We study the parameterized complexity of maximum temporal connected components (tccs) in temporal graphs, that is, graphs whose edges are available only at specific points in time. In a tcc, every pair of vertices must be able to reach one another via time-respecting paths. We consider both maximum open tccs (openTCC), which allow temporal paths through vertices outside the component, and closed tccs (closedTCC), which require at least one temporal path entirely within the component for every pair of vertices. We perform a comprehensive study of the openTCC and closedTCC problems with respect to both structural parameters (treewidth, pathwidth, vertex cover number) and a temporal parameter (temporal path number). We show that the exact complexity, i.e., paraNP-hardness vs XP-tractability, depends on both whether we seek an open or closed tcc and on whether the temporal graph is directed or not. Vertex cover number suffices for XP algorithms for both openTCC and closedTCC on undirected temporal graphs only, while temporal path number suffices only for openTCC in both directed and undirected temporal graphs. Our results are tight: every XP algorithm is complemented by a matching W[1]-hardness result, and for every other case we prove NP-hardness for small constant values of the parameters even on planar graphs. Finally, we prove that both problems become fixed-parameter tractable on both directed and undirected graphs when parameterized by treewidth and temporal path number together.

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