Past exam of the computer science course of the University of Oxford / 2023 / Artificial intelligence / 1 / e

For incompleteness, use the two-row, three-column board from part (c), with exit $(1,2)$. If stay is the first action explored, depth-first tree search follows the repeating state forever. Moving the rabbit down instead survives the round: under legal horizontal-then-vertical movement the fox moves to $(1,0)$ and then $(1,1)$, leaving the rabbit safely at the exit. Thus a one-round escape exists but is never reached in this action order.

For nonoptimality, use an unobstructed one-row board with columns $0,\dots ,6$, initial fox column $0$, initial rabbit column $4$, and exits at columns $3$ and $6$. Moving left once escapes at column $3$ with the fox at column $2$. Moving right twice escapes at column $6$ with successive fox positions $2,4$. A depth-first tree search that explores the latter route first returns two rounds although one suffices. This counterexample supplies an actual returned, suboptimal escape independently of the infinite-branch example.