93,103 research outputs found
Large circulant graphs of fixed diameter and arbitrary degree
We consider the degree-diameter problem for undirected and directed circulant graphs. To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case. We present a direct product construction yielding improved bounds for small diameters and introduce a new general technique for “stitching” together circulant graphs which enables us to improve the current best known asymptotic orders for every diameter. As an application, we use our constructions in the directed case to obtain upper bounds on the minimum size of a subset A of a cyclic group of order n such that the k-fold sumset kA is equal to the whole group. We also present a revised table of largest known circulant graphs of small degree and diameter
Large butterfly Cayley graphs and digraphs
We present families of large undirected and directed Cayley graphs whose
construction is related to butterfly networks. One approach yields, for every
large and for values of taken from a large interval, the largest known
Cayley graphs and digraphs of diameter and degree . Another method
yields, for sufficiently large and infinitely many values of , Cayley
graphs and digraphs of diameter and degree whose order is exponentially
larger in than any previously constructed. In the directed case, these are
within a linear factor in of the Moore bound.Comment: 7 page
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
On a relation between the domination number and a strongly connected bidirection of an undirected graph
As a generalization of directed and undirected graphs, Edmonds and Johnson introduced bidirected graphs. A bidirected graph is a graph each arc of which has either two positive end-vertices (tails), two negative end-vertices (heads), or one positive end-vertex (tail) and one negative end-vertex (head). We extend the notion of directed paths, distance, diameter and strong connectivity from directed to bidirected graphs and characterize those undirected graphs that allow a strongly connected bidirection. Considering the problem of finding the minimum diameter of all strongly connected bidirections of a given undirected graph, we generalize a result of Fomin et al. about directed graphs and obtain an upper bound for the minimum diameter which depends on the minimum size of a dominating set and the number of bridges in the undirected graph
Quadratic diameter bounds for dual network flow polyhedra
Both the combinatorial and the circuit diameters of polyhedra are of interest
to the theory of linear programming for their intimate connection to a
best-case performance of linear programming algorithms.
We study the diameters of dual network flow polyhedra associated to -flows
on directed graphs and prove quadratic upper bounds for both of them:
the minimum of and for the combinatorial
diameter, and for the circuit diameter. The latter
strengthens the cubic bound implied by a result in [De Loera, Hemmecke, Lee;
2014].
Previously, bounds on these diameters have only been known for bipartite
graphs. The situation is much more involved for general graphs. In particular,
we construct a family of dual network flow polyhedra with members that violate
the circuit diameter bound for bipartite graphs by an arbitrary additive
constant. Further, it provides examples of circuit diameter
Degree/diameter problem for mixed graphs
The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of the degree/diameter problem for mixed graphs
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