On the complexity of computing the \(k\)-restricted edge-connectivity of a graph

The \emph{\(k\)-restricted edge-connectivity} of a graph \(G\), denoted by \(\lambda_k(G)\), is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least \(k\) vertices. This graph invariant, which can be seen as a generalization of a m...

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Bibliographic Details
Published inarXiv.org
Main Authors Luis Pedro Montejano, Sau, Ignasi
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 17.09.2016
Subjects
Algorithms
Apexes
Combinatorial analysis
Complexity
Computation
Connectivity
Graph theory
Invariants
Parameterization
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Summary:The \emph{\(k\)-restricted edge-connectivity} of a graph \(G\), denoted by \(\lambda_k(G)\), is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least \(k\) vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing \(\lambda_k(G)\). Very recently, in the parameterized complexity community the notion of \emph{good edge separation} of a graph has been defined, which happens to be essentially the same as the \(k\)-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.
ISSN:2331-8422