Chapter 3

1. Chapter 3 Matchings and Covers Algorithms and Applications Matchings in General Graph Matchings and Factors Ch. 3. Matchings and Factors

2. Matching 3.1.1 • A matching in a simple graph G is a set of edges with no shared endpoints. • The vertices incident to the edges of a matching Mare said to be saturated by M; the others are unsaturated. • Aperfect matching in a graph is a matching that saturates every vertex. Saturated Unsaturated A matching in bipartite graph A matching in general graph Ch. 3. Matchings and Factors

3. Matching 3.1.1 • Aperfect matching in a graph is a matching that saturates every vertex. Perfect Matching Maximum Matching Not a perfect matching Ch. 3. Matchings and Factors

4. Example: Perfect matching in Kn,n3.1.2 • ConsiderKn,nwith partite sets X={x1,…, xn} andY={y1,…, yn}. • A perfect matching defines a bijection from X to Y. • Successively finding mates for x1, x2,… yieldsn! perfect matchings. • Each matching is represented by a permutation of [n], mappingitojwhenxiis matched to yj. Ch. 3. Matchings and Factors

5. Example: Perfect matching in Kn,n3.1.2 • ForK3,3, there are: 123, 132, 213, 231, 312, 321. • 312: (x1-y3) (x2-y1) (x3-y2) Ch. 3. Matchings and Factors

6. 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 Example: Perfect matching in Kn,n3.1.2 • We can express the matchings as matrices. • WithXandYindexing the rows and columns, we let positioni, j be 1 for each edge xiyjin a matchingMto obtain the corresponding matrix. • There is one 1 in each row and each column. Y1 Y2 Y3 Y4 X1 X2 X3 X4 X1 X2 X3 X4 X Y Y1 Y2 Y3 Y4 Ch. 3. Matchings and Factors

7. Perfect Matchings in Complete graphs • K2n+1has no perfect matching, Consider K2n. • Let fnbe the number of ways to pair up 2ndistinct people in K2n • There are 2n-1 choices for partner of v2n, and for each such choices there are fn-1ways to complete the matching (see example in the next page) • Hencefn=(2n-1)fn-1forn1. Withf0=1, it follows by induction thatfn=(2n-1)·(2n-3) · · ·(1). Ch. 3. Matchings and Factors

8. Example: Perfect Matchings in K6 • ForK6 , n=3, • f3is the number of perfect matchings • f3= (2n-1) * f2 = 5* f2=5*3*f1 Ch. 3. Matchings and Factors

9. Maximal Matching and Maximum Matching 3.1.4 • A maximum matchingin a graph is matching that can not be enlarged by adding an edge. • A maximum matchingis a matching of maximum size among all matchings in the graph. • A matchingM is maximal if every edge not in Mis incident to an edge already in M. • Every maximum matching is a maximal matchings, but the converse need not hold. Ch. 3. Matchings and Factors

10. Maximal  Maximum 3.1.5 • The smallest graph having a maximal matching that is not a maximum matching is P4. • If we take the middle edge, then we can add no other, but the two end edges form a larger matching. • Below we show this phenomenon in P4. Maximal Maximum Ch. 3. Matchings and Factors

11. Alternating path & Augmenting path 3.1.6 • Given a matching M, anM-alternating pathis a path that alternate between edges in Mand edges not inM. • AnM-alternating path whose endpoints are unsaturated byMis aM-augmenting path. B A C D B A C D Augmenting path: 1) E-B-F-D 2) A-F-B-G-C-H F E H G F E H G Ch. 3. Matchings and Factors

12. Symmetric Difference 3.1.7 • If GandHare graphs with vertex set V, then thesymmetric differenceGHis the graph with vertex setVwhose edges are those edges appearing in exactly one ofGand H. • We also use this notation for sets of edges ; in particular, if MandM’are matchings, then MM’ = (M-M’)(M’-M). • Very similar to Exclusive-OR Ch. 3. Matchings and Factors

13. Example of Symmetric Difference M M’ B B A C A C D E F G D E F G I I H K L M N H K L M N J J B A C D E F G MM’ I H K L M N J Ch. 3. Matchings and Factors

14. Lemma: Every component of the symmetric difference of two matchings is a path or an even cycle3.1.9 Proof: • Let Mand M’ be matchings, and F beMM’. • SinceMandM’ are matchings, every vertex has at most one incident edge from each of them • ThusFhas at most two edges at each vertex. • Since (F)2, every component of Fis a path or a cycle.  B A C D E F G MM’ I H K L M N J Ch. 3. Matchings and Factors

15. Lemma: Every component of the symmetric difference of two matchings is a path or an even cycle3.1.9 Proof: (continue) • Furthermore, every path or cycle inFalternates between edgesM-M’and edges of M’-M. • Thus each cycle has even length, with an equal number of edges from Mand from M’. Ch. 3. Matchings and Factors

16. Theorem: A matching M in a graph G is a maximum matching in G if and only if G has no M-augmenting path. 3.1.10 Proof: • We prove the contrapositive of each direction; Ghas a matching larger than Mif and only ifGhas anM-augmenting path. • (sufficiency) It is seen that an M-augmenting path can be used to produce a matching larger than M. Ch. 3. Matchings and Factors

17. Theorem: A matching M in a graph G is a maximum matching in G if and only if G has no M-augmenting path. 3.1.10 Proof: (continue) • (necessity) For the converse, • Let M’be a matching in Glarger than M • LetF=MM’. By Lemma 3.1.9, Fconsists of paths and even cycles; the cycles have the same number of edges fromMandM’ • Since |M’|>|M|, Fmust have a component with more edges of M’ than of M. Such a component can only be a path that starts and ends with an edge of M’; thus it is a M-augmenting path in G. Ch. 3. Matchings and Factors

18. Theorem: An X,Y-bigraph G has a matching that saturates X if and only if |N(S)|  |S| for all SX. 3.1.11 • Necessity.The |S| vertices matched to Smust lie in N(S). • Sufficiency.To prove that Hall’s Condition is sufficient, we prove the contra-positive: If Mis a maximum matching in Gand Mdoes not saturate X, then there is a set SX such that |N(S)|<|S|. Ch. 3. Matchings and Factors

19. Theorem: An X,Y-bigraph G has a matching that saturates X if and only if |N(S)|  |S| for all SX. 3.1.11 Sufficiency • LetuXbe a vertex unsaturated by M. • Among all the vertices reachable from ubyM-alternating paths in G, letSconsist of those in X, and that Tconsist of those in Y. → S Given u, Let ‘~’ represent alternating Let R ={v| (v,u) M~Pathexists} S={v|vX and vR} T={v|vY and vR} X u Y T=N(S) Ch. 3. Matchings and Factors

20. Theorem 3.1.11Continue SufficicyContinue • We claim that MmatchesTwithS-{u}. TheM-alternating paths from ureachYalong edges not in Mand return to Xalong edges inM. • Hence every vertex of S-{u} is reached by an edge in Mfrom a vertex in T. S X u Y T=N(S) Ch. 3. Matchings and Factors

21. Theorem 3.1.11Continue Sufficiencycontinue • Since there is no M-augmenting path, every vertex of Tis saturated; thus anM-augmenting path reaching yTextends via Mto a vertex of S. • Hence these edges ofMyield a bijection fromTto S-{u}, and we have |T|=|S-{u}|. S X u Y T=N(S) Ch. 3. Matchings and Factors

22. Theorem 3.1.11Continue Sufficiencycontinue • With T=N(S), we have proved that |N(S)|=|T|=|S|-1<|S| for this choice of S. This completes the proof of the contrapositive. S X u Y T=N(S) Ch. 3. Matchings and Factors

23. Corollary: for k>0, every k-regular bipartite graph has a perfect matching. 3.1.13 Proof : • LetGbe ak-regular X,Y- bigraph • Counting the edges by endpoints in Xand by endpoints inYshows thatk|X| = k|Y| • So|X|=|Y|. • Hence it suffices to verify Hall’s Condition; a matching that saturatesXwill also saturate Yand be perfect matching. ► Ch. 3. Matchings and Factors

24. Corollary: for k>0, every k-regular bipartite graph has a perfect matching. 3.1.13 Proof : Continued • ConsiderSX. Letmbe the number of edges from StoN(S) • Since Gisk-regular, m=k|S|. Thesemedges are incident toN(S), somk|N(S)| N(S) S m=k|S|, mk|N(S)| Ch. 3. Matchings and Factors

25. Corollary: for k>0, every k-regular bipartite graph has a perfect matching. 3.1.13 Proof : Continued • Hencek|S|k|N(S)|, which yields |N(S)||S| whenk>0. • Having chosen SXarbitrarily, we have established Hall’s condition. N(S) S m=k|S|, mk|N(S)| Ch. 3. Matchings and Factors

26. Vertex Cover 3.1.14 • A vertex cover of a graph Gis a set QV(G) that contains at least one endpoint of every edge. The vertices inQcover E(G). Ch. 3. Matchings and Factors

27. Vertex Cover 3.1.14 • In a graph that represents a road network (with straight roads and no isolated vertices). • Finding a minimum vertex cover = Placing the minimum number of policemen to guard the entire road network.   Ch. 3. Matchings and Factors

28. Matchings and Vertex covers • In the graph underneath • We mark a vertex cover of size 2 and show a matching of size 2 in bold. • |vertex cover|  | matching| Ch. 3. Matchings and Factors

29. Matchings and Vertex covers • As the graph illustrated underneath • the optimal values differ by 1 for an odd cycle • The difference can be arbitrarily large Ch. 3. Matchings and Factors

30. Theorem: If G is a bipartite graph, then the maximum size of a matching in G equals the minimum size of a vertex cover of G. 3.1.16 Green: Vertex cover Red: Matching |Q||M| B B A C A C F F E E D D G G Ch. 3. Matchings and Factors

31. Theorem 3.1.16 Proof : • LetGbe an X, Y-bigraph. • Since distinct vertices must be used to cover the edges of a matching, |Q|  |M| wheneverQis a vertex cover andMis a matching in G • Given a smallest vertex cover Qof G, we construct a matching of size |Q| to prove that equality can always be achieved. Ch. 3. Matchings and Factors

32. Theorem 3.1.16 Continue • PartitionQby letting R=QXand T=QY. • LetHandH’ be the subgraphs of Ginduced by R(Y-T) andT(X-R), respectively. R X H H’ Y T Ch. 3. Matchings and Factors

33. Theorem 3.1.16 Continue • We use Hall’s Theorem to show that Hhas a matching that saturatesRintoY-TandH’ has a matching that saturated T. • SinceHandH’ are disjoint, the two matchings together form a matching of size |Q| inG. R X H H’ Y T Ch. 3. Matchings and Factors

34. Theorem 3.1.16 Continue • SinceRTis a vertex cover, Ghas no edge from Y-TtoX-R. • Otherwise, an edge between Y-TtoX-Ris not covered X S R H H’ Y T NH(S) Ch. 3. Matchings and Factors

35. Theorem 3.1.16 Continue • For each SR, we consider NH(S), which is contained in Y-T. • If |NH(S)|<|S|, then we can substitute NH(S) for S in Q to obtain a smaller vertex cover • since NH(S) cover all edges incident toSthat are not covered by T R |NH(S)||S| X S H H’ Y T NH (S) Ch. 3. Matchings and Factors

36. Theorem 3.1.16 continue • The minimality of Qthus yields Hall’s Condition in H, and hence Hhas a matching that saturates R • Applying the same argument toH’yields the matching that saturates T R X S H H’ Y T NH(S) A matching exists in H of size |R| A matching exists in H’ of size |T| Ch. 3. Matchings and Factors

37. Min-Max relation 3.1.17 • A min-max relationis a theorem stating equality between the answers to a minimization problem and maximization problem over a class of instances. • The Konig-Egervary Theorem is such a relation for vertex covering and matching in bipartite graphs. Ch. 3. Matchings and Factors

38. Min-Max relation 3.1.17 • For the discussions in this text, we think of a dual pair of optimization problems • as a maximization problemMand minimization problemN, defined on the same instances (such as graphs), • such that for every candidate solutionMtoMand every candidateNtoN, the number of Mis less than or equal to the value of N. • Often the “value” is cardinality, as above where Mis maximum matching and Nis minimum vertex cover. Ch. 3. Matchings and Factors

39. Independent sets and covers • The independence number of a graph is the maximum size of an independent set of vertices. • The independence number of a bipartite graph does notalways equal the size of a partite set. • In the graph bellow, both partite sets have size 3, but we have marked an independent set of size 4. Ch. 3. Matchings and Factors

40. Edge cover 3.1.19 • An edge coverof Gis a set Lof edges such that every vertex of Gis incident to some edge ofL. • The four bold edges in thee following graph form an edge cover. Ch. 3. Matchings and Factors

41. Definitions • For the optimal sizes of the sets of the independence and covering problems we have defined, we use the notation below. • Maximum size of independent set(G) • Maximum size of matching’(G) • Minimum size of vertex cover (G) • Minimum size of edge cover ’(G) Ch. 3. Matchings and Factors

42. Lemma: In a graph G, SV(G) is an independent set if and only if Ŝ is a vertex cover, and hence (G)+ (G) = n(G).3.1.21 Proof : • IfSis an independent set, then every edge is incident to at least one vertex of Ŝ. • Conversely, if Ŝcovers all the edges, then there are no edges joining vertices of S.   Independent set Ch. 3. Matchings and Factors

43. Graph Theory Lemma: In a graph G, SV(G) is an independent set if and only if Ŝ is a vertex cover, and hence (G)+ (G) = n(G).3.1.21 Proof : continued Hence every maximum independent set is the complement of a minimum vertex cover, and (G)+(G)=n(G) (G) = 4 (G) = 2 n(G) = 6 43 Ch. 3. Matchings and Factors

44. Theorem: If G is a graph without isolated vertices, then α’(G)+ β’(G) =n(G).3.1.22’(G): Maximum size of matching ’(G): Minimum size of edge cover Proof: • From a maximum matching M, we will construct an edge cover of sizen(G)-|M|.(see next page) • Since a smallest edge cover is not bigger than this cover, this will imply that ’(G)n(G)-’(G). • Also, from a minimum edge cover L, we will construct a matching of sizen(G)-|L|.(see next page) • Since a largest matching is not smaller than this matching, this will imply that ’(G)n(G)-’(G). • These two inequalities complete the proof. (detail) Ch. 3. Matchings and Factors

45. Theorem 3.1.22 Continue • GivenM, we construct an edge cover of size n(G)-|M|. • Add to Mone edge incident to each unsaturated vertex. • We have used one edge for each vertex, except that each edge ofMtakes care of two vertices, • So the total size of this edge cover isn(G)-|M|, as desired.  Select one edge for each unsaturated vertex |(unsaturated Vertices)| = n(G)-2|M| Ch. 3. Matchings and Factors

46. Theorem 3.1.22 Continue • Given a minimum edge cover L, we construct a matching of sizen(G)-|L|. • If both ends of an edge e belong to edges in Lother thane, theneL, sinceL-{e} is also an edge cover. • Hence each component formed by edges of Lhas at most one vertex of degree exceeding 1 and is a star (a tree with at most one non-leaf).  can be deleted Assume k components, then |L|=n(G)-k Impossible case Onecomponent Ch. 3. Matchings and Factors

47. Theorem 3.1.22 Continue • Letkbe the number of these components • SinceLhas one edge for each non-central vertex in each star, we have |L|=n(G)-k • We form a matching Mof size k =n(G)-|L| by choosing one edge from each star inL Ch. 3. Matchings and Factors

48. Maximum Bipartite Matching3.2 • To find a maximum matching, we iteratively seek augmenting paths to enlarge the current matching. • Given a matching Min an X, Y –bigraph G, we search forM-augmenting paths from each M-unsaturated vertex in X. Ch. 3. Matchings and Factors

49. Augmenting Path Algorithm – for matching 3.2.1 Input:AnX, Y-bigraph G, a matching MinG, and the setUofM-unsaturated vertices in X. Ideal: • ExploreM-alternating paths from U • LetSXandTYbe the sets of vertices reached • Mark vertices ofSthat have been explored for path extensions • As a vertex is reached, record the vertex from which it is reached Ch. 3. Matchings and Factors

50. Augmenting Path Algorithm – for matching 3.2.1 Initialization:S=UandT=. Iteration: • IfS has no unmarked vertex, stop and report • T(X-S)as a minimum cover and • M as a maximum matching  Ch. 3. Matchings and Factors