The cat and the noisy mouse

We consider a variant of a pursuit and evasion game studied independently by Britnell and Wildon as well as Haslegrave. In their game, a cat has to catch an invisible mouse that moves along the edges of some graph G. In our version, the cat receives partial information about its distance to the mouse, and we show that the cat has a winning strategy if and only if G is a forest. Seager proposed a similar game with complete distance information whose rules cause some small yet important differences to the game we consider.