Patrolling a perimeter

In this paper we study the problem of patrolling a perimeter. The general situation considered here can correspond to different tactical problems and it is studied from the point of view of game theory. To put the ideas in a context we describe it as follows. An intruder seeks to carry out a sabotage on the perimeter of a protected zone. He has to perform the action along n consecutive days and has to position himself each day at one of m strategic points placed on this border. The first day he can take his place at any of the m points, but on successive days he can move only to adjacent points. Furthermore, the perimeter is protected by a patroller, who will select each day one of the m points to inspect. The strategic situation is modeled as a two-person zero-sum game, which is developed on a cyclic set of m points over n time units. We prove some interesting properties of the strategies, solve the game in closed form under certain constraints and obtain bounds for the value of the game in several non-solved cases.