The Generalized Stable Set Problem for Perfect Bidirected Graphs

Bidirected graphs are a generalization of undirected graphs. For bidirected graphs, we can consider a problem whichi is a natural extension of the maximum weighted stable set problem for undirected graphs. Here we call this problem the generalized stable set problem. It is well known that the maximum weighted stable set problem is solvable in polynomial time for perfect undirected graphs. Perfectness is naturally extended to bidirected graphs in terms of polytopes. Furthermore, it has been proved that a bidirected graph is perfect if and only if its underlying graph is perfect. Thus it is natural to expect that the generalized stable set problem for perfect bidirected graphs can be solved in polynomial time. In this paper, we show that the problem for any bidirected graph is reducible to the maximum weighted stable set problem for a certain undirected graph is in time polynomial in the number of vertices, and moreover, prove that this reduction preserves perfectness. That is, this paper gives an affirmative answer to our expectation.