# Matching Hall's and Konig-Egervary Theorems R. Inkulu http://www ...

Ηλεκτρονική - Συσκευές

8 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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Matching
Hall's and Konig-Egervary Theorems
R.Inkulu
http://www.iitg.ac.in/rinkulu/
(Hall's and Konig-Egervary Theorems)
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Hall's Theorem(a.k.a.Marriage Problem)
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U'
N(U')
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(A, B)
(A', B')

Let G(U,V,E) be a bipartite graph with |U| = |V|.Also,N(A

) be
dened for any subset A

of U as {v ∈ V|∃u ∈ A

and (u,v) ∈ E}.The
bipartite graph G has a perfect matching if and only if for every
U

⊆ U,|N(U

)| ≥ |U

|.
⇒:Direct proof (immediate fromthe perfect matching deniti on).
⇐:Proof by contrapositive.A-B min-cut of size less than |U| is ensured
fromlast Theorem.Let U

be the set of vertices in A that belong to U.
Then the cut capacity of (A

,B

) is
|B

∩ U| +|A

∩ V| ≤ (|U| −|U

|) +(|N(U

)|).
(Hall's and Konig-Egervary Theorems)
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Konig-Egervary Theorem
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VC'
VC''
min s-t cut
The maximumcardinality of a matching in a bipartite graph G is equal to the
minimumcardinality of a vertex cover of its edges.

fromdirect (and obvious) proof,|MM| ≤ |VC|

choose vertics (T ∩U) ∪(S∩V) as vertex cover;leading to |VC| ≤ |MM|
(Hall's and Konig-Egervary Theorems)
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Relation between Max-Flow Min-Cut,Menger's
(edge-disjoint),Hall's,and Konig-Egervary Theorems
Min-cut
Max-flow
Uses in proving
Konig-Egervary
Menger's
Hall's

Max-Flow Min-Cut Thm:For any ow network G,MaximumFlow in G =
MinimumCut in G.
• Menger's Thm:For any graph G,MaximumEdge-Disjoint s-t paths in G =
Cardinality of MinimumEdge-Cut in G
′′
that sepeartes s fromt;and uses - for
any graph G,MaximumEdge-Dijoint s-t paths in G = MaximumFlow in ow
network G
′′
.

Hall's Thm:Characterizes perfect matching in bipartite graphs;and uses - for
any bipartite graph G,MaximumMatching cardinality in G = MaximumFlow
in in ow network G

.

Konig-Egervary Thm:For any bipartite graph G,MinimumVertex Cover in G =
MaximumFlow in ow network G

= Min Cut in ow network G

.
(Hall's and Konig-Egervary Theorems)
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