A topological sorting is the task of constructing a topological order of a finite directed graph, or reporting that a directed cycle makes one impossible. Kahn's algorithm repeatedly takes a vertex with no unfinished prerequisites. This is useful whenever an operation consumes results produced by earlier operations: once a vertex is reached in a topological order, every prerequisite along a directed path has already been processed. [ref]
The reason a choice is always available in a nonempty finite Directed acyclic graph is simple. If every vertex had an incoming directed edge, repeatedly following predecessors would eventually revisit a vertex, giving a directed cycle. Removing a vertex and its incident directed edges cannot create a directed cycle, so the argument repeats until every vertex is listed. This proves existence of a topological order and explains why finiteness matters to this procedure.
// Target: combinatorics.bigb
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Topological sorting is a linear ordering of the vertices of a directed acyclic graph (DAG) such that for every directed edge \( u \rightarrow v \), vertex \( u \) comes before vertex \( v \) in the ordering. This concept is particularly useful in scenarios where there is some dependency or precedence represented by the directed edges.