Patrolling environments by means of autonomous mobile robots has received an increasing attention in the last few years. The interest of the agent community is mainly in the development of effective patrolling strategies. Approaches based on game theory have been demonstrated to be very effective. They model the patrolling situation as a two-player leader-follower game, where the patroller is the leader and the intruder is the follower. These models present several limitations that prevent their use in realistic settings. In this paper, we extend the most general model from the state of the art along two directions, we propose algorithms to solve efficiently our extensions, and we experimentally evaluate them.