On computing large temporal (unilateral) connected components

A temporal (directed) graph is one where edges are available only at specific times during its lifetime, τ. Paths in these graphs are sequences of adjacent edges whose appearing times are either strictly increasing or non-strictly increasing. Classical concepts of connected and unilateral components can be naturally extended to temporal graphs. We address fundamental questions in temporal graphs. (i) What is the complexity of deciding the existence of a component of size k, parameterized by τ, k, and k+τ? The answer depends on the component definition and whether the graph is directed or undirected. (ii) What is the minimum running time to check if a subset of vertices is pairwise reachable? A quadratic-time algorithm exists, but a faster time is unlikely unless SETH fails. (iii) Can we verify if a subset of vertices forms a component in polynomial time? This is NP -complete depending on the temporal component definition.