The Steiner Subgraph Problem Revisited

The Steiner Subgraph Problem (PS for short) is a natural generalization of the classical Steiner Tree Problem. Instead of only one terminal net there is given a set of terminal nets. The task is to choose a susbet F of edges at minimum cost such that vertices of the same net lie in the same connected component in the graph induced by F . Richey and Parker [7] posed the question whether PS is hard on series-parallel graphs. We partially answer this question by showing that PS is strongly NP-hard on graphs with treewidth 3. On the other hand, a quadratic time algorithm for the special case on outerplanar graphs is suggested. Since series-parallel graphs have treewidth 2 and outerplanar graphs are series-parallel, we almost close the gap between polynomially solvable and hard cases.