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 \F\ is even. Here F denotes the set of edges incident with s. Lovasz (Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979) proved that if k greater than or equal to 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 boolean OR Q = V is given and every split edge must connect R and Q (SIAM J. Discrete Math. 12 (2) (1999) 160). In this paper, we investigate an even more general problem, where two disjoint subsets R,Q subset of V are given and the goal is to split off (the 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, the 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. (C) 2003 Elsevier B.V. All rights reserved.