The continuous patrolling game studied here was first proposed in Alpern et al. (2011), which studied a discrete time game where facilities to be protected were modeled as the nodes of a graph. Here we consider protecting roads or pipelines, modeled as the arcs of a continuous network $Q$. The Attacker chooses a point of $Q$ to attack during a chosen time interval of fixed duration (the attack time, $alpha$). The Patroller chooses a unit speed path on $Q$ and intercepts the attack (and wins) if she visits the attacked point during the attack time interval. Solutions to the game have previously been given in certain special cases. Here, we analyze the game on arbitrary networks. Our results include the following: (i) a solution to the game for any network $Q$, as long as $alpha$ is sufficiently short, generalizing the known solutions for circle or Eulerian networks and the network with two nodes joined by three arcs; (ii) a solution to the game for all tree networks that satisfy a condition on their extremities. We present a conjecture on the solution of the game for arbitrary trees and establish it in certain cases.
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