Temporal graph realization problems ask whether it is possible to construct a temporal graph that satisfies certain prescribed structural or temporal properties.
The input may specify a particular static graph or a class of static graphs (to be used as the footprint), or place no restriction on the footprint at all. In the first case, the task is to construct a time labeling of the edges in a way that meets the desired temporal constraints, and in the latter case, the goal is to construct both the footprint and the labeling from scratch.
The properties to be realized vary widely. Common objectives include ensuring specific connectivity goals (e.g., being $\mathsf{TC}$), having a maximal or minimal number of temporal edges, enforcing shortest temporal distances according to some matrix $D$, matching prescribed reachability patterns, or respecting degree sequences across time steps.
All of these objectives may be achieved under temporal restrictions such as simplicity (at most one label per edge), properness (no two edges incident to a vertex are active at the same time), or upper bounds on the number of labels (per edge or globally).