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Knowledge
The chapter introduces the concept of bipartite graphs and their application in job assignment problems. It explains various types of matchings, such as maximum, maximal, and complete matching, along with a necessary condition for the existence of a complete matching in bipartite graphs as elucidated by Hall's marriage theorem. Through examples, the chapter illustrates how these concepts can help in modeling assignments fairly and effectively in different organizational setups.
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Sections
References
ch46.pdfClass Notes
Memorization
What we have learnt
- Bipartite graphs can be use...
- A matching in a graph helps...
- Hall's marriage theorem pro...
Final Test
Revision Tests
What we have learnt
- Bipartite graphs can be used to model job assignments between two sets of entities.
- A matching in a graph helps ensure that certain conditions, such as employees not being assigned multiple jobs, are met.
- Hall's marriage theorem provides a necessary and sufficient condition for the existence of complete matching in bipartite graphs.
Key Concepts
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Term: Bipartite Graph
Definition: A graph whose vertices can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent.
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Term: Matching
Definition: A set of edges in a graph such that no two edges share a vertex.
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Term: Maximum Matching
Definition: A matching that contains the largest possible number of edges.
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Term: Maximal Matching
Definition: A matching that cannot be extended by adding an edge.
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Term: Complete Matching
Definition: A matching where every vertex in one set is matched with a vertex in the other set.
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Term: Hall's Marriage Theorem
Definition: A condition that must be satisfied for a complete matching to exist in a bipartite graph, stating that for any subset of vertices the number of neighbors must be greater than or equal to the number of vertices in the subset.