Constrained Edge-Splitting Problems
Splitting off two edges in a graph means deleting and adding a new edge . Let be -edge-connected in () and let be even. Lovász proved that the edges incident to can be split off in pairs in a such a way that the resulting graph on vertex set is -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 of neighbours of for which splitting off destroys -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 and be two graphs on the same set of vertices and suppose that their sets of edges incident to coincide. Let () be -edge-connected (-edge-connected, respectively) in and let be even. Then there exists a pair which can be split off in both graphs preserving -edge-connectivity (-edge-connectivity, resp.) in , provided . If and 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