Maximum Bipartite Graph Problem

Introduction

• Topic: Maximum Bipartite Graph Problem (MBGP)
• Method: Step-by-step explanation using an example

Steps to Solve MBGP

Initial Setup

• Vertices labeled 1 to 6
• Bipartite Graph: Two sets of vertices, connections between sets
• Goal: Find maximum matching, i.e., pair as many vertices as possible

Step 1: Starting Point

• Vertices: 1, 2, 3, 4, 5, 6
• Initial Pairing: Pair vertex 1 with vertex 5, vertex 1 with vertex 6
• Connections: Try all possibilities to ensure maximum pairings

Step 2: Additional Pairings

• Focus on other connections after initial pairing
• Reconfigure pairs between 1, 5, and 6

Step 3: Optimize Pairing

• Vertices for Rethinking: 2, 3, 4
• Review Connections: Vertex 1 to 5, Vertex 2 to 6, etc.
• Diagram Representation: Showing connections and their optimizations

Step 4: Consider Alternate Options

• Check connection from vertex 3 to vertex 4
• Attempt to augment connections for maximum pairing
• Example: Vertex 2 to 5, 3 to 4, etc.

Step 5: Diagram Updates

• New Diagram: New connections with augmented paths
• Review all possible connections and ensure to cover maximum
• Example: Vertex 1 to 5, 2 to 6, 3 to 4

Step 6: Advanced Scenarios

• Vertices: 7, 8, 9, 10
• New Connections: Extend problem to show robustness
• Optimize Again: Example paths vertex 1 to 7, vertex 2 to 8

Step 7: Validation

• Final Diagram: Maximum connections achieved
• Verification: Cross-check diagram to ensure no connections missed
• Efficiency: Ensure no repetitive paths or wasted connections

Final Thoughts

• Review Concept: Recap of MBGP steps and importance
• Step-by-Step Method: Emphasize methodical approach ensures maximum connections
• Complex Examples: Extend understanding by considering more vertices and possibilities

Conclusion

• Learning Outcome: Ability to solve MBGP with assorted examples
• Future Applications: Use learnings in larger, more complex bipartite graphs