Given an undirected unweighted graph G = (V, E) and an integer k ≥ 1, we consider the problem of computing the edge connectivities of all those (s, t) vertex pairs, whose edge connectivity is at most k. We present an algorithm with expected running time Õ(m + nk) for this problem, where |V | = n and |E| = m. Our output is a weighted tree T whose nodes are the sets V1, V2, . . . , V` of a partit...

Convexity is a notion that has been defined for subsets of R and for subsets of general graphs. A convex cut of a graph G = (V,E) is a 2-partition V1∪̇V2 = V such that both V1 and V2 are convex, i. e., shortest paths between vertices in Vi never leave Vi, i ∈ {1, 2}. Finding convex cuts is NP-hard for general graphs. To characterize convex cuts, we employ the Djoković relation, a reflexive and s...

We study graphs G = (V,E) containing a long cycle which for given integers a1, a2, ..., ak ∈ N have an edge cut whose removal results in k components with vertex sets V1, V2, ..., Vk such that |Vi| ≥ ai for 1 ≤ i ≤ k. Our results closely relate to problems and recent research in network sharing and network reliability.