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Updated 2024-04-25 03:56

Why must a graph with a topological sort be acyclic and why must an acyclic graph have a topological sort?

Why must a graph with a topological sort be acyclic?

A directed graph with no directed cycles is called an acyclic graph or directed acyclic graph (DAG). A topological sort of a DAG is a linear ordering of its vertices such that for every directed edge from vertex u to vertex v, u comes before v in the ordering. If a graph has a topological sort, then it must be acyclic. This is because if there were a directed cycle in the graph, then there would be no way to order the vertices in a way that satisfies the topological sort definition.

Why must an acyclic graph have a topological sort?

Conversely, if a graph is acyclic, then it must have a topological sort. This is because an acyclic graph can be decomposed into a series of strongly connected components (SCCs). Each SCC is a set of vertices that are all reachable from each other. The SCCs of an acyclic graph can be ordered in a way such that for every directed edge from vertex u in SCC A to vertex v in SCC B, SCC A comes before SCC B in the ordering. Once the SCCs have been ordered, the vertices within each SCC can be ordered in any way. The resulting ordering is a topological sort of the graph.

Related Questions and Brief Answers

1. What is the difference between a directed and an undirected graph?
   • A directed graph has edges with directions, while an undirected graph does not.
2. What is the definition of a cycle in a graph?
   • A cycle is a path in a graph that starts and ends at the same vertex.
3. What is a strongly connected component (SCC)?
   • An SCC is a set of vertices in a graph that are all reachable from each other.
4. Can a graph have multiple topological sorts?
   • Yes, a graph can have multiple topological sorts if it contains no directed cycles.
5. What is the time complexity of finding a topological sort of a graph?
   • The time complexity of finding a topological sort of a graph is O(V + E), where V is the number of vertices and E is the number of edges.

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