 Solution

Write $F=|V|\le mn$. The ordered position pairs give at most $F^2$ states, or $F(F-1)$ live states after excluding capture. Thus the usual Big O notation estimate is $O(m^2{n}^2)$. An additional terminal-failure marker changes this by only one.

For $k$ distinguishable foxes, retain $(r,f_1,\dots ,f_k)$. Different foxes may share a square; none may share the rabbit's square in a live state. There are at most $F(F-1{)}^k$ live position tuples, giving $O((mn{)}^{k+1})$. These counts describe possible encodings; obstacles, starting positions and the fixed fox rule can make the reachable state space much smaller.