On the complexity of computing the $k$-restricted edge-connectivity of a graph

TL;DR

This paper investigates the computational complexity of calculating the $k$-restricted edge-connectivity in graphs, revealing NP-hardness, W[1]-hardness, and providing fixed-parameter tractable algorithms.

Contribution

It systematically studies the parameterized complexity of $k$-restricted edge-connectivity, establishing hardness results and developing FPT algorithms.

Findings

Proves NP-hardness of computing $mbda_k(G)$.

Shows W[1]-hardness for certain parameterizations.

Provides fixed-parameter tractable algorithms for specific cases.

Abstract

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.