Matching

Description:

• $M$, a subset $\in E$ such that no vertex in 𝑉 is on more than one edge in 𝑀
  • i.e. one $V$ has maximum one edge

Perfect matching:

• Every vertex in $V$ is on an edge in $M$

Maximum matching:

• Largest possible of $|M|$, meaning the maximum number of connections between V1 and V2 as possible

Complete matching:

• $M$ such that every nodes in $V_1$ is matched (assumed $|V_1|\le |V_2|$)
• A complete match is a maximum match (as we have found the max matches for set V1), reverse is not true

Hall’s marriage theorem:

• The bipartite graph 𝐺 = (𝑉, 𝐸) with bipartition $(V_1,V_2)$ has a complete matching from $V_1$ to $V_2$ if and only if $|N(A)|\ge |A|$ (number of neighbor vertices of A is $\ge$ than A) for all subsets of $A$ of $V_1$.theorem
  • Any subset of $V_1$ must have more matchable nodes then the number of nodes in $V_1$
  • note: $|N(A)|$ is neighbor of A
• Proof: lecture 14, slide 16

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