A topological ordering of a directed acyclic graph places its vertices in a sequence such that every directed edge

$$u\to v$$

places $u$ before $v$.

Such an ordering exists exactly when the directed graph has no directed cycle.

One algorithm repeatedly removes a vertex with indegree zero and then removes its outgoing edges. Another uses depth-first search and orders vertices by decreasing finish time.

With adjacency lists, either approach can run in

$$O(|V|+|E|)$$

time.

A topological order is not necessarily unique. It represents one valid linear schedule consistent with a partial set of dependencies, making it useful for build systems, prerequisite graphs and task scheduling.