2,307 research outputs found
On the probability of planarity of a random graph near the critical point
Consider the uniform random graph with vertices and edges.
Erd\H{o}s and R\'enyi (1960) conjectured that the limit
\lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists
and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994)
proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower
and upper bounds for this probability.
In this paper we determine the exact probability of a random graph being
planar near the critical point . For each , we find an exact
analytic expression for
In particular, we obtain .
We extend these results to classes of graphs closed under taking minors. As
an example, we show that the probability of being
series-parallel converges to 0.98003.
For the sake of completeness and exposition we reprove in a concise way
several basic properties we need of a random graph near the critical point.Comment: 10 pages, 1 figur
Local tree-width, excluded minors, and approximation algorithms
The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that
associates with every natural number r the maximal tree-width of an
r-neighborhood in G. Our main graph theoretic result is a decomposition theorem
for graphs with excluded minors that essentially says that such graphs can be
decomposed into trees of graphs of bounded local tree-width.
As an application of this theorem, we show that a number of combinatorial
optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set,
and Maximum Independent Set have a polynomial time approximation scheme when
restricted to a class of graphs with an excluded minor
Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion
The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016].
Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
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