Computation and algorithm for the minimum k-edge-connectivity of graphs

Abstract

Boesch and Chen (SIAM J Appl Math 34:657–665, 1978) introduced the cut-version of the generalized edge-connectivity, named k-edge-connectivity. For any integer k with $2\le k\le n$, the k-edge-connectivity of a graph G, denoted by $\lambda _k(G)$, is defined as the smallest number of edges whose removal from G produces a graph with at least k components. In this paper, we first compute some exact values and sharp bounds for $\lambda _k(G)$ in terms of n and k. We then discuss the relationships between $\lambda _k(G)$ and other generalized connectivities. An algorithm in $\mathcal {O}(n^2)$ time will be provided such that we can compute a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function f(n, k, t) which is defined as the minimum size of a connected graph G with order n and $\lambda _k(G)=t$.