A temporal graph is a graph whose edges appear at certain points in time. These graphs are temporally connected (in class TC) if all vertices can reach each other by temporal paths (traversing the edges in chronological order). Reachability based on temporal paths is not transitive, with important consequences. For instance, TC graphs do not always admit TC spanning trees. In this paper, we show that deciding if a given temporal graph admits a TC spanning tree is actually NP-complete. Then, we explore possible relaxations. A key feature of TC spanning trees is to support reachability along the same paths in both directions. We show that this property is not equivalent to TC spanning trees, it is more general and it can be tested in polynomial time. Still, minimizing the size of a spanner preserving this property -- a bidirectional spanner -- is \textsf{NP}-hard even more generally than TC spanning tree, including the setting of simple temporal graphs. Along the way, we show that deciding the existence of TC spanning tree is FPT when parameterized by the feedback edge set number (fes) of the underlying graph, and deciding bidirectional spanners of size $k$ is FPT when parameterized by fes + $\ell$ (the maximum number of labels per edge). On the structural side, we show that TC trees always admit a pivot vertex or a pivot edge -- reachable by all vertices by a certain time and able to reach all vertices afterward -- a fact that may be of independent interest.

翻译：时间图是一种边在特定时间点出现的图。若所有顶点都能通过时间路径（按时间顺序遍历边）相互到达，则这类图具有时间连通性（属于TC类）。基于时间路径的可达性不具有传递性，这会产生重要影响。例如，TC图并不总是存在TC生成树。本文证明，判定给定时间图是否存在TC生成树实际上是NP完全的。随后，我们探讨了可能的松弛方案。TC生成树的关键特征在于支持沿相同路径的双向可达性。我们证明这一性质不等价于TC生成树，它更为一般且可在多项式时间内判定。然而，最小化保留该性质的展形图——即双向展形图——的规模是NP难的，其难度甚至超过TC生成树，包括简单时间图的场景。在此过程中，我们证明：当以底层图的反馈边集数（fes）为参数时，判定TC生成树的存在性属于FPT；当以fes + ℓ（每条边的最大标签数）为参数时，判定规模为$k$的双向展形图属于FPT。在结构方面，我们证明TC树总存在一个枢轴顶点或枢轴边——该顶点/边在特定时间前可被所有顶点到达，并在之后可达所有顶点——这一发现可能具有独立的研究价值。