195 research outputs found
Approximating survivable networks with β-metric costs
AbstractThe Survivable Network Design (SND) problem seeks a minimum-cost subgraph that satisfies prescribed node-connectivity requirements. We consider SND on both directed and undirected complete graphs with β-metric costs when c(xz)⩽β[c(xy)+c(yz)] for all x,y,z∈V, which varies from uniform costs (β=1/2) to metric costs (β=1).For the k-Connected Subgraph (k-CS) problem our ratios are: 1+2βk(1−β)−12k−1 for undirected graphs, and 1+4β3k(1−3β2)−12k−1 for directed graphs and 12⩽β<13. For undirected graphs this improves the ratios β1−β of Böckenhauer et al. (2008) [3] and 2+βkn of Kortsarz and Nutov (2003) [11] for all k⩾4 and 12+3k−22(4k2−7k+2)⩽β⩽k2(k+1)2−2. We also show that SND admits the ratios 2β1−β for undirected graphs, and 4β31−3β2 for directed graphs with 1/2⩽β<1/3. For two important particular cases of SND, so-called Subset k-CS and Rooted SND, our ratios are 2β31−3β2 for directed graphs and β1−β for subset k-CS on undirected graphs
Cop and robber game and hyperbolicity
In this note, we prove that all cop-win graphs G in the game in which the
robber and the cop move at different speeds s and s' with s'<s, are
\delta-hyperbolic with \delta=O(s^2). We also show that the dependency between
\delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger
condition. This solves an open question from the paper (J. Chalopin et al., Cop
and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25
(2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and
s'=r+2\delta for any r>0, this establishes a new - game-theoretical -
characterization of Gromov hyperbolicity. We also show that for weakly modular
graphs the dependency between \delta and s is linear for any s'<s. Using these
results, we describe a simple constant-factor approximation of the
hyperbolicity \delta of a graph on n vertices in O(n^2) time when the graph is
given by its distance-matrix
Recommended from our members
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
Topology of random simplicial complexes: a survey
This expository article is based on a lecture from the Stanford Symposium on
Algebraic Topology: Application and New Directions, held in honor of Gunnar
Carlsson, Ralph Cohen, and Ib Madsen.Comment: After revisions, now 21 pages, 5 figure
- …