M augmenting path Edges in bold are in

M augmenting path Edges in bold, are in the matching M M-augmenting path has been used to create a new matching M’ which has Cardinality one more than in M.

Bipartite Graph 1 2 3 4 5 6 X Y 1 2 3 4 5 6

Bipartite Graph 1 S 2 3 4 5 6 S’ X |N(S’)| <|S’| Therefore we can Never find a matching Which saturates All vertices in X. N(s) Y 1 2 3 4 5 N(S’) 6

Initial Matching Bipartite Graph u 1 2 3 4 5 6 X Y 1 2 3 4 5 6

Construct H tree at node u x 3 y 1 S={x 1, x 3} N(S)={y 1, y 2, y 4} T={y 1} T is not equal to N(S) But T is a subset of N(S), case b), So grow the tree… U =x 1

Construct H tree at node u We added y 4, x 2. Now S={x 1, x 3, x 2}, N(S)={y 1, y 2, y 4}, T={y 1, y 4} T is a subset of N(S), but not equal to N(S). x 2 x 3 y 4 y 1 U =x 1

Construct H tree at node u We have found a M-unsaturated vertex y 2. which was in N(S)T. So we can flip this M-augmenting path. x 2 x 3 y 1 y 2 y 4 U =x 1

update Matching u 1 2 3 4 5 6 X Y 1 2 3 4 5 6

Construct H tree at node u x 1 y 2 U =x 5

Construct H tree at node u x 3 y 1 x 1 y 2 U =x 5

Construct H tree at node u x 3 y 1 x 1 y 2 x 2 y 4 S={x 5, x 1, x 2, x 3} N(S)={y 1, y 2, y 4} T={y 2, y 1, y 4} T is equal to N(S) Therefore a matching which saturates All vertices in X does not exist. U =x 5