Temporal dominating set and temporal vertex cover under the lens of degree restrictions

We study the Temporal Dominating Set problem, in which one asks whether a temporal graph G = ( G 1 , ⋯ , G T ) given as a sequence of snapshot graphs over the same vertex set V has a set S of at most k temporal vertices such that each vertex v of V is dominated by some w ∈ S in the snapshot that contains w . Additionally, we consider Temporal Partial Dominating Set , where one asks whether at least t (and not necessarily all) vertices of V can be dominated by S and a further generalization in which the solution may only contain a bounded number of temporal vertices from each snapshot. We analyze how the complexity of Temporal (Partial) Dominating Set is influenced by the maximum snapshot degree and the structure of the underlying graph, the graph with vertex set V whose edge set is the union of all snapshot edge sets. For example, we obtain a complexity dichotomy for the maximum snapshot degree and show that Temporal Partial Dominating Set is fixed-parameter tractable for tw + Δ , where tw and Δ denote the treewidth and the maximum degree of the underlying graph of G , respectively. We also study which of our results transfer to the well-studied Temporal Vertex Cover problem. For example, we show that Temporal Vertex Cover is also fixed-parameter tractable for tw + Δ which substantially extends the previously known polynomial-time algorithms for the case that the underlying graph is a path or cycle.