In this paper, we consider a patrolling problem in adversarial environments where intruders use the information about a patrolling path to increase chances of successful attacks. We use Markov chains to design randomized patrolling paths on graphs. We present four different intruder models, each of which use information about the patrolling path in a different manner. We characterize the expected rewards for those intruder models as a function of the Markov chain that is being used for patrolling. We show that minimizing the reward functions is a non-convex optimization problem. We propose a pattern search based algorithm to determine a locally optimal patrolling strategy. We also show that for a certain type of intruder, a deterministic patrolling policy given by an orienteering tour of the graph is the optimal patrolling strategy.