1. khabbazian1 Copyright 1996-2001 © Dale Carnegie & Associates, Inc. Mohammad Khabbazian Department of computer engineering,sharif University,Tehran,iran m_khabbazian@ce.sharif.edu Forced matching number of bipartite graphs

2. khabbazian2 What’s Matching? Definition 1:Matching,  A matching in graph G is a set of non loop edges with no shared endpoints. Definition 2:Perfect matching,  The matching which saturates every vertex in graph G

3. khabbazian3 M_alternating path Definition 3 M_alternating path: Let M be a matching in graph G. M_alternating path is the path that edges alternate between edges in M and edge not in M.

4. khabbazian4 P.Hall’s theorem in bipartite graph Definition 4:  If s is a subset of vertices of graph G then N(s) is all of the vertices that have edges with s. Proposition 1:  If G is a bipartite graph with bipartition X,Y then G has a matching that saturates X if and only if

5. khabbazian5 Marriage Theorem For k>0,any K-regular bipartite graph has a perfect matching.  Then graph G has at least k matching.

6. khabbazian6 Forcing set for a perfect matching Suppose M is a perfect matching for graph G.Then :  A forcing set for perfect matching is a subset S of M, such that S is contained in no other perfect matching of G  The cardinality of a forcing set of M with smallest size is called the forcing number of M, and is denoted by ƒ(G,M).

7. khabbazian7 Jobs and Workers A master want with at least command indicate that every workers do special job he can build bipartite Graph and fined forcing set of matching worker and job

8. khabbazian8 Forcing set for a perfect matching Proposition 2:  Let M be a matching in bipartite graph G and S M be forcing set for M. Then there is a vertex in G that is forced immediately by S, that is,there is an edge uv of M \ S such that all of the neighbors of v except u are in V(S).( set of vertexes in S)  Proof : If this is not the case, after removing the set of all endpoints of the edges in S from G. we will obtain a bipartite graph in which every vertex has degree at least two. Therefore by a generalization of the marriage theorem of P.Hall,this graph has more than one matching.thus S can be complete in more than one way, which is a contradiction.

9. khabbazian9 Forcing set for a perfect matching Proposition 3:  Let G be a graph and M be perfect matching in G. A subset S of M is a forcing set for M if and only if it contains at least one edge form each M-alternating cycle.  Proof.

10. khabbazian10 Algorithmic results. Definition.  let G be a bipartite graph with bipartition X and Y and let M={x1 y1,x2 y2,…,xn, yn} be perfect matching in G.The digraph D(G,M) is defined as follows: the vertex (xi yi) is joined to another vertex (xj yj) if and only if yi joined to xj in G. 

11. khabbazian11 Algorithmic results. Proposition 3 :  Let G be a bipartite graph, M a perfect matching in G, and S a subset of M. then S is a forcing set for M if and only if D(G,M) \ S is an acyclic digraph.  Remark 1 By proposition 3.finding the smallest forcing set for a given matching M in a graph G is equivalent to finding the smallest number of vertices of D(G,M) whose removal vertex leaves no directed cycle in D(G,M).

12. khabbazian12 Job and Worker sample