Bipartite Graph to solve the wolf river crossing problem
I have never really imagined solving this problem with a graph, but it is an interesting point of view.
My guess is that the graph you have to create is a bipartite graph with 16 vertices, each vertex representing the state of the situation, that is representing where is each of the different actors of the problem: the wolf, the goat, the cabbage and the boat. Each actor have two positions possibles (each side of the river), so $2^4 = 16$ different configurations. You might then delete forbidden configurations, representing a case where the wolf is alone with the goat or the goat alone with the cabbage.
Now, you can add an edge between two vertices $v$ and $w$ if and only if you can go from $v$ to $w$ by crossing the river with the boat, with one actor from wolf/goat/cabbage on board.
The graph is indeed bipartite, because there are no edges between two vertices where the boat is on the same side.
Solving the problem now results in finding a path between the two vertices where the four actors are on each side of the river.