This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Graph Theory: Matchings and Hall’s Theorem COS 341 Fall 2004 Definition 1 A matching M in a graph G ( V, E ) is a subset of the edge set E such that no two edges in M are incident on the same vertex, i.e. if { w, x } , { y, z } ∈ M , then the vertices w, x, y, z are distinct. The size of a matching M is the number of edges in M . For a graph G ( V, E ), a matching of maximum size is called a maximum matching. Definition 2 If M is a matching in a graph G , a vertex v is said to be Msaturated if there is an edge in M incident on v . Vertex v is said to be Munsaturated if there is no edge in M incident on v . If G ( V 1 , V 2 , E ) is a bipartite graph than a matching M of G that saturates all the vertices in V 1 is called a complete matching (also called a perfect matching). When does a bipartite graph have a complete matching ? Given a graph, if we wanted to prove that the graph has a complete matching, we can simply give the edges in the matching....
View
Full Document
 Fall '08
 Jamespotvein
 Graph Theory, Bipartite graph, Perfect graph, Hall’s Theorem

Click to edit the document details