Splitting Off Edges within a Specified Subset Preserving the Edge-Connectivity of the Graph

Splitting off a pair su; sv of edges in a graph G means the operation that deletes su and sv and adds a new edge uv. Given a graph G = (V + s;E) which is k-edge-connected (k 2) between vertices of V and a specified subset R V , first we consider the problem of finding a longest possible sequence of disjoint pairs (splittings) of edges sx; sy, x; y 2 R which can be split off preserving k-edge-connectivity in V . If R = V and d(s) is even then the well-known splitting off theorem of Lovász asserts that a complete R-splitting exists, that is, all the edges connecting s to R can be split off in pairs. This is not the case in general. We characterize the graphs possessing a complete R-splitting and give a formula for the length of a longest R-splitting sequence. The main result of our paper is a solution for the following optimization problem: given G and R as above, find a smallest set F of new edges incident to s such that G0 = (V + s;E + F ) has a complete R-splitting. We give a min-max formula for jF j as well as a polynomial algorithm to find a smallest F . The motivation of our research is the well-known strong connection between splitting off results and connectivity augmentation problems. Here we propose a general framework for solving edge-connectivity augmentation problems where the set of new edges has to satisfy some extra property. This shows that our main result may be an important step towards the solution of the following open problem, raised in [2]: given a graph H = (V;E), an integer k 2 and a set R V , find a smallest set F 0 of new edges for which H 0 = (V;E + F 0) is k-edge-connected and no edge of F 0 crosses R.