Cops, robbers, and infinite graphs

We study the simplest variant of the cops and robbers game, where one cop tries to catch one robber by moving along the edges of a graph. A well known result connected to this game is that on a finite graph the cop has a winning strategy if and only if the graph is constructible. Chastand et al. proposed the notion of weakly cop-win graphs which they thought could lead to a generalisation of this result. We show, that this is not the case (not even for locally finite graphs). However, with a very similar winning criterion we can show that every constructible graphs is weakly cop-win. We also characterise the locally finite constructible graphs as the graphs for which the cop has a so-called protective strategy and prove that the existence of such a strategy implies constructibility for arbitrary graphs.