1. 5.8 Graph Matching  Example: Set of worker assign to a set of task  Four tasks are to be assigned to four workers.  – Worker 1 is qualified to do tasks B and C  – Worker 2 is qualified to do tasks A,C and D  – Worker 3 is qualified to do tasks B and D  – Worker 4 is qualified to do task A and C.  Can all 4 workers be assigned to different tasks for which they are qualified?

2.  Example 2: The Marriage Problem:Given a set of men, each of whom knows some women from a given set of women, under what conditions is it possible for all men to marry women they know?  Four men each know some of four women  Peter knows Mary and Ann  Kevin knows Mary, Ann, Rose and Tina  Brian knows Mary and Ann  Fred knows Ann  Is it possible for all the men to marry women they know?  Graph Matching

3.  Definition 36: 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. 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. M1={e1,e7},M2={e1,e2,e5},M3 ={e1,e2,e5,e6}and M4= {e1,e2,e7,e8} are matching. M3 and M4 are maximum matching.

4.  Definition 37: If M is a matching in a graph G, a vertex v is said to be M-saturated if there is an edge in M incident on v. Vertex v is said to be M-unsaturated if there is no edge in M incident on v. M1={e1,e7}, M3={e1,e2,e5,e6} M1-saturated : v M1-unsaturated: u M3-saturated:u,v

5.  Definition 38: A matching M of G is perfect if all vertices of G are M-saturated. If G(V1;V2) is a bipartite graph then a matching M of G that saturates all the vertices in V1 is called a complete matching from V1 to V2. M={e1,e3,e7,e8}, M1={e1,e3,e5,e6} M1 and M are perfect matching.

6.  M={{v1,u1},{v2,u2},{v3,u3}}  M is a complete matching from V1 to V2, but it is not a complete matching from V2 to V1.

7.  Definition 39: Given a matching M in a graph G, a M-alternating path (cycle) is a path (cycle) in G whose edges are alternately in M and outside of M (i.e. if an edge of the path is in M, the next edge is outside M and vice versa). A M-alternating path whose end vertices are M- unsaturated is called an M-augmenting path.

8.  Theorem 5.25 ： M is a maximal matching of G iff there is no augmenting path relative to M.  Proof: (1) There is no augmenting path relative to M, we prove M is a maximal matching of G.  Suppose M and N are matching with |M|<|N|.  To see that M  N contains an augmenting path relative to M we consider G’ = (V’, M  N )  1≤d G’ (v)≤2  Since |M|<|N|, M  N has more edges from N than M and hence has at least one augmenting path relative to M  contradiction

9.  (2)If M is a maximal matching of G then there is no augmenting path relative to M  Assume that there exists an M-augmenting path p.  To see that M  p is a matching of G and |M  p|>|M|  1)M  p is a matching of G   e1,e2  M  p, e1 and e2 are not adjacent.  2) |M  p|>|M|  |M  p|=|(M ∪ p)-(M∩p)|= |(M ∪ p)|-|(M∩p)|  =|M|+|p|-2|(M∩p)|  =|M|+1

10.  Definition 40: Given a bipartite graph G(V1;V2), and a subset of vertices S  V, the neighborhood N(S) is the subset of vertices of V that are adjacent to some vertex in S, i.e.  N(S) ={v  V|  u  S,{u,v}  E(G)} A={v1,v3},N(A)={v2,v6,v4} A 1 ={v1,v4},N(A 1 )={v2,v6,v4,v3, v5,v1}

11.  Theorem 5.26: Let G(V 1,V 2 ) be a bipartite graph with |V 1 |=|V 2 |. Then a complete matching of G from V 1 to V 2 is a perfect matching

12.  Theorem 5.27 (Hall's Theorem) Let G(V 1 ; V 2 ) be a bipartite graph with |V 1 |≤|V 2 |. Then G has a complete matching saturating every vertex of V 1 iff |S|≤|N(S)| for every subset S  V 1  Example: Let G be a k-regular bipartite graph. Then there exists a perfect matching of G, where k>0.  k-regular  For A  V 1,E 1 ={e|e incident a vertex of A}, E 2 ={e|e incident a vertex of N(A)}  For  e  E 1, e  E 2. Thus E 1  E 2. Therefore |E 1 |≤|E 2 |.  Because k|A|=|E 1 |≤|E 2 |=k|N(A)|, |N(A)|≥|A|.  By Hall’s theorem, G has a complete matching M from V 1 to V 2.  Because |V 1 |=|V 2 |, Thus M is a perfect matching.

13. 5.9 Planar Graphs  5.9.1 Euler’s Formula  Definitions 41: Intuitively, a graph G is planar if it can be embedded in the plane, that is, if it can be drawn in the plane without any two edges crossing each other. If a graph is embedded in the plane, it is called a planar graph.

16.  Definition 42:A planar embedded of a graph splits the plane into connected regions, including an unbounded region. The unbounded region is called outside region, the other regions are called inside regions.

17.  Exercise:  1. Let G be a bipartite graph. Then G has a perfect matching iff |N(A)|≥|A| for  A  V.