Given a directed graph G = (V, E) and two nodes s, t, an st-walk is a sequence of nodes s = v0, v1, . . . , vk = t where (vi , vi+1) is an edge of G for 0 ? i < k. Note that a node may be visited multiple times in a walk ? this is how it differs from a path. Given G, s, t and an integer k ? n, design a linear time algorithm to check if there is an st-walk in G that visits at least k distinct nodes including s and t.

Solve the problem when G is a DAG.

Let s ? v1, s ? v2, . . . , s ? v` be all outgoing edges of s. Describe a recursive solution for this problem.

Modify DFS to solve this problem on a DAG.

Hint: it is easier to think about this problem if you view the vertices in topological order.

Solve the problem when G is a an arbitrary directed graph. Hint: If G is strongly connected then there is always such a walk even for k = n (do you see why?).