3 research outputs found
The -connectivity augmentation problem: Algorithmic aspects
Durand de Gevigney and Szigeti \cite{DgGSz} have recently given a min-max
theorem for the -connectivity augmentation problem. This article
provides an algorithm to find an
optimal solution for this problem
Splitting-off in Hypergraphs
The splitting-off operation in undirected graphs is a fundamental reduction
operation that detaches all edges incident to a given vertex and adds new edges
between the neighbors of that vertex while preserving their degrees. Lov\'asz
(1974) and Mader (1978) showed the existence of this operation while preserving
global and local connectivities respectively in graphs under certain
conditions. These results have far-reaching applications in graph algorithms
literature. In this work, we introduce a splitting-off operation in
hypergraphs. We show that there exists a local connectivity preserving complete
splitting-off in hypergraphs and give a strongly polynomial-time algorithm to
compute it in weighted hypergraphs. We illustrate the usefulness of our
splitting-off operation in hypergraphs by showing two applications:
(1) we give a constructive characterization of -hyperedge-connected
hypergraphs and
(2) we give an alternate proof of an approximate min-max relation for max
Steiner rooted-connected orientation of graphs and hypergraphs (due to Kir\'aly
and Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)). Our proof of the
approximate min-max relation for graphs circumvents the Nash-Williams' strong
orientation theorem and uses tools developed for hypergraphs