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.

翻译：时间图是一种边仅在特定时间点出现的图。这类图中的可达性依赖于时间路径，即边需按时间顺序遍历。若时间图支持所有顶点对之间的可达性，则称其为时间连通（TC）。对于非TC的时间图，自然的问题是它们是否存在给定大小$k$的TC子图（即闭合时间分量）。该问题最早由Bhadra与Ferreira于2003年提出，并被证明是NP难问题。我们显著强化了这一结论：通过简单且适当的归约，证明在所有标准时间图设定（有向/无向、严格/非严格）中，判定是否存在至少包含3个顶点的TC子图已经是NP难的。这意味着闭合时间分量与开放时间分量（允许路径穿越分量外部）之间存在显著差异——后者的包含极大分量可在多项式时间内找到。作为副产品，我们的归约强化了多项现有结论并推导出新结果：在有向图中，最大TC子图的大小无法在$(1-ε)n$近似因子内逼近；在无向图中，则无法在$(1-ε)\frac{n}{2}$近似因子内逼近。其中一项归约还完善了以参数$k$表示精确大小为$k$的TC子图问题的复杂度图景（填补了缺失的非严格情形）。最后，在结构方面，我们的结果表明存在任意大的恒定生存期TC图（其中不含非平凡TC子图），并证明了存在任意围长的TC图——这两项事实均具有独立研究价值。