A temporal graph $\mathcal{G}=(G,\lambda)$ can be represented by an underlying graph $G=(V,E)$ together with a function $\lambda$ that assigns to each edge $e\in E$ the set of time steps during which $e$ is present. The reachability graph of $\mathcal{G}$ is the directed graph $D=(V,A)$ with $(u,v)\in A$ if only if there is a temporal path from $u$ to $v$. We study the Reachability Graph Realizability (RGR) problem that asks whether a given directed graph $D=(V,A)$ is the reachability graph of some temporal graph. The question can be asked for undirected or directed temporal graphs, for reachability defined via strict or non-strict temporal paths, and with or without restrictions on $\lambda$ (proper, simple, or happy). Answering an open question posed by Casteigts et al. (Theoretical Computer Science 991 (2024)), we show that all variants of the problem are NP-complete, except for two variants that become trivial in the directed case. For undirected temporal graphs, we consider the complexity of the problem with respect to the solid graph, that is, the graph containing all edges that could potentially receive a label in any realization. We show that the RGR problem is polynomial-time solvable if the solid graph is a tree and fixed-parameter tractable with respect to the feedback edge set number of the solid graph. As we show, the latter parameter can presumably not be replaced by smaller parameters like feedback vertex set or treedepth, since the problem is W[2]-hard with respect to these parameters.

翻译：时态图 $\mathcal{G}=(G,\lambda)$ 可由一个底层图 $G=(V,E)$ 和一个函数 $\lambda$ 表示，该函数为每条边 $e\in E$ 分配其存在的时间步集合。$\mathcal{G}$ 的可达图是一个有向图 $D=(V,A)$，其中 $(u,v)\in A$ 当且仅当存在一条从 $u$ 到 $v$ 的时态路径。我们研究可达图可实现性（RGR）问题，即判断一个给定的有向图 $D=(V,A)$ 是否为某个时态图的可达图。该问题可以针对无向或有向时态图提出，其可达性可通过严格或非严格时态路径定义，并且对 $\lambda$ 可以有或没有限制（proper、simple 或 happy）。针对 Casteigts 等人（Theoretical Computer Science 991 (2024)）提出的一个开放性问题，我们证明了该问题的所有变体都是 NP 完全的，除了在有向情况下变得平凡的两个变体。对于无向时态图，我们基于实体图（即包含所有在任何实现中可能获得标签的边的图）来研究该问题的复杂度。我们证明，如果实体图是一棵树，则 RGR 问题可在多项式时间内求解；并且相对于实体图的反馈边集数，该问题是固定参数可处理的。正如我们所展示的，后一个参数可能无法被更小的参数（如反馈顶点集或树深度）所替代，因为相对于这些参数，该问题是 W[2]-难的。