On the complexity of computing the k-restricted edge-connectivity of a graph
The k-restricted edge-connectivity of a graph G, denoted by λk(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 exte...
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| Published in: | Theoretical computer science Vol. 662; pp. 31 - 39 |
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| Main Authors: | , |
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| Language: | English |
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01.02.2017
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| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | The k-restricted edge-connectivity of a graph G, denoted by λk(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 λk(G). Very recently, in the parameterized complexity community the notion of 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. |
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| AbstractList | The k-restricted edge-connectivity of a graph G, denoted by λ 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 λ k (G). Very recently, in the parameterized complexity community the notion of 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. The k-restricted edge-connectivity of a graph G, denoted by λk(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 λk(G). Very recently, in the parameterized complexity community the notion of 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. The k-restricted edge-connectivity of a graph G, denoted by , 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 . Very recently, in the parameterized complexity community the notion of 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. |
| Author | Montejano, Luis Pedro Sau, Ignasi |
| Author_xml | – sequence: 1 givenname: Luis Pedro surname: Montejano fullname: Montejano, Luis Pedro email: lpmontejano@gmail.com organization: Département de Mathématiques, Université de Montpellier, Montpellier, France – sequence: 2 givenname: Ignasi surname: Sau fullname: Sau, Ignasi email: ignasi.sau@lirmm.fr organization: AlGCo project team, CNRS, LIRMM, Montpellier, France |
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| Keywords | Good edge separation Graph cut k-Restricted edge-connectivity Polynomial kernel Parameterized complexity FPT-algorithm k-restricted edge-connectivity |
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| Snippet | The k-restricted edge-connectivity of a graph G, denoted by λk(G), is defined as the minimum size of an edge set whose removal leaves exactly two connected... The k-restricted edge-connectivity of a graph G, denoted by , is defined as the minimum size of an edge set whose removal leaves exactly two connected... The k-restricted edge-connectivity of a graph G, denoted by λ k (G), is defined as the minimum size of an edge set whose removal leaves exactly two connected... |
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| SubjectTerms | Algorithms Combinatorial analysis Communities Complexity Computation FPT-algorithm Good edge separation Graph cut Graph theory Graphs Invariants k-Restricted edge-connectivity Mathematics Parameterized complexity Polynomial kernel |
| Title | On the complexity of computing the k-restricted edge-connectivity of a graph |
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