In the field of data structures, a **strongly connected graph** is a type of directed graph in which there is a path from any vertex to any other vertex. This concept is crucial in understanding the connectivity and reachability within a graph.

## Definition

A __directed graph__, also known as a digraph, consists of a set of vertices and a set of directed edges. Each directed edge connects an origin vertex to a destination vertex. In a **strongly connected graph**, every vertex is reachable from every other vertex through directed edges.

## Properties

A strongly connected graph has several notable properties:

**Cycle Property:**Every strongly connected graph contains at least one cycle. This means that starting from any vertex, we can traverse the graph and return back to the starting vertex.**Path Property:**For every pair of vertices u and v, there exists both a path from u to v and a path from v to u. This ensures that all vertices are interconnected.**Strong Connectivity Property:**A directed graph G is considered strongly connected if there is a path between every pair of vertices in G.

## Determining Strong Connectivity

To determine whether a given graph is strongly connected, we can use various algorithms such as Tarjan’s algorithm or Kosaraju’s algorithm.

### Tarjan’s Algorithm

Tarjan’s algorithm is based on the concept of depth-first search (DFS). It performs DFS on the given graph and keeps track of the lowest reachable node for each node encountered during the search. If all nodes can be reached from any other node, the graph is strongly connected.

### Kosaraju’s Algorithm

Kosaraju’s algorithm also utilizes DFS but in two passes. In the first pass, it performs a DFS on the graph and assigns a finishing time to each vertex.

In the second pass, it reverses the graph and performs a DFS again, visiting vertices in decreasing order of their finishing times. If every vertex is visited in the second pass, the graph is strongly connected.

## Applications

Strongly connected graphs have various applications in different fields:

**Network Routing:**Strong connectivity plays a crucial role in determining efficient routes between nodes in communication networks.**Web Crawling:**When crawling web pages, it is essential to ensure that all pages are reachable from any starting point. Strong connectivity helps in achieving this.**Software Verification:**Strong connectivity analysis is used to ensure that all parts of a software system are reachable during testing and verification processes.

In conclusion, a strongly connected graph is a directed graph where there exists a path between every pair of vertices. It possesses important properties such as cycle property and path property.

Determining strong connectivity can be done using algorithms like Tarjan’s algorithm or Kosaraju’s algorithm. The concept finds application in various domains such as network routing, web crawling, and software verification.