Graph Theory - Strongly Connected Graphs

Strongly Connected Graphs

A strongly connected graph is a directed graph in which there is a directed path from any vertex to every other vertex in the graph. In other words, for any pair of vertices u and v in the graph, there exists a directed path from u to v and a directed path from v to u.

Properties of Strongly Connected Graphs

Strongly connected graphs have several important properties, such as −

• Bidirectional Paths: For any two vertices u and v, there are directed paths from u to v and from v to u. This property makes the graph strongly connected.
• Single Component: A strongly connected graph is a single strongly connected component. Any pair of vertices is reachable from each other.
• Closure: The graph forms a "closed" structure in the sense that no matter which vertex you start from, you can reach all other vertices and return to the starting point through directed paths.
• Recurrence: In a strongly connected graph, there are recurring cycles that allow traversal between vertices in any direction, ensuring the graph remains interconnected.

Strongly Connected Components (SCCs)

In any directed graph, it is useful to break the graph into smaller subgraphs called strongly connected components (SCCs). An SCC is a maximal subgraph where each vertex is reachable from every other vertex in the same component.

The following image shows a graph broken into strongly connected components −

• Maximal: An SCC is the largest subgraph where all vertices are mutually reachable.
• Independence: Each SCC is strongly connected, and no vertex in one SCC is reachable from a vertex in another SCC.
• Partition: The entire graph can be partitioned into SCCs.

Conditions for Strong Connectivity

A directed graph is strongly connected if and only if it satisfies the following condition −

• There is a directed path between any pair of vertices u and v in the graph. That is, for every pair of vertices u and v, there is a path from u to v and a path from v to u.

Example: Strongly Connected Graph

In the below graph, for any pair of vertices, there are directed paths in both directions. Therefore, the graph is strongly connected.

Applications of Strongly Connected Graphs

Strongly connected graphs have many practical applications in areas where traversal between nodes in both directions is essential −

• Web Crawling: In web crawling, strongly connected components represent groups of web pages that can be reached from each other. Understanding SCCs can help identify closely linked web pages or websites.
• Social Networks: Strongly connected graphs model networks where users can interact with each other in both directions. If a network is strongly connected, users can connect with each other regardless of their starting points.
• Dependency Management: In software development and project management, strongly connected graphs represent dependencies between tasks or components that must be handled in both directions. SCCs ensure that dependencies are correctly handled.
• Communication Systems: Strongly connected graphs are used in communication systems to ensure that any message can be passed between nodes, even in the case of node failures, by using alternate paths.

Testing Strong Connectivity

There are various ways to test if a directed graph is strongly connected, such as −

• DFS-Based Algorithm: A depth-first search (DFS) can be used to check for reachability between all vertices. If a DFS traversal from any vertex visits all other vertices, and the reverse DFS traversal also visits all vertices, the graph is strongly connected.
• Kosaraju's Algorithm: Kosaraju's algorithm finds strongly connected components (SCCs) in a graph. If the graph contains only one SCC, it is strongly connected.
• Tarjan's Algorithm: Tarjan's algorithm also finds SCCs in a directed graph. Like Kosaraju's, if there is only one SCC in the graph, it is strongly connected.

Strong Connectivity in Various Graphs

Following are the lists of various types of directed graphs and their strong connectivity status −