  We study an optimization problem that arises in the design of covering
strategies for multi-robot systems. Consider a team of $n$ cooperating robots
traveling along predetermined closed and disjoint trajectories. Each robot
needs to periodically communicate information to nearby robots. At places where
two trajectories are within range of each other, a communication link is
established, allowing two robots to exchange information, provided they are
"synchronized", i.e., they visit the link at the same time. In this setting a
communication graph is defined and a system of robots is called
\emph{synchronized} if every pair of neighbors is synchronized.
  If one or more robots leave the system, then some trajectories are left
unattended. To handle such cases in a synchronized system, when a live robot
arrives to a communication link and detects the absence of the neighbor, it
shifts to the neighboring trajectory to assume the unattended task. If enough
robots leave, it may occur that a live robot enters a state of
\emph{starvation}, failing to permanently meet other robots during flight. To
measure the tolerance of the system under this phenomenon we define the
\emph{$k$-resilience} as the minimum number of robots whose removal may cause
$k$ surviving robots to enter a state of starvation. We show that the problem
of computing the $k$-resilience is NP-hard if $k$ is part of the input, even if
the communication graph is a tree. We propose algorithms to compute the
$k$-resilience for constant values of $k$ in general communication graphs and
show more efficient algorithms for systems whose communication graph is a tree.
