Bipartite graph
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In the mathematical field of graph theory, a bipartite graph is a special graph where the set of vertices can be divided into two disjoint sets with two vertices of the same set never sharing an edge.
Bipartite graphs are useful for modelling matching problems. An example is a job matching problem. Suppose we have a set of people P and a set of jobs J, with not all people suitable for all jobs. We can model this as a graph with P + J the set of vertices. If a person <math>p_i<math> is suitable for a certain job <math>j_i<math> there is a edge between <math>p_i<math> and <math>j_i<math> in the graph.
The marriage theorem provides a characterization of bipartite graphs which allow perfect matchings.
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Definitions
A simple undirected graph <math>G:=(V,E)<math> is called bipartite if there exists a partition of the vertex set <math>V=V_1 \cup V_2 <math> so that both <math>V_1<math> and <math>V_2<math> are independent sets. We often write <math>G:=(V_1 + V_2, E)<math> to denote a bipartite graph with partitions <math>V_1<math> and <math>V_2<math>.
Examples
- every tree is bipartite
- cycle graphs with an even number of vertices are bipartite
- graphs with a vertex chromatic number of 2 are bipartite
Properties
- a graph is bipartite iff it does not contain an odd cycle
- the size of minimum vertex cover is equal to the size of the maximum matching
- the size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices.
- for a connected bipartite graph the size of the minimum edge cover is equal to the size of the maximum independent set
- for a connected bipartite graph the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices.