What is the definition of a strongly connected graph in a directed graph?
Conditions
- The graph is directed.
- The criterion requires mutual reachability between every pair of vertices.
Reasoning, step by step
- Identify the graph as a directed graph.
- Check if there is a directed path from u to v for every pair of vertices.
- Check if there is a directed path from v to u for every pair of vertices.
- If both conditions are met for all pairs, the graph is strongly connected.
Example
In the example directed graph, A can reach B and then C, and C can reach D and then A. Since every pair of vertices is mutually reachable, the graph is strongly connected.
Common misconceptions
- Assuming that one-way reachability is sufficient for strong connectivity.
- Confusing strong connectivity with weak connectivity in directed graphs.
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Related questions
To determine if a directed graph is strongly connected, you must verify that every pair of vertices is mutually reachable. If you find even one pair of vertices where one cannot reach the other, the graph is not strongly connected.
Conditions: The graph is directed.; You need to check mutual reachability for all vertex pairs.
The maximum number of edges in a strongly connected directed graph with n vertices is . This corresponds to a complete directed graph where every pair of distinct vertices has edges in both directions.
Conditions: The graph is directed.; The graph is strongly connected.; The graph has n vertices.
An undirected connected graph with n vertices has at least n-1 edges. This minimum is achieved when the graph forms a tree structure.
Conditions: The graph is undirected.; The graph is connected.; The graph has n vertices.
The maximum number of edges in an undirected simple graph with n vertices is . This maximum is achieved when the graph is a complete graph.
Conditions: The graph is undirected.; The graph is simple (no self-loops or multiple edges between the same pair of vertices).; The graph has n vertices.
A connected graph is an undirected graph in which there exists a path between every pair of vertices. If even one pair of vertices is not connected by a path, the graph is called disconnected.
Conditions: The graph is undirected.; The criterion for judgment is whether a path exists between any two vertices.
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