Path-search in the pyramid and in other graphs

We are given an acyclic directed graph with one source, and a subset of its edges which contains exactly one outgoing edge for every non-sink vertex. These edges determine a unique path from the source to a sink. We can think of it as a switch in every vertex, which determines which way the water arriving to that vertex flows further. We are interested in determining either the sink the flow arrives, or the whole path, with as few questions as possible. The questions we can ask correspond to the vertices of the graph, and the answer describes the switch, i.e. tells which outgoing edge is in our