Periodic Patrols on the Line and Other Networks

We consider a patrolling game on a graph recently introduced by Alpern et al. (2011) where the Patroller wins if he is at the attacked node while the attack is taking place. This paper studies the periodic patrolling game in the case that the attack duration is two periods. We show that if the Patroller's period is even, the game can be solved on any graph by finding the fractional covering number and fractional independence number of the graph. We also give a complete solution to the periodic patrolling game on line graphs of arbitrary size, extending the work of Papadaki et al. (2016) to the periodic domain. This models the patrolling problem on a border or channel, which is related to a classical problem of operational research going back to Morse and Kimball (1951). A periodic patrol is required to start and end at the same location, for example the place where the Patroller leaves his car to begin a foot patrol.