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Matchings Matching: A matching in a graph G is a set of non-loop edges with no shared endpoints Maximal Matching: A maximal matching in a graph is a matching that cannot enlarged by adding an edge Maximum Matching: A maximum matching is a matching of maximum size among all matchings in the graph

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Matching

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Symmetric Difference ∆ If M and M’ are matchings, then M ∆ M’=(M- M’) (M’-M).

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Lemma 3.19 Every component of the symmetric difference of two matchings is a path or an even cycle. 1. Let F=M∆M’. 2. ∆(F)<=2 Every component is a path or cycle. 3. Every path or cycle alternates between M-M’ and M’-M. The cycle has even length.

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M-saturated M-saturated Vertices: The vertices incident to the edges of a matching M M-unsaturated Vertices: The vertices which are not saturated by M

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M-augmenting Path M-alternating path: Given a matching M, and an M-alternating path is a path that alternates between edges in M and edges not in M M-augmenting path: An M-alternating path whose endpoints are unsaturated by M

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Theorem 3.10 A matching M is a maximum matching in G iff G has no M-augmenting path ( ) Suppose G has an M-augmenting path. Then, we can produce a matching larger than M. ( ) Let M’ be a matching in G larger than M. Then, we can construct an M-augmenting path. 1. Let F=M∆M’. F consists paths and even cycles. 2. F must have a path with more edges of M’ than of M. 3. Such a path is an M-augmenting path in G.

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Bipartite Graph Independent Set: a set of pairwise nonadjacent vertices in graph Bipartite Graph: A graph is bipartite if V is the union of two disjoint independent sets called partite sets of G X,Y-bigraph: A bipartite graph with partite sets X and Y

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Theorem 3.1.11 An X,Y-bigraph G has a matching that saturates X iff |N(S)|>=|S| for all S X. ( ) If |N(S)|<|S| for some S X, then no matching can saturate S. ( ) 1. Let M is a maximum matching in G, we need to prove no node v X is M-unsaturated. 2. Suppose that node v X is M-unsaturated.

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Theorem 3.1.11 3. Let X* and Y* be the sets of M-saturated vertices in X and Y, respectively. 4. v has at least one neighbor v* in Y. 5. v* Y*. Otherwise, M’=M {vv*} is a matching larger than M. 6. We can construct an M-augmenting path starting at v. M is not a maximum matching. no node v X is M-unsaturated.

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The Marriage Theorem A X,Y-bigraph G with |X|=|Y| has a perfect matching iff |N(S)|>=|S| for all S X

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Corollary 3.1.13 For k>0, any k-regular bipartite graph has a perfect matching 1. k|X|=k|Y| |X|=|Y|. 2. Let m be the number of edges from S to N(S). 3. m=k|S| and m<=k|N(S)| |S|<= |N(S)|.

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Vertex Cover Vertex Cover: A vertex cover of a graph G is a set Q V(G) that contains at least one endpoint of every edge.

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Theorem 3.1.16 If G is a bipartite graph, then the maximum size of a matching in G equals to the minimum size of a vertex cover in G 1. |Q|>=|M|. 2. Let Q be the smallest vertex cover of G. 3. Let R=Q X and T=Q Y. 4. Let H and H’ be sugraphs of G induced by R (Y-T) and T (X-R), respectively.

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Theorem 3.1.16 (2/2) 5. Let S R. Let N H (S)=N(S) (Y-T). 6. |N H (S)|>=|S|. Otherwise, T N H (S) is a vertex cover smaller than Q. 7. H has a matching that saturates R. 8. Similarly, H’ has a matching that saturates T.

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Dual Optimization Problem A maximization problem M and a minimization problem N, defined on the same instances, such that: (1) for every candidate solution M to M and every candidate solution N to N, the value of M is less than or equal to the value of N. (2) obtaining candidate solutions M and N that have the same value PROVES that M and N are optimal solutions for that instance.

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Edge Cover Edge Cover: An edge cover of G is a set L of edges such that every vertex of G is incident to some edge of L (G): maximum size of independent set ’(G): maximum size of matching (G): minimum size of vertex cover ’(G): minimum size of edge cover

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Lemma 3.1.21 In a graph G, S V(G) is an independent set iff V(G)-S is a vertex cover, and hence (G)+ (G)=n(G) ( )1. No edge joins two nodes in S. Every edge has an endpoint in V(G)-S. V(G)-S is a vertex cover. ( ) Every edge has an endpoint in V(G)-S. No edge joins two nodes in S. S is an independent set

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Theorem 3.1.22 If G is a graph without isolated vertices, then ’(G)+ ’(G)=n(G). 1. (1) Let M be a maximum matching M. (2) We can construct an edge cover of size n(G)-|M| by adding to M one edge incident to each unsaturated vertex. ’(G)<=n(G)- ’(G).

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Theorem 3.1.22 (2/2) 2. (1) Let L be a minimum edge cover. (2) Only one endpoint of e L belong to edges in L. (Otherwise, L-{e} is also an edge cover) L is a star. (3) Let k be the number of components in L. |L|=n(G)-L. (4) We can form a matching M of size k=n(G)-|L|. ’(G)>=n(G)- ’(G).

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Corollary 3.1.24 If G is a bipartite graph with no isolated vertices, then (G)= ’(G). 1. ’(G)+ ’(G)=n(G) by Theorem 3.1.22. 2. (G)+ (G)=n(G) by Lemma 3.1.21. 3. ’(G)= (G) by Theorem 3.1.16.

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Dominating Set Dominating Set: In a graph G, a set S V(G) is a dominating set if every vertex not in S has a neighbor in S. Dominating Number (G): the minimum size of of a dominating set in G.

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Theorem For any graph G, (G)<= (G). 1. Let I be a maximum independent set. 2. For all v V(G)-I, v must be adjacent to some vertex in I. Otherwise, I {v} is an independent set larger than I. I is a dominating set of G. (G)<= (G).

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