18,416 research outputs found
On Reconfiguration Graphs of Independent Sets under Token Sliding
An independent set of a graph is a vertex subset such that there is no edge joining any two vertices in . Imagine that a token is placed on each vertex of an independent set of . The - (-) reconfiguration graph of takes all non-empty independent sets (of size ) as its nodes, where is some given positive integer. Two nodes are adjacent if one can be obtained from the other by sliding a token on some vertex to one of its unoccupied neighbors. This paper focuses on the structure and realizability of these reconfiguration graphs. More precisely, we study two main questions for a given graph : (1) Whether the -reconfiguration graph of belongs to some graph class (including complete graphs, paths, cycles, complete bipartite graphs, connected split graphs, maximal outerplanar graphs, and complete graphs minus one edge) and (2) If satisfies some property (including -partitedness, planarity, Eulerianity, girth, and the clique's size), whether the corresponding - (-) reconfiguration graph of also satisfies , and vice versa. Additionally, we give a decomposition result for splitting a -reconfiguration graph into smaller pieces
Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings
Consider a directed or an undirected graph with integral edge weights from
the set [-W, W], that does not contain negative weight cycles. In this paper,
we introduce a general framework for solving problems on such graphs using
matrix multiplication. The framework is based on the usage of Baur-Strassen's
theorem and of Strojohann's determinant algorithm. It allows us to give new and
simple solutions to the following problems:
* Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time
algorithm for finding shortest cycles in undirected and directed graphs. For
directed graphs (and undirected graphs with non-negative weights) this matches
the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the
other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously
known for undirected graphs with negative weights. Furthermore our algorithm
for a given directed or undirected graph detects whether it contains a negative
weight cycle within the same running time.
* Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega})
time algorithm for computing a diameter and radius of an undirected or directed
graphs. To the best of our knowledge no algorithm with this running time was
known for undirected graphs with negative weights.
* Finding Minimum Weight Perfect Matchings -- We present an
\tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect
matchings in undirected graphs. This resolves an open problem posted by
Sankowski in 2006, who presented such an algorithm but only in the case of
bipartite graphs.
In order to solve minimum weight perfect matching problem we develop a novel
combinatorial interpretation of the dual solution which sheds new light on this
problem. Such a combinatorial interpretation was not know previously, and is of
independent interest.Comment: To appear in FOCS 201
TDMA is Optimal for All-unicast DoF Region of TIM if and only if Topology is Chordal Bipartite
The main result of this work is that an orthogonal access scheme such as TDMA
achieves the all-unicast degrees of freedom (DoF) region of the topological
interference management (TIM) problem if and only if the network topology graph
is chordal bipartite, i.e., every cycle that can contain a chord, does contain
a chord. The all-unicast DoF region includes the DoF region for any arbitrary
choice of a unicast message set, so e.g., the results of Maleki and Jafar on
the optimality of orthogonal access for the sum-DoF of one-dimensional convex
networks are recovered as a special case. The result is also established for
the corresponding topological representation of the index coding problem
A Note on the Sparing Number of Graphs
An integer additive set-indexer is defined as an injective function
such that the induced function defined by is also
injective. An IASI is said to be a weak IASI if
for all . A graph which admits a
weak IASI may be called a weak IASI graph. The set-indexing number of an
element of a graph , a vertex or an edge, is the cardinality of its
set-labels. The sparing number of a graph is the minimum number of edges
with singleton set-labels, required for a graph to admit a weak IASI. In
this paper, we study the sparing number of certain graphs and the relation of
sparing number with some other parameters like matching number, chromatic
number, covering number, independence number etc.Comment: 10 pages, 10 figures, submitte
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