Edge-splitting problems with demands

Abstract

Splitting off two edges su, sv in a graph G means deleting su, sv and adding a new edge uυ. Let G = (V + s, E) be κ-edge-connected in V (κ ≥ 2) and let d(s) be even. Lovász [8] proved that the edges incident to s can be split off in pairs in a such a way that the resulting graph remains κ-edge-connected. In this paper we investigate the existence of such complete splitting sequences when the set of split edges has to satisfy some additional requirements. We prove structural properties of the set of those pairs u, υ of neighbours of s for which splitting off su, sυ destroys κ-edge-connectivity. This leads to a new method for solving problems of this type. By applying this new approach first we obtain a short proof for a recent result of Nagamochi and Eades [9] on planarity-preserving complete splitting sequences. Then we apply our structural result to prove the following: let G and H be two graphs on the same set V + s of vertices and suppose that their sets of edges incident to s coincide. Let G (H) be κ-edge-connected (l-edge-connected, respectively) in V and let d(s) be even. Then there exists a pair su, sv which is allowed to split off in both graphs simultaneously provided d(s) ≥ 6. If k and l are both even then such a pair exists for arbitrary even d(s). Using this result and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a smallest (common) set of new edges.