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Advanced Graph Problems

Complex graph algorithms and applications for advanced problem-solving.

Network flow​

  • Maximum Flow: Maximum flow from source to sink
  • Minimum Cut: Minimum capacity cut separating source and sink
  • Ford-Fulkerson Algorithm: Finding maximum flow
  • Edmonds-Karp: BFS-based implementation of Ford-Fulkerson

Bipartite matching​

  • Bipartite Graph: Graph with two sets of vertices
  • Maximum Matching: Maximum number of edges in matching
  • Hungarian Algorithm: Assignment problem solution
  • Hopcroft-Karp: Efficient bipartite matching algorithm

Graph coloring​

  • Vertex Coloring: Assign colors to vertices with constraints
  • Edge Coloring: Assign colors to edges with constraints
  • Chromatic Number: Minimum number of colors needed
  • Applications: Scheduling, register allocation

Hamiltonian & Eulerian paths​

  • Hamiltonian Path: Path visiting each vertex exactly once
  • Hamiltonian Cycle: Hamiltonian path that forms cycle
  • Eulerian Path: Path using each edge exactly once
  • Eulerian Cycle: Eulerian path that forms cycle

Graph isomorphism​

  • Definition: Two graphs are isomorphic if they have same structure
  • Detection: Determining if two graphs are isomorphic
  • Applications: Chemical compound analysis, network comparison
  • Complexity: No known polynomial-time algorithm