4.69 A city can be divided into several rectangular blocks: it has $n$ streets running parallel to each other, and another set of $m$ streets intersecting the first set at right angles. Some police officers are on duty on the city's streets (not at the intersections). Each police officer reports the number of the bus that passes by, its direction, and the time it passes. To be able to reconstruct the route of a bus traveling along a loop (without retracing any street segment) based on the information provided by the police, what is the minimum number of police officers needed on the streets?
［Solution］Suppose several police officers stand on duty at required positions. We consider the road between two adjacent intersections as a segment and remove the road segments where police officers are stationed. According to the problem, the road network is then divided into $k$ connected segments without any loops, and these segments are not connected to each other.

If a connected segment of the road network contains $p$ intersections, then it has exactly $p-1$ road segments. Since the entire city has $m n$ intersections, the $k$ segments of the road network have at most $m n-k$ road segments, and these segments do not have police officers on duty, otherwise they would have been removed. Since the entire city has $2 m n-m-n$ road segments, the number of road segments where police officers are stationed is at least
$$
m n-m-n+k \geqslant(m-1)(n-1) \text {. }
$$

In the diagram, small circles represent the positions of the police officers, which clearly meet the requirements of the problem and involve a total of $(m-1)(n-1)$ police officers. Therefore, the minimum number required is $(m-1)(n-1)$.