Journal ArticleDOI
Bisecting sparse random graphs
TLDR
In this article, the bisection width of a graph G is the minimum over all partitions of the number of "cross edges" between the parts of the graph G with edge probability c n.Abstract:
Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ‘‘cross edges’’ between the parts. We are interested in sparse random graphs Ž . G with edge probability c n. We show that, if c ln 4, then the bisection width is n n, c n with high probability; while if c ln 4, then it is equal to 0 with high probability. There are corresponding threshold results for partitioning into any fixed number of parts. 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 31 38, 2001read more
Citations
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Journal IssueDOI
The phase transition in inhomogeneous random graphs
TL;DR: A very general model of an inhomogeneous random graph with (conditional) independence between the edges is introduced, which scales so that the number of edges is linear in thenumber of vertices.
Journal ArticleDOI
A survey of graph layout problems
TL;DR: A complete view of the current state of the art with respect to layout problems from an algorithmic point of view is presented.
Journal ArticleDOI
The phase transition in inhomogeneous random graphs
TL;DR: The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees tend to be concentrated around a typical value.
Book ChapterDOI
Metrics for sparse graphs
Béla Bollobás,Oliver Riordan +1 more
TL;DR: This paper deals mainly with graphs with $o(n^2)$ but $\omega(n)$ edges: a companion paper [arXiv:0812.2656] will discuss the (more problematic still) case of {\em extremely sparse} graphs, with O( n) edges.
Journal ArticleDOI
Sparse graphs: Metrics and random models
Béla Bollobás,Oliver Riordan +1 more
TL;DR: Bollobas, Janson and Riordan as mentioned in this paper introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [ 0, 1]2 → [0, ∞).
References
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Book
The Probabilistic Method
TL;DR: A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Journal ArticleDOI
The evolution of random graphs
Journal ArticleDOI
Some simplified NP-complete graph problems
TL;DR: This paper shows that a number of NP - complete problems remain NP -complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP -complete.
Journal ArticleDOI
Improved approximation algorithms for MAX k-CUT and MAX BISECTION
Alan Frieze,Mark Jerrum +1 more
TL;DR: Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of partitioning the vertices of a weighted graph intok blocks so as to maximize the weight of crossing edges.