Upgrading edge-disjoint paths in a ring

In this paper we introduce the upgrading problem for edge-disjoint paths. In the off-line upgrading problem a supply graph G and two demand graphs H1 and H2 are given on the same vertex set. What is the maximum size of a set F ⊆ E(H1)∩E(H2) such that F has a routing in G which can be extended to a routing of Hi in G, for i = 1, 2? In the online upgrading problem we are given a supply graph G, a demand graph H with a routing and another demand graph H2 such that E(H) ⊆ E(H2). What is the maximum size of a set F ⊆ E(H) such that the restriction of the given routing to F can be extended to routing of H2? Thus, depending on whether the graphs are directed or undirected, we have four different versions. In this paper we give full solution for the case when G is a ring and the demand graphs are stars. All four versions are NP-complete in general.