Temporal reachability minimization: Delaying vs. deleting

We study spreading processes in temporal graphs, that is, graphs whose connections change over time. More precisely, we investigate how such a spreading process, emerging from a given set of sources, can be contained to a small part of the graph. We consider two ways of modifying the graph, which are (1) deleting and (2) delaying connections. We show a close relationship between the two associated problems. It is known that both problems are W[1]-hard when parameterized by the number of modifications. We consider the number of vertices to which the spread is contained as a parameter. Surprisingly, we prove W[1]-hardness for the deletion variant but fixed-parameter tractability for the delaying variant. Furthermore, we give a polynomial time algorithm for both problem variants when the graph has a tree structure and show how to generalize this result to an FPT-algorithm for the so-called timed feedback vertex number as a parameter.