Constrained Edge-Splitting Problems

Splitting off two edges su, sv in a graph G means deleting su, sv and adding a new edge uv. Let G = (V +s,E) be k-edge-connected in V (k >= 2) and let d(s) be even. Lov´asz proved that the edges incident to s can be split off in pairs in such a way that the resulting graph on vertex set V is k-edge-connected. In this paper we investigate the existence of such complete splitting sequences when the set of split edges has to meet additional requirements. We prove structural properties of the set of those pairs u, v of neighbours of s for which splitting off su, sv destroys k-edge-connectivity. This leads to a new method for solving problems of this type. By applying this method we obtain a short proof for a recent result of Nagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: 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 k-edge-connected (l-edge-connected, respectively) in V and let d(s) be even. Then there exists a pair su, sv which can be split off in both graphs preserving k-edge-connectivity (l-edge-connectivity, resp.) in V , provided d(s) >= 6. If k and l are both even then such a pair always exists. Using these edge-splitting results 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 (common) set of new edges of (almost) minimum size.