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?

 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

 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.

 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

 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.

 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.

 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.

 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

 (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

 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}

 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

 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.

5.9 Planar Graphs  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.

 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.

 Theorem 5.28(Euler’s formula) If G is a connected plane graph with n vertices, e edges and f regions, then n -e+f= 2.  Proof. Induction on e, the case e = 0 being as in this case n = 1, e = 0 and f =1  n-e+f=1-0+1=2

 Assume the result is true for all connected plane graphs with fewer than e edges,  e ≥ 1, and suppose G has e edges.  If G is a tree, then n =e+1 and f= 1, so the result holds. If G is not a tree, let a be an edge of a cycle of G and consider G-a. Clearly, G-a is a connected plane graph with n vertices, e-1 edges and f-1 regions, so by the induction hypothesis, n- (e-1) + (f- 1) = 2, from which it follows that n -e +f = 2.

 Corollary 5.1 If G is a plane graph with n vertices, e edges, k components and f regions, then n-e +f= 1+k.  Corollary 5.2: If G is a connected planar simple graph with e edges and n vertices where n ≥ 3, then e≤3n-6. Proof: A connected planar simple graph drawn in the plane divides the plane into regions, say f of them. The degree of each region is at least three(Since the graphs discussed here are simple graphs, no multiple edges that could produce regions of degree two, or loops that could produce regions of degree one, are permitted). The degree of a region is defined to be number of edges on the boundary of this region. We denoted the sum of the degree of the regions by s.

 Suppose that K 5 is a planar graph, by the Corollary 5.2,  n=5,e=10 ，  10  3*5-6=9, contradiction  K 3,3 ， n=6,e=9,  3n-6=3*6-6=12>9=e ，  But K 3,3 is a nonplanar graph

 Corollary 5.3: If a connected planar simple graph G has e edges and n vertices with n ≥ 3 and no circuits of length three, then e≤2n- 4.  Proof: Now, if the length of every cycle of G is at least 4, then every region of (the plane embodied of) G is bounded by at least 4 edges.  K 3,3 is a nonplanar graph  Proof: Because K 3,3 is a bipartite graph, it is no odd simple circule.

 Corollary 5.5: Every connected planar simple graph contains at least three vertices of degree at most five, where n≥3.  Next: Characterizations of Planar Graphs  Graph Colourings P320

 Exercise:  1. Let G be a bipartite graph. Then G has a perfect matching iff |N(A)|≥|A| for  A  V.  2.Suppose that G is a planar simple graph. If the number of edges of G less than 30, then there exists a vertex so that its degree less than 5.  3.Let G be a connected planar graph with δ(G)≥3 and f<12. Then G has a region with the degree less than 5.  4.Prove corollary 5.1  5.Prove figure 1 is a nonplanar graph  figure 1