Bipartite Graphs

A bipartite graph is a special type of graph that can be divided into two distinct sets of vertices, such that no two vertices within the same set are adjacent. Here's an example:

Formally, a graph $$G = (V, E)$$ is bipartite if its vertex set $$V$$ can be split into two disjoint sets, $$U$$ and $$W$$, with the property that every edge in $$E$$ connects a vertex in $$U$$ to a vertex in $$W$$. This means that there are no edges connecting vertices within the same set.

Applications

Bipartite graphs are particularly important in various fields due to their unique structure. They are commonly used to model relationships between two different types of entities. For example, in a bipartite graph representing a job allocation system, one set of vertices might represent workers, and the other set represents jobs, with edges indicating which workers are eligible for which jobs.

Similarly, in a social network analysis, one set might represent people and the other set could represent activities, with edges denoting participation. This structure is also prevalent in fields such as scheduling, matching problems, and network flow, where bipartite graphs facilitate the representation and solution of complex relational problems.

Summary

Watch the following videos to learn more about bipartite graphs: