Bipartition constrained edge-splitting in directed graphs Original Research Article

Let G=(V+s,E) be a digraph with ρ(s)=δ(s) which is k-edge-connected in V. Mader (European J. Combin. 3 (1982) 63) proved that there exists a pair vs, st of edges which can be “split off”, that is, which can be replaced by a new edge vt, preserving k-edge-connectivity in V. Such a pair is called admissible. We extend this theorem by showing that for more than ρ(s)/2 edges vs there exist at least ρ(s)/2 edges st such that vs and st form an admissible pair. We apply this result to the problem of splitting off edges in G when a prespecified bipartition V=A∪B is also given and no edge can be split off with both endvertices in A or both in B. We prove that an admissible pair satisfying the bipartition constraints exists if ρ(s)⩾2k+1. Based on this result we develop a polynomial algorithm which gives an almost optimal solution to the bipartition constrained edge-connectivity augmentation problem. In this problem we are given a directed graph H=(V,E), a bipartition V=A∪B and a positive integer k; the goal is to find a smallest set F of edges for which H′=(V,E∪F) is k-edge-connected and no edge of the augmenting set F has both endvertices in A or both in B. Our algorithm adds at most k edges more than the optimum.