Solving infinite games on trees with back-edges

We study the computational complexity of solving the following problem: Given a game G played on a finite directed graph G, output all nodes in G from which a specific player wins the game G. We provide algorithms for solving the above problem when the games have Büchi and parity winning conditions and the graph G is a tree with back-edges. The running time of the algorithm for Büchi games is O(min{r ·m, l+m}) where m is the number of edges, l is the sum of the distances from the root to all leaves and the parameter r is bounded by the height of the tree. The algorithm for parity has a running time of O(l +m).