  A temporal (directed) graph is a graph whose edges are available only at
specific times during its lifetime, $\tau$. Paths are sequences of adjacent
edges whose appearing times are either strictly increasing or non-strictly
increasingly (i.e., non-decreasing) depending on the scenario. Then, the
classical concept of connected components and also of unilateral connected
components in static graphs and digraphs naturally extends to the temporal
setting.
  In this paper, we answer to the following fundamental questions in temporal
graphs. (i) What is the complexity of deciding the existence of a component of
size $k$, parameterized by $\tau$, by $k$, and by $k+\tau$? We show that this
question has a different answer depending on the considered definition of
component and whether the temporal graph is directed or undirected. (ii) What
is the minimum running time required to check whether a subset of vertices are
pairwise reachable? A quadratic algorithm is known but, contrary to the static
case, we show that a better running time is unlikely unless SETH fails. (iii)
Is it possible to verify whether a subset of vertices is a component in
polynomial time? We show that depending on the definition of temporal component
this test is NP-complete.
