decompositions. The methods that we use for obtaining such
edge-orders differ considerably in almost all details from the ones used for
their vertex-counterparts, as different graph-theoretic constructions are used
in the inductive proof and standard reductions from edge- to
vertex-connectivity are bound to fail.
As a first application, we consider the famous Edge-Independent Spanning Tree
Conjecture, which asserts that every k-edge-connected graph contains k rooted
spanning trees that are pairwise edge-independent. We illustrate the impact of
the above edge-orders by deducing algorithms that construct 2- and 3-edge
independent spanning trees of 2- and 3-edge-connected graphs, the latter of
which improves the best known running time from O(n^2) to linear time
Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)
Although the ``scale-free'' literature is large and growing, it gives neither
a precise definition of scale-free graphs nor rigorous proofs of many of their
claimed properties. In fact, it is easily shown that the existing theory has
many inherent contradictions and verifiably false claims. In this paper, we
propose a new, mathematically precise, and structural definition of the extent
to which a graph is scale-free, and prove a series of results that recover many
of the claimed properties while suggesting the potential for a rich and
interesting theory. With this definition, scale-free (or its opposite,
scale-rich) is closely related to other structural graph properties such as
various notions of self-similarity (or respectively, self-dissimilarity).
Scale-free graphs are also shown to be the likely outcome of random
construction processes, consistent with the heuristic definitions implicit in
existing random graph approaches. Our approach clarifies much of the confusion
surrounding the sensational qualitative claims in the scale-free literature,
and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet
Mathematics (2005
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