Complete bipartite graph

Definition

A complete bipartite graph is a special type of graph that consists of two disjoint sets of vertices, where every vertex from one set is connected to every vertex in the other set. This structure means there are no edges connecting vertices within the same set, making it distinct and useful for modeling relationships between two different groups. Complete bipartite graphs are denoted as K_{m,n}, where m and n represent the number of vertices in each set.

5 Must Know Facts For Your Next Test

1. In a complete bipartite graph K_{m,n}, there are exactly m*n edges connecting the two sets of vertices.
2. Complete bipartite graphs are often used in computer science and combinatorial problems, particularly in matching problems and network flows.
3. The complete bipartite graph K_{1,n} is also known as a star graph, which has one central vertex connected to n outer vertices.
4. Complete bipartite graphs are always bipartite, meaning they can be colored with two colors without adjacent vertices sharing the same color.
5. The complete bipartite graph is a regular graph, as every vertex in the first set has the same degree (n) and every vertex in the second set has the same degree (m).

Review Questions

• How does a complete bipartite graph differ from a general bipartite graph?
  • A complete bipartite graph is a specific type of bipartite graph where every vertex in one set is connected to every vertex in the other set. In contrast, a general bipartite graph may have some vertices that are not connected to all vertices in the opposite set. This means that while all complete bipartite graphs are bipartite, not all bipartite graphs are complete.
• Discuss the applications of complete bipartite graphs in real-world scenarios.
  • Complete bipartite graphs have numerous applications, especially in fields like computer science, economics, and social sciences. They are particularly useful for modeling relationships between two distinct groups, such as job applicants and job positions, where each applicant can apply to multiple positions. Additionally, they play a crucial role in algorithms related to matching problems, network flows, and resource allocation tasks.
• Evaluate how the properties of complete bipartite graphs influence their use in algorithms and computational methods.
  • The properties of complete bipartite graphs, such as their regularity and the ability to be colored with two colors, greatly enhance their utility in algorithms. These properties allow for efficient matching algorithms like the Hopcroft-Karp algorithm, which finds maximum matchings in polynomial time. Their structured connectivity also simplifies the analysis of network flows, enabling solutions to complex optimization problems by clearly defining relationships between nodes.

Related terms

Bipartite Graph: A graph whose vertices can be divided into two distinct sets such that no two graph vertices within the same set are adjacent.

Vertex: A fundamental unit of a graph, representing an endpoint or a node where edges meet.

Edge: A connection between two vertices in a graph, which may represent various types of relationships.