What is Topological sorting in graph theory?

Topological sorting is a method of arranging vertices in a directed acyclic graph in a linear order.

In graph theory, a directed acyclic graph (DAG) is a graph that has directed edges and no cycles. A cycle is a path that starts and ends at the same vertex. Topological sorting is a method of arranging the vertices of a DAG in a linear order such that for every directed edge (u, v), vertex u comes before vertex v in the ordering. This ordering is called a topological order.

Topological sorting can be used to solve a variety of problems, such as scheduling tasks that have dependencies on each other. For example, if task A must be completed before task B can start, and task B must be completed before task C can start, then the tasks can be scheduled in the order A, B, C.

To perform a topological sort, we can use a depth-first search algorithm. Starting at any vertex, we visit all of its neighbours and mark them as visited. We then recursively visit all of the unvisited neighbours of the visited vertices. Once we have visited all of the neighbours of a vertex, we add it to the front of the topological order.

If the graph contains a cycle, then it is not possible to perform a topological sort. This is because a cycle creates a circular dependency, which cannot be resolved by a linear ordering. Therefore, it is important to check for cycles before attempting to perform a topological sort.