3/24/03Tucker, Section 4.31 Tucker, Applied Combinatorics, Sec. 4.3, prepared by Jo E-M Bipartite GraphMatching Some Definitions X-Matching Maximal Matching A set of independent edges Edges only between X and Y X Y All vertices in X are used The largest possible number of edges Note: an X-matching is necessarily maximal.

3/24/03Tucker, Section 4.32 More Definitions Edge Cover A set of vertices so that every edge is incident to at least one of them R(A) Range of a set A of vertices R(A) is the set of vertices adjacent to at least one vertex in A A = red, R(A) = blue

3/24/03Tucker, Section 4.33 Matching Network z a All edges capacity 1 All edges infinite capacity By converting the bipartite graph to a network, we can use network flow techniques to find matchings.

3/24/03Tucker, Section 4.34 Matching Network z a All edges capacity 1 All edges infinite capacity Matchinga – z flow

3/24/03Tucker, Section 4.35 Matching Network z a All edges capacity 1 All edges infinite capacity X – matching saturation at a

3/24/03Tucker, Section 4.36 Matching Network Maximal matching Maximal flow z a All edges capacity 1 All edges infinite capacity

3/24/03Tucker, Section 4.37 Matching Network Edge cover Finite a – z cut z a All edges capacity 1 All edges infinite capacity A = red on left, B = red on right S = red vertices, an edge cover P = a, black on left, red on right P = z, red on left, black on right Edge cover means that all edges have at least one red endpoint: P to P, to, to P. Thus only uncovered edges, would go from P to.

3/24/03Tucker, Section 4.38 Matching Network Not an edge cover Infinite a – z cut z a All edges capacity 1 All edges infinite capacity A = red on left, B = red on right S = red vertices, NOT an edge cover P = a, black on left, red on right P = z, red on left, black on right An infinite cut would go through an edge with two black endpoints, which corresponds to an uncovered edge.

3/24/03Tucker, Section 4.39 Theorem 1 Recall Corollary 3a from Section 4.2: The size of a maximal flow is equal to the capacity of a minimal cut. Since…. Then, The size of a maximal matching is equal to the size of a minimal edge cover. Edge cover Finite a – z cut Matchinga – z flow and

3/24/03Tucker, Section Finding Matchings z a All edges capacity 1 All edges infinite capacity Take any matching, convert to a network, and then use the augmenting flow algorithm to find a maximal flow, hence a maximal matching.

3/24/03Tucker, Section An easy way to think about it 1.Start with any matching. 2.From an unmatched vertex in X, alternate between nonmatching and matching edges until you hit an unmatched vertex in Y. 3.Then switch between the nonmatching and matching edges along to path to pick up one more matching edge. 4.Continue this until no unmatched vertex in X leads to an unmatched vertex in Y.

3/24/03Tucker, Section Example Start with any matching Find an alternating path Start at an unmatched vertex in X End at an unmatched vertex in Y Switch matching to nonmatching and vice versa A maximal matching!

3/24/03Tucker, Section Theorem 2 Hall’s Matching Theorem A bipartite graph has an X-matching if and only if for every subset A of X, |R(A)| is greater than or equal to |A|. X – matching True for all A in X

3/24/03Tucker, Section Proof of Hall’s Matching Theorem “ ”: An X-matching implies that for every subset A of X, |R(A)| is greater than or equal to |A|. X – matching For any A, the X – matching gives at least one vertex in R(A) for every vertex in A.

3/24/03Tucker, Section Proof of Hall’s Matching Theorem “ ”: If |R(A)| is greater than or equal to |A| for every subset A of X, then there is an X-matching. First note that if M is a maximal matching, then. Taking A = X, we have that since the range of X is contained in Y. Thus. Also, by Theorem 1, if S is a minimal edge cover, then. Note that if, then M must be an X – matching. Thus, it suffices to show that for all edge covers S.

3/24/03Tucker, Section Since,. Now let A be the vertices in X but not in S, so that. Thus,. Now, S is an edge-cover, so if it doesn’t contain on the X side, it must contain all the vertices a goes to on the Y side in order to cover the edges in between. Thus, so. Thus, But, hence, so as needed. /// Proof of Hall’s Matching Theorem “ ”: Continued… we need to show that for all edge covers S.

3/24/03Tucker, Section Example To Try Make up a random bipartite graph with about 20 vertices total. Swap with someone, and then both of you find an X- matching