Java Program to Find a Mother Vertex in a Graph

A vertex in a graph that is the starting point for all other vertices to reach is known as the mother vertex. Stated otherwise, if vertex v is a mother vertex, then a path connects v to every other vertex in the network. Finding a mother vertex is useful in several applications, such as analyzing the connectivity of graphs or optimizing certain graph-based algorithms.

In this section, we will cover how to find a mother vertex in a directed graph, explain the approach, and provide a detailed Java implementation.

Approach to Finding a Mother Vertex

The key idea behind finding a mother vertex revolves around Depth First Search (DFS). The steps to find the mother vertex are as follows:

1. Perform DFS Traversal
   Traverse the graph using DFS and keep track of the last finished vertex. This vertex is a potential candidate for being a mother vertex because, if it finishes last, it means that it is part of the largest possible DFS tree that can be rooted at any vertex.
2. Verify the Potential Mother Vertex
   Once a candidate vertex is found, perform a DFS starting from that vertex to check if all the vertices are reachable. If all vertices can be reached, this candidate is indeed a mother vertex.
3. Optimization
   If there is more than one connected component in the graph, there would not be any mother vertex. We handle this by ensuring that we cover all vertices in the DFS traversal.

Steps

1. DFS Traversal
   We first traverse the graph using DFS. To track if a given node as already been visited, there is a visited array maintained as well. The strategy is to take any node and label it as a visited node and then apply the process to the nodes connected to it.
2. Finding the Last Finished Vertex
   The last finished vertex in a DFS traversal may be a mother vertex. It is because, in a strongly connected graph, all nodes can be reached from this vertex if it finishes last.
3. Verification
   Once we find a potential mother vertex, we perform DFS from this vertex. If we can visit all vertices, then it is the mother vertex.

File Name: MotherVertex.java

Output:

 The mother vertex is: 5 

Complexity

The time complexity of this solution is O(V + E), where V is the number of vertices and E is the number of edges. Because we perform DFS twice: once to find the candidate mother vertex and once to verify it.

Conclusion

Finding a mother vertex in a directed graph is a useful operation in graph theory, often used to identify the reachability of a node in network analysis. The approach described here leverages depth-first search (DFS) to efficiently determine the mother vertex, ensuring that the solution is both optimal and easy to implement.