Detachments Preserving Local Edge-Connectivity of Graphs

Let G = (V + s,E) be a graph and let S = (d1, ..., dp) be a set of positive integers with ∑ dj = d(s). An S-detachment splits s into a set of p independent vertices s1, ..., sp with d(sj) = dj , 1 ≤ j ≤ p. Given a requirement function r(u, v) on pairs of vertices of V , an S-detachment is called r-admissible if the detached graph G satisfies λG′(x, y) ≥ r(x, y) for every pair x, y ∈ V . Here λH(u, v) denotes the local edge-connectivity between u and v in graph H . We prove that an r-admissible S-detachment exists if and only if (a) λG(x, y) ≥ r(x, y), and (b) λG−s(x, y) ≥ r(x, y) − ∑ ⌊dj/2⌋ hold for every x, y ∈ V . The special case of this characterization when r(x, y) = λG(x, y) for each pair in V was conjectured by B. Fleiner. Our result is a common generalization of a theorem of W. Mader on edge splittings preserving local edge-connectivity and a result of B. Fleiner on detachments preserving global edge-connectivity. Other corollaries include previous results of L. Lovász and C.J.St.A. Nash-Williams on edge splittings and detachments, respectively. As a new application, we extend a theorem of A. Frank on local edge-connectivity augmentation to the case when stars of given degrees are added.