A two-player zero-sum game is considered, where one player (the patroller) chooses a route to patrol an object that is a lattice graph. The other player (the hider) chooses a vertex of the graph to attack, with the goal of remaining undetected for a certain time T. The patroller wins if, during the time T, its route passes through the vertex chosen by the other player. The equilibrium in this game is derived in the class of mixed strategies.