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A search algorithm is complete for a specified class of problems if it finds a solution whenever a solution exists in that class. This definition need not require termination on unsolvable instances; an additional termination guarantee should be stated separately. A finite branching factor makes breadth-first tree search complete for finite-depth solutions, while depth-first tree search additionally needs a reason it cannot be trapped on an infinite branch.

For a finite constraint satisfaction problem with $v$ variables, all complete assignments have depth $v$ when each step assigns one new variable. Thus the shallowest-solution property of breadth-first search offers no advantage: every complete solution is already at that depth.

Depth-first search with backtracking stores a path and pending choices, using $O(vd)$ auxiliary space with ordinary explicit choice bookkeeping, or $O(v)$ when each level stores only a current value index. Breadth-first search may store an exponentially large frontier, up to $O(d^v)$ partial assignments. Depth-first search can also combine naturally with immediate constraint propagation and undoing changes. The finite depth avoids the infinite-branch obstruction of unrestricted depth-first tree search. Neither traversal removes the possible exponential time complexity.

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.

Let $d_t$ be the graph distance between fox and rabbit after round $t$, with $d_0$ their initial graph distance. Initial reachability makes $d_0$ finite, and ordinary moves preserve membership in their common connected component of a graph. After the rabbit's next move, the triangle inequality gives intermediate separation at most $d_t+1$. If that separation is at most two, the fox catches the rabbit that round. Otherwise the fox takes two steps along a shortest path, leavingThus a successful surviving history has at most $d_0-1$ rounds; if it does not escape by then, the next round captures it. The shortest-path pursuit distance bound excludes infinite live branches, even when the rabbit stays. Together with the finite branching factor, it makes the whole relevant search tree finite, so depth-first tree search eventually finds an escape whenever one exists.

For nonoptimality, take the corridor of part (g) and let depth-first tree search try left first. It immediately returns the one-round escape with bonus $1$, whereas trying right twice earns $2$. The decreasing separation proves termination, not optimality of the first returned escape.

Suppose two pieces occupy a common connected component of a graph in an unweighted undirected graph. For integers $a>b\ge 0$, in each round an evader moves at most $b$ edges, then a pursuer moves up to $a$ edges along a shortest path to the new evader position, stopping upon capture. Let $d_t$ be their finite graph distance at a surviving round boundary.

The triangle inequality bounds the separation after the evader moves by $d_t+b$. If that separation is at most $a$, the pursuer captures it. Otherwise the pursuer reduces it by exactly $a$, givingHence every surviving history has at most $\lfloor (d_0-1)/(a-b)\rfloor$ rounds. With a finite branching factor, the surviving search tree is finite and depth-first tree search becomes a complete search algorithm. This does not make its first goal optimal for arbitrary rewards. Initial reachability is essential: an infinite graph distance supplies no decreasing finite bound.