  In a seminal paper, Chen, Roughgarden and Valiant studied cost sharing
protocols for network design with the objective to implement a low-cost Steiner
forest as a Nash equilibrium of an induced cost-sharing game. One of the most
intriguing open problems to date is to understand the power of budget-balanced
and separable cost sharing protocols in order to induce low-cost Steiner
forests. In this work, we focus on undirected networks and analyze topological
properties of the underlying graph so that an optimal Steiner forest can be
implemented as a Nash equilibrium (by some separable cost sharing protocol)
independent of the edge costs. We term a graph efficient if the above stated
property holds. As our main result, we give a complete characterization of
efficient undirected graphs for two-player network design games: an undirected
graph is efficient if and only if it does not contain (at least) one out of few
forbidden subgraphs. Our characterization implies that several graph classes
are efficient: generalized series-parallel graphs, fan and wheel graphs and
graphs with small cycles.
