16,098 research outputs found
Long Paths and Hamiltonian paths in Inhomogenous Random Graphs
In this paper, we study long paths and Hamiltonian paths in inhomogenous
random graphs. In the first part of the paper, we consider an inhomogenous
Erd\H{o}s-R\'enyi random graph with average edge density We prove
that if as then the
longest path contains at least nodes with high
probability (i.e., with probability converging to one as ), for some constant In particular, if for some constant large, then is Hamiltonian with high
probability; i.e., the longest path contains all the nodes of
In the second part of the paper, we consider a random geometric graph
consisting of nodes, each independently distributed according to a (not
necessarily uniform) density If is the connectivity radius and
then with high probability, the longest cycle
contains at least nodes for some constant As a consequence of our proof, we obtain that if and as then with high probability contains a Hamiltonian
cycle
On some upper bounds on the fractional chromatic number of weighted graphs
Given a weighted graph G_\bx, where is a non-negative,
real-valued weight assigned to the vertices of G, let B(G_\bx) be an upper
bound on the fractional chromatic number of the weighted graph G_\bx; so
\chi_f(G_\bx) \le B(G_\bx). To investigate the worst-case performance of the
upper bound , we study the graph invariant \beta(G) = \sup_{\bx \ne 0}
\frac{B(G_\bx)}{\chi_f(G_\bx)}.
\noindent This invariant is examined for various upper bounds on the
fractional chromatic number. In some important cases, this graph invariant is
shown to be related to the size of the largest star subgraph in the graph. This
problem arises in the area of resource estimation in distributed systems and
wireless networks; the results presented here have implications on the design
and performance of decentralized communication networks
Fault tolerant supergraphs with automorphisms
Given a graph on vertices and a desired level of fault-tolerance ,
an objective in fault-tolerant system design is to construct a supergraph
on vertices such that the removal of any nodes from leaves a
graph containing . In order to reconfigure around faults when they occur, it
is also required that any two subsets of nodes of are in the same orbit
of the action of its automorphism group. In this paper, we prove that such a
supergraph must be the complete graph. This implies that it is very expensive
to have an interconnection network which is -fault-tolerant and which also
supports automorphic reconfiguration. Our work resolves an open problem in the
literature. The proof uses a result due to Cameron on -homogeneous groups
Duality in percolation via outermost boundaries III: Plus connected components
Tile into disjoint unit squares with the
origin being the centre of and say that and are star adjacent
if they share a corner and plus adjacent if they share an edge. Every square is
either vacant or occupied. In this paper, we use the structure of the outermost
boundaries derived in Ganesan (2017) to alternately obtain duality between star
and plus connected components in the following sense: There is a star connected
cycle of vacant squares attached to and surrounding the finite plus connected
component containing the origin
Randomized detection and detection capacity of multidetector networks
In this paper, we study the following detection problem. There are
detectors randomly placed in the unit square assigned to detect the presence of a
source located at the origin. Time is divided into slots of unit length and
represents the (random) decision of the
detector in time slot . The location of the source is unknown to the
detectors and the goal is to design schemes that use the decisions
and detect the presence of the source in as short time as
possible.
We first determine the minimum achievable detection time and show
the existence of \emph{randomized} detection schemes that have detection times
arbitrarily close to for almost all configuration of detectors,
provided the number of detectors is sufficiently large. We call such
schemes as \emph{capacity achieving} and completely characterize all capacity
achieving detection schemes
Automorphism group of the modified bubble-sort graph
The modified bubble-sort graph of dimension is the Cayley graph of
generated by cyclically adjacent transpositions. In the present paper, it
is shown that the automorphism group of the modified bubble sort graph of
dimension is , for all . Thus, a complete
structural description of the automorphism group of the modified bubble-sort
graph is obtained. A similar direct product decomposition is seen to hold for
arbitrary normal Cayley graphs generated by transposition sets
Graph extensions, edit number and regular graphs
A graph G on n vertices is said to be extendable if G can be modified to form
a new graph H on more than n vertices, while preserving the degrees of the
vertices common to G and H. The added vertices all have the same degree and we
define edit numbers to quantify the amount of modification needed to obtain the
extended graph. Characterizing graphs with least possible edit numbers, we
obtain that graphs with zero edit number can be extended using regular graphs.
We also describe an iterative algorithm to construct connected regular graphs
on arbitrarily large vertex sets, starting from the complete graph on a fixed
set of vertices
Existence of connected regular and nearly regular graphs
For integers and , we prove the following: If is even, there is a connected -regular graph on vertices. If is odd, there is a connected nearly -regular graph on vertices
ROUGE 2.0: Updated and Improved Measures for Evaluation of Summarization Tasks
Evaluation of summarization tasks is extremely crucial to determining the
quality of machine generated summaries. Over the last decade, ROUGE has become
the standard automatic evaluation measure for evaluating summarization tasks.
While ROUGE has been shown to be effective in capturing n-gram overlap between
system and human composed summaries, there are several limitations with the
existing ROUGE measures in terms of capturing synonymous concepts and coverage
of topics. Thus, often times ROUGE scores do not reflect the true quality of
summaries and prevents multi-faceted evaluation of summaries (i.e. by topics,
by overall content coverage and etc). In this paper, we introduce ROUGE 2.0,
which has several updated measures of ROUGE: ROUGE-N+Synonyms, ROUGE-Topic,
ROUGE-Topic+Synonyms, ROUGE-TopicUniq and ROUGE-TopicUniq+Synonyms; all of
which are improvements over the core ROUGE measures
Duality in percolation via outermost boundaries II: Star connected components and left right crossings
Tile into disjoint unit squares with
the origin being the centre of and say that and are
star adjacent if they share a corner and plus adjacent if they share an edge.
Every square is either vacant or occupied. In this paper, we use the structure
of the outermost boundaries derived in Ganesan (2015) to alternately obtain
duality between star and plus connected components in the following sense:
There is a plus connected cycle of vacant squares attached to surrounding the
finite star connected component containing the origin. We also obtain the
mutual exclusivity of left right crossings and top down crossings of star and
plus connected components in rectangles
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