Catching a Fast Robber on Interval Graphs

We analyse the Cops and ∞-fast Robber game on the class of interval graphs and prove it to be polynomially decidable on such graphs. This solves an open problem posed in paper “Pursuing a fast robber on a graph” by Fomin et al. [4] The game is known to be already NP-hard on chordal graphs and split-graphs. The game is played by two players, one controlling k cops, the other a robber. The players alternate in turns, all the cops move at once to distance at most one, the robber moves along any cop-free path. Cops win by capturing the robber, the robber by avoiding capture. The analysis relies on the properties of an interval representation of the graph. We show that while the game-state graph is generally exponential, every cops’ move can be decomposed into simple moves of three types, and the states are reduced to those defined by certain cuts of the interval representation. This gives a restricted game equivalent to the original one together with a winning strategy computable in polynomial time.