Abstract

In the vertex pursuit game of cops and robbers on finite graphs, the cop has a winning strategy if and only if the graph admits a dominating order. Such graphs are called constructible in the graph theory literature. This equivalence breaks down for infinite graphs, and variants of the game have been proposed to reestablish partial connections between constructibility and being cop-win. We answer an open question of Lehner about one of these variants by giving examples of weak cop-win graphs which are not constructible. We show that the index set of computable constructible graphs is Π11 hard and the index set of computable constructible locally finite graphs is Π40 hard. Finally, we give an example of a computable constructible graph for which every dominating order computes 0″.