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<br />Sum dj = d(s). An S-detachment splits s into a set of p independent vertices s1, ..., sp with<br />d(sj) = dj, 1 <= j <= p. Given a requirement function r(u, v) on pairs of vertices of V , an<br />S-detachment is called r-admissible if the detached graph G' satisfies lambda_G' (x, y) >= r(x, y)<br />for every pair x, y in V . Here lambda_H(u, v) denotes the local edge-connectivity between u and v<br />in graph H.<br />We prove that an r-admissible S-detachment exists if and only if (a) lambda_G(x, y) >= r(x, y),<br />and (b) lambda_G−s(x, y) >= r(x, y) − Sum |dj/2| hold for every x, y in V .<br />The special case of this characterization when r(x, y) = lambda_G(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. Lov´asz 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.