  A temporal graph is a dynamic graph where every edge is assigned a set of
integer time labels that indicate at which discrete time step the edge is
available. In this paper, we study how changes of the time labels,
corresponding to delays on the availability of the edges, affect the
reachability sets from given sources. The questions about reachability sets are
motivated by numerous applications of temporal graphs in network epidemiology,
which aim to minimise the spread of infection, and scheduling problems in
supply networks in manufacturing with the opposite objectives of maximising
coverage and productivity. We introduce control mechanisms for reachability
sets that are based on two natural operations of delaying. The first operation,
termed merging, is global and batches together consecutive time labels into a
single time label in the whole network simultaneously. This corresponds to
postponing all events until a particular time. The second, imposes independent
delays on the time labels of every edge of the graph. We provide a thorough
investigation of the computational complexity of different objectives related
to reachability sets when these operations are used. For the merging operation,
i.e. global lockdown effect, we prove NP-hardness results for several
minimization and maximization reachability objectives, even for very simple
graph structures. For the second operation, independent delays, we prove that
the minimization problems are NP-hard when the number of allowed delays is
bounded. We complement this with a polynomial-time algorithm for minimising the
reachability set in case of unbounded delays.
