3,710 research outputs found
Approximating subset -connectivity problems
A subset of terminals is -connected to a root in a
directed/undirected graph if has internally-disjoint -paths for
every ; is -connected in if is -connected to every
. We consider the {\sf Subset -Connectivity Augmentation} problem:
given a graph with edge/node-costs, node subset , and
a subgraph of such that is -connected in , find a
minimum-cost augmenting edge-set such that is
-connected in . The problem admits trivial ratio .
We consider the case and prove that for directed/undirected graphs and
edge/node-costs, a -approximation for {\sf Rooted Subset -Connectivity
Augmentation} implies the following ratios for {\sf Subset -Connectivity
Augmentation}: (i) ; (ii) , where
b=1 for undirected graphs and b=2 for directed graphs, and is the th
harmonic number. The best known values of on undirected graphs are
for edge-costs and for
node-costs; for directed graphs for both versions. Our results imply
that unless , {\sf Subset -Connectivity Augmentation} admits
the same ratios as the best known ones for the rooted version. This improves
the ratios in \cite{N-focs,L}
Connectivity keeping spiders in k-connected graphs
W. Mader [J. Graph Theory 65 (2010), 61--69] conjectured that for any tree
of order , every -connected graph with
contains a tree such
that remains -connected. In this paper, we confirm Mader's
conjecture for all the spider-trees.Comment: 9 page
Proof a conjecture on connectivity keeping odd paths in k-connected bipartite graphs
Luo, Tian and Wu (2022) conjectured that for any tree with bipartition
and , every -connected bipartite graph with minimum degree at
least , where max, contains a tree such that
is still -connected. Note that when
the tree is the path with order . In this paper, we proved that every
-connected bipartite graph with minimum degree at least contains a path of order such that
remains -connected. This shows that the conjecture is true for paths with
odd order. And for paths with even order, the minimum degree bound in this
paper is the bound in the conjecture plus one
The Cost of Global Broadcast in Dynamic Radio Networks
We study the single-message broadcast problem in dynamic radio networks. We
show that the time complexity of the problem depends on the amount of stability
and connectivity of the dynamic network topology and on the adaptiveness of the
adversary providing the dynamic topology. More formally, we model communication
using the standard graph-based radio network model. To model the dynamic
network, we use a generalization of the synchronous dynamic graph model
introduced in [Kuhn et al., STOC 2010]. For integer parameters and
, we call a dynamic graph -interval -connected if for every
interval of consecutive rounds, there exists a -vertex-connected stable
subgraph. Further, for an integer parameter , we say that the
adversary providing the dynamic network is -oblivious if for constructing
the graph of some round , the adversary has access to all the randomness
(and states) of the algorithm up to round .
As our main result, we show that for any , any , and any
, for a -oblivious adversary, there is a distributed
algorithm to broadcast a single message in time
. We further show that even for large interval -connectivity,
efficient broadcast is not possible for the usual adaptive adversaries. For a
-oblivious adversary, we show that even for any (for any constant ) and for any , global broadcast in -interval -connected networks requires at least
time. Further, for a oblivious adversary,
broadcast cannot be solved in -interval -connected networks as long as
.Comment: 17 pages, conference version appeared in OPODIS 201
T-Pebbling in k-connected graphs with a universal vertex
The t-pebbling number is the smallest integer m so that any initially distributed supply of m pebbles can place t pebbles on any target vertex via pebbling moves. The 1-pebbling number of diameter 2 graphs is well-studied. Here we investigate the t-pebbling number of diameter 2 graphs under the lens of connectivity.Fil: Alcón, Liliana Graciela. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Gutierrez, Marisa. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: Hulbert, Glenn. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentin
Chordal graphs with bounded tree-width
Given and , we prove that the number of labelled
-connected chordal graphs with vertices and tree-width at most is
asymptotically , as , for some constants
depending on and . Additionally, we show that the number
of -cliques () in a uniform random -connected chordal
graph with tree-width at most is normally distributed as .
The asymptotic enumeration of graphs of tree-width at most is wide open
for . To the best of our knowledge, this is the first non-trivial
class of graphs with bounded tree-width where the asymptotic counting problem
is solved. Our starting point is the work of Wormald [Counting Labelled Chordal
Graphs, Graphs and Combinatorics (1985)], were an algorithm is developed to
obtain the exact number of labelled chordal graphs on vertices.Comment: 23 pages, 5 figure
Chordal graphs with bounded tree-width
Given and , we prove that the number of labelled -connected chordal graphs with vertices and tree-width at most is asymptotically , as , for some constants depending on and . Additionally, we show that the number of -cliques () in a uniform random -connected chordal graph with tree-width at most is normally distributed as . The asymptotic enumeration of graphs of tree-width at most is wide open for . To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, \textit{Graphs and Combinatorics} (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on vertices.Peer ReviewedPostprint (author's final draft
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