We study pursuit-evasion games on graphs with a single pursuer and an invisible evader. The pursuer may assign integer travel times to the edges of the graph and specify a finite sequence of vertices to query, one per time step. The evader then chooses a walk over the same time horizon, aiming to elude all queries. With unit travel times, the setting in which the evader must move at every time step is known as the hunter and rabbit game, while the variant in which the evader can wait at a vertex can be phrased as a firefighting game: the vertices of a burning graph must be extinguished, and any vertex left burning reignites its neighbors. For both settings, we show that the power to choose travel times allows a single pursuer to succeed in polynomial time on any graph. This contrasts with unweighted graphs, where the number of hunters or firefighters needed can grow linearly in the number of vertices. If the evader, in addition to waiting, may start at an earlier time unknown to the pursuer, we show that the pursuer still wins on any graph given exponential time.

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