  We present solutions to a continuous patrolling game played on network. In
this zero-sum game, an Attacker chooses a time and place to attack a network
for a fixed amount of time. A Patroller patrols the network with the aim of
intercepting the attack with maximum probability. Our main result is the proof
of a recent conjecture on the optimal patrolling strategy for trees. The
conjecture asserts that a particular patrolling strategy called the
E-patrolling strategy is optimal for all tree networks. The conjecture was
previously known to be true in a limited class of special cases. The
E-patrolling strategy has the advantage of being straightforward to calculate
and implement. We prove the conjecture by presenting $\varepsilon$-optimal
strategies for the Attacker which provide upper bounds for the value of the
game that come arbitrarily close to the lower bound provided by the
E-patrolling strategy. We also solve the patrolling game in some cases for
complete networks.
