What is Bipartite Graph?

What is a Bipartite Graph? | Comprehensive Guide

A bipartite graph is a type of graph where the set of vertices can be divided into two distinct sets such that no two vertices within the same set are adjacent. This concept is widely used in computer science, especially in graph theory, to solve problems related to matching, scheduling, and network flow.

Formal Definition:

A graph G=(V,E)G = (V, E)G=(V,E) is bipartite if its vertex set VVV can be partitioned into two disjoint sets UUU and WWW, such that every edge in EEE connects a vertex in UUU to a vertex in WWW, and there are no edges between vertices within the same set.

Key Characteristics of a Bipartite Graph:

Two Distinct Sets:

• The graph is divided into two sets UUU and WWW, where edges only exist between vertices from different sets.

No Edges Within the Same Set:

• In a bipartite graph, vertices within the same set are not connected directly by an edge.

No Odd-Length Cycles:

• A graph is bipartite if and only if it contains no odd-length cycles. If a graph contains any odd-length cycles, it cannot be bipartite.

2-Colorable:

• A bipartite graph can be colored with two colors such that no two adjacent vertices share the same color. One color is assigned to all vertices in set UUU, and another color is assigned to all vertices in set WWW.

Example of a Bipartite Graph:

Consider a bipartite graph with the following sets and edges:

• Set U={1,2}U = \{1, 2\}U={1,2}
• Set W={A,B}W = \{A, B\}W={A,B}
• Edges: (1,A),(1,B),(2,A)(1, A), (1, B), (2, A)(1,A),(1,B),(2,A)

In this graph, vertices in UUU are connected only to vertices in WWW, and there are no edges between vertices in the same set.

How to Identify a Bipartite Graph:

BFS or DFS (Breadth-First Search or Depth-First Search):

• You can use BFS or DFS to check if a graph is bipartite by trying to color the graph using two colors. Start by assigning one color to a vertex, and assign the opposite color to all adjacent vertices. If any two adjacent vertices end up with the same color, the graph is not bipartite.

Cycle Detection:

• A graph is not bipartite if it contains any odd-length cycles. Therefore, checking for odd-length cycles in the graph can help determine if it’s bipartite.

Applications of Bipartite Graphs:

Job Assignment Problems:

• Bipartite graphs are widely used in matching problems where one set represents workers and the other set represents jobs. Edges represent which workers are qualified for which jobs, and the goal is to assign workers to jobs efficiently.

Scheduling:

• In scheduling, tasks and resources can be represented as two sets of vertices, and bipartite graphs help model optimal assignments between tasks and resources.

Recommendation Systems:

• Bipartite graphs are used in recommendation systems where one set represents users and the other set represents items (e.g., movies). Edges represent interactions between users and items, which can be used to generate recommendations.

Network Flow Problems:

• Bipartite graphs are often used to solve network flow problems, where the goal is to find the maximum flow in a network by efficiently matching vertices between the two sets.

Why Learn About Bipartite Graphs?

Bipartite graphs are crucial in solving many real-world problems, including optimization, matching, and network flow. Their simple yet powerful structure allows for efficient solutions to complex computational problems, making them an essential topic in computer science and graph theory.

Topics Covered:

Definition and Properties: A clear understanding of what makes a graph bipartite, including two-set partitioning and 2-colorability.

Identifying Bipartite Graphs: Methods such as BFS, DFS, and cycle detection to determine if a graph is bipartite.

Applications: Practical use cases, including job assignments, recommendation systems, scheduling, and network flow.

For more details and examples, check out the full article on GeeksforGeeks: https://www.geeksforgeeks.org/what-is-bipartite-graph/.