On the Hardness of Finding Temporally Connected Subgraphs of Any Size
For researchers in temporal graph theory and algorithms, this paper resolves open questions and provides strong hardness results that separate closed from open temporal components.
The paper strengthens the NP-hardness result for finding temporally connected subgraphs of size at least 3 in temporal graphs, showing it is NP-hard in all standard settings. It also proves inapproximability bounds and completes the parameterized complexity landscape for exact size k.
Temporal graphs are graphs whose edges are only present at certain points in time. Reachability in these graphs relies on temporal paths, where edges are traversed chronologically. A temporal graph that offers all-pairs reachability is said to be temporally connected (or TC). For temporal graphs that are not TC, a natural question is whether they admit a TC subgraph (a.k.a. closed temporal component) of a given size $k$. This question was one of the earliest in the field, shown to be NP-hard by Bhadra and Ferreira in 2003. We strengthen this result dramatically, showing that deciding if a TC subgraph exists on at least $3$ vertices is already NP-hard in all the standard temporal graph settings (directed/undirected and strict/non-strict through simple and proper reductions). This implies a strong separation between closed temporal components and open temporal components (where temporal paths can travel outside the component), for which inclusion-maximal components can be found in polynomial time. As a by-product, our reductions strengthen a number of existing results and establish new derived results. They imply that the size of the largest TC subgraph cannot be approximated within a factor of $(1-ε)n$ in directed graphs, and within a factor of $(1-ε)\frac{n}{2}$ in undirected graphs. One of the reductions also completes the complexity landscape for TC subgraphs of size exactly $k$ when parameterized by $k$ (answering the missing non-strict case). Finally, on the structural side, our results imply that there exist arbitrarily large TC graphs of constant lifetime without nontrivial TC subgraphs, and we also show that there exist TC graphs of arbitrary girth, both facts being of independent interest.