Shortest-path pursuit distance bound ($d_{t+1}\le d_t-a+b$)

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.