Hunter & Mole

We consider a variation of a cops and robbers game in which the cop---here referred to as "hunter"---is not constrained by the graph but must play in the dark against a "mole." We characterize the graphs---which we will call "hunter-win"---on which the hunter can guarantee capture of the mole in bounded time. We also define an optimal hunter strategy (and consequently an upper bound on maximum game time on hunter-win graphs) and note that an optimal hunter strategy need not take advantage of the hunter's unconstrained movement! This game comes from a puzzle of unknown origin which was told to the authors by Dick Hess.