On finding separators in temporal split and permutation graphs

We study the NP-hard Temporal $(s,z)$-Separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle Temporal $(s,z)$-Separation by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general Temporal $(s,z)$-Separation remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of “change over time”.