Splitting off edges between two subsets preserving the edge-connectivity of the graph

Splitting off a pair of edges su; sv in a graph G means replacing these two edges by a new edge uv. This operation is well-known in graph theory. Let G = (V + s; E + F ) be a graph which is k-edge-connected in V and suppose that jF j is even. Here F denotes the set of edges incident with s. Lovász [12] proved that if k 2 then the edges in F can be split off in pairs preserving the k-edge-connectivity in V . This result was recently extended to the case where a bipartition R [ Q = V is given and every split edge must connect R and Q [4]. In this paper we investigate an even more general problem, where two disjoint subsets R;Q V are given and the goal is to split off (a largest possible subset of) the edges of F preserving k-edge-connectivity in V in such a way that every split edge incident with a vertex from R has the other end-vertex in Q. Motivated by connectivity augmentation problems, we introduce another extension, the so-called split completion version of our problem. Here a smallest set F of edges incident to s has to be found for which all the edges of F + F can be split off in the augmented graph G = (V + s; E + F + F ) preserving k-edge-connectivity and in such a way that every split edge incident with a vertex from R has the other end-vertex in Q. We solve each of the above extensions when k is even: we give min-max formulae and polynomial algorithms to find the optima. For the case when k is odd we show how to find a solution to the split completion problem using at most two edges more than the optimum.