Recognizing and Realizing Temporal Reachability Graphs

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.