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 of edges sx, sy, (x ,y  R) which can be split off preserving k-edge-connectivity in V. If R = V and d(s) is even then a well-known 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. Motivated by the connection between splitting off results and connectivity augmentation problems we also investigate the following problem that we call the split completion problem: given G and R as above, find a smallest set F of new edges incident to s such that G′ = (V + s, E + F) has a complete R-splitting. We give a min-max formula for F as well as a polynomial algorithm to find a smallest F. As a corollary we show a polynomial algorithm which finds a solution of size at most k/2 + 1 more than the optimum for the following augmentation problem, raised in 2]]: given a graph H = (V, E), an integer k ≥ 2, and a set R  V, find a smallest set F′ of new edges for which H′ = (V, E + F′) is k-edge-connected and no edge of F′ crosses R.