On the Hardness of Finding Temporally Connected Subgraphs of Any Size

Abstract

Temporal graphs are graphs whose edges are present only at certain points in time. Reachability in these graphs is defined via temporal paths, in which edges are traversed in chronological order. A temporal graph is temporally connected (or TC) if every ordered pair of vertices is connected by a temporal path. When the graph itself is not TC, a natural question is whether it admits a TC subgraph (a.k.a.~closed temporal component) of a given size $k$. This question was one of the earliest and most studied in the field, shown to be NP-hard by Bhadra and Ferreira in 2003. We strengthen this result dramatically, showing that deciding if a temporal graph admits a TC subgraph \emph{of any size} (beyond the trivial case of a single vertex in the directed and a single edge in the undirected case) is already NP-hard. Our result holds for all standard temporal graph settings, answering a series of open questions in the field and strengthening several existing results. This sharply separates closed components from open ones (where temporal paths can travel outside the component), for which the analogous problem is trivially solvable in polynomial time. More precisely, our reductions imply that the size of the largest TC subgraph cannot even be approximated within a factor of~$(1-\epsilon)n$ in directed graphs, and within a factor of~$(1-\epsilon)\frac{n}{2}$ in undirected graphs. They also complete the complexity landscape for TC subgraphs of size exactly $k$ when parameterized by $k$ (answering the missing ``non-strict'' case). Our results also have structural implications. In particular, they imply that there exist arbitrarily large TC graphs without nontrivial TC subgraphs, and that there exist TC graphs of arbitrary girth, both facts being of independent interest.