Constrained Edge-Splitting Problems

Abstract

<p>Splitting off two edges su, sv in a graph G means deleting su, sv and<br />adding a new edge uv. Let G = (V +s,E) be k-edge-connected in V<br />(k >= 2) and let d(s) be even. Lov´asz proved that the edges incident to s<br />can be split off in pairs in such a way that the resulting graph on vertex<br />set V is k-edge-connected. In this paper we investigate the existence of<br />such complete splitting sequences when the set of split edges has to meet<br />additional requirements. We prove structural properties of the set of those<br />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.</p><p>By applying this method we obtain a short proof for a recent result of<br />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.</p>