# On the Complexity of Computing the *k*-restricted Edge-connectivity of a Graph

## Abstract

The *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 *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.

## Keywords

Graph cut*k*-restricted edge-connectivity Good edge separation Parameterized complexity FPT-algorithm polynomial kernel

## Supplementary material

## References

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