Graphs in which some and every maximum matching is uniquely restricted
Lucia Draque Penso, Dieter Rautenbach, Ueverton dos Santos Souza
aa r X i v : . [ m a t h . C O ] A p r Graphs in which some and every maximum matching is uniquely restricted
Lucia Draque Penso , Dieter Rautenbach , U´everton dos Santos Souza Institute of Optimization and Operations Research, Ulm University, Ulm, Germany [email protected], [email protected] Instituto de Computa¸c˜ao, Universidade Federal Fluminense, Niter´oi, Brazil [email protected]
Abstract
A matching M in a graph G is uniquely restricted if there is no matching M ′ in G that is distinctfrom M but covers the same vertices as M . Solving a problem posed by Golumbic, Hirst, andLewenstein, we characterize the graphs in which some maximum matching is uniquely restricted.Solving a problem posed by Levit and Mandrescu, we characterize the graphs in which everymaximum matching is uniquely restricted. Both our characterizations lead to efficient recognitionalgorithms for the corresponding graphs. Keywords:
Maximum matching; uniquely restricted matching
MSC2010:
We consider finite and simple graphs as well as digraphs, and use standard terminology and notation.A matching in a graph G is a set of disjoint edges of G . A matching in G of maximum cardinality is maximum . A matching M in G is perfect if each vertex of G is incident with an edge in M , and near-perfect if each but exactly one vertex of G is incident with an edge in M . A graph G is factor-critical if G − u has a perfect matching for every vertex u of G . For a matching M in G , let V G ( M ) denote theset of vertices of G that are incident with an edge in M . A path or cycle in G is M -alternating if oneof every two adjacent edges belongs to M . For two sets M and N , the symmetric difference M ∆ N isthe set ( M \ N ) ∪ ( N \ M ). Note that ∆ is commutative and associative, that is, M ∆ N = N ∆ M and( M ∆ N )∆ O = M ∆( N ∆ O ). For a digraph D and a vertex u of D , let V + D ( u ) be the set of vertices v of D such that D contains a directed path from u to v . Similarly, let V − D ( u ) be the set of vertices w of D such that D contains a directed path from w to u . For a directed path or cycle ~P , let P denotethe underlying undirected path or cycle. For a positive integer k , let [ k ] denote the set of positiveintegers at most k . A set I of vertices of a graph is independent if no two vertices in I are adjacent.An independent set of maximum cardinality is maximum . Classical results of K˝onig [4] and Gallai [2]imply that | I | + | M | = n for a bipartite graph G of order n , a maximum matching M in G , and amaximum independent set I in G .Golumbic, Hirst, and Lewenstein [3] define a matching M in a graph G to be uniquely restricted if there is no matching M ′ in G with M ′ = M and V G ( M ′ ) = V G ( M ), that is, M is the uniqueperfect matching in the subgraph G [ V G ( M )] of G induced by V G ( M ). In [3] they show that it isNP-hard to determine a uniquely restricted matching of maximum size in a given bipartite graph that1as a perfect matching. Furthermore, they ask for which graphs the maximum size of a uniquelyrestricted matching equals the size of a maximum matching, that is, for which graphs some maximummatching is uniquely restricted. In [5] Levit and Mandrescu ask how to recognize the graphs for whichevery maximum matching is uniquely restricted. We answer both these questions completely givingstructural characterizations of both these classes of graphs that lead to efficient recognition algorithms. Let I be an independent set in a bipartite graph G , and let σ : x , . . . , x k be a linear ordering of theelements of I .For j ∈ [ k ], let I σ ≤ j = { x i : i ∈ [ j ] } .For y ∈ N G ( I ), let p ( y ) = x i , where i = min n j ∈ [ k ] : y ∈ N G (cid:16) I σ ≤ j (cid:17)o , that is, the index i is suchthat y N G ( x ) ∪ . . . ∪ N G ( x i − ) but y ∈ N G ( x i ). Let M σ = { yp ( y ) : y ∈ N G ( I ) } . Note that in the graph ( V ( G ) , M σ ), every vertex in N G ( I ) has degree exactly one.If E is a subset of the set E ( G ) of edges of G , then σ is E -good if M σ ⊆ E .The linear ordering σ is an accessibility ordering for I [5] if (cid:12)(cid:12) N G (cid:0) I σ ≤ j (cid:1)(cid:12)(cid:12) − (cid:12)(cid:12) N G (cid:0) I σ ≤ j − (cid:1)(cid:12)(cid:12) ≤ j ∈ [ k ]. Note that the definitions immediately imply that σ is an accessibility ordering ifand only if M σ is a matching in G .A partial accessibility ordering for I is an accessibility ordering σ ′ for a subset I ′ of I .We summarize some results from [3] that will be used. Theorem 1 (Golumbic, Hirst, and Lewenstein [3])
A matching M in a bipartite graph G isuniquely restricted if and only if G contains no M -alternating cycle. The following result slightly extends Theorem 3.2 in [5].
Lemma 2
Let G be a bipartite graph, and let E be a set of edges of G .The following statements are equivalent.(i) There is a maximum independent set I in G that has an E -good accessibility ordering σ .(ii) There is a maximum matching M in G such that M is uniquely restricted and M ⊆ E .(iii) Every maximum independent set I in G has an E -good accessibility ordering σ .Proof: (i) ⇒ (ii). Let I and σ : x , . . . , x k be as in (i). As noted above, M σ is a matching. Since σ is E -good, we have M σ ⊆ E . By construction, | M σ | = | N G ( I ) | , and, since I is a maximumindependent set in G , we have | V ( G ) | = | I | + | N G ( I ) | = | I | + | M σ | . This implies that M σ is amaximum matching in G . Let σ ′ : x ′ , . . . , x ′ ℓ be the subordering of σ formed by those x j where j ∈ [ k ] is such that (cid:12)(cid:12)(cid:12) N G (cid:16) I σ ≤ j (cid:17)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) N G (cid:16) I σ ≤ j − (cid:17)(cid:12)(cid:12)(cid:12) = 1, that is, σ ′ arises from σ by removing the x j with N G ( x j ) ⊆ N G (cid:16) I σ ≤ j − (cid:17) . Let N G ( I ) = { y , . . . , y ℓ } be such that p ( y i ) = x ′ i for i ∈ [ ℓ ], that is, M σ = { x ′ i y i : i ∈ [ ℓ ] } . For a contradiction, we assume that M σ is not uniquely restricted. By Theorem2, there is an M σ -alternating cycle C . Since every edge of C is incident with a vertex in I , and I is independent, C alternates between I and N G ( I ), that is, C has the form y r x ′ r y r x ′ r . . . y r t x ′ r t y r .Since y r i ∈ N G ( x ′ r i − ) ∩ N G ( x ′ r i ) for i ∈ [ t ], where we identify indices modulo t , the definition of p ( · )implies the contradiction r > r > r > . . . > r t > r . Hence, M σ is uniquely restricted, and G satisfies (ii).(ii) ⇒ (iii). Let M = { x y , . . . , x ℓ y ℓ } be a maximum matching in G such that M is uniquelyrestricted and M ⊆ E . Let I be a maximum independent set in G . As noted in the introduction,we have | I | + | M | = | V ( G ) | . Since I contains at most one vertex from each edge in M , this impliesthat I contains all vertices in V ( G ) \ V G ( M ), and exactly one vertex from each edge in M . Wemay assume that I = { x . . . , x ℓ , x ℓ +1 . . . , x k } , where V ( G ) \ V G ( M ) = { x ℓ +1 . . . , x k } . Note that thevertices x , . . . , x ℓ not necessarily belong to the same partite set of the bipartite graph G . If there issome set J ⊆ [ ℓ ] such that | N G ( x j ) ∩ { y i : i ∈ J }| ≥ j ∈ J , then, since I is independent, G contains an M -alternating cycle, which is a contradiction. Hence, for every set J ⊆ [ ℓ ], there issome j ∈ J with N G ( x j ) ∩ { y i : i ∈ J } = { y j } . Therefore, we may assume that x , . . . , x ℓ are orderedin such a way that i ≥ j for every i, j ∈ [ ℓ ] with x i y j ∈ E ( G ). This implies that σ : x . . . x k is anaccessibility ordering for I such that M σ = M ⊆ E , that is, G satisfies (iii).(iii) ⇒ (i). This implication is trivial. (cid:3) Lemma 3
Let G be a bipartite graph, let E be a set of edges of G , and let I be a maximum independentset in G . I has an E -good accessibility ordering σ : x , . . . , x k if and only if for every E -good partial acces-sibility ordering σ ′ : x ′ , . . . , x ′ ℓ − for I with ≤ ℓ − < | I | , there is an E -good partial accessibilityordering σ ′′ : x ′ , . . . , x ′ ℓ − , x ′ ℓ for I , that is, every E -good partial accessibility ordering that does notcontain all of I can be extended.Proof: Since the sufficiency is trivial, we only prove the necessity. Let σ and σ ′ be as in the statement.If { x , . . . , x ℓ − } = { x ′ , . . . , x ′ ℓ − } , then N G ( x ℓ ) \ N G ( { x ′ , . . . , x ′ ℓ − } ) = N G ( x ℓ ) \ N G ( { x , . . . , x ℓ − } ).Furthermore, if N G ( x ℓ ) \ N G ( { x ′ , . . . , x ′ ℓ − } ) contains a vertex y , then, since σ is E -good, we have x ℓ y ∈ E . Therefore, σ ′′ : x ′ , . . . , x ′ ℓ − , x ℓ is an E -good partial accessibility ordering for I .If { x , . . . , x ℓ − } 6 = { x ′ , . . . , x ′ ℓ − } , then { x , . . . , x ℓ − } 6⊆ { x ′ , . . . , x ′ ℓ − } . For j = min { i ∈ [ ℓ −
1] : x i
6∈ { x ′ , . . . , x ′ ℓ − }} , we have x , . . . , x j − ∈ { x ′ , . . . , x ′ ℓ − } , and hence, N G ( x j ) \ N G ( { x ′ , . . . , x ′ ℓ − } ) ⊆ N G ( x j ) \ N G ( { x , . . . , x j − } ). Furthermore, if N G ( x j ) \ N G ( { x ′ , . . . , x ′ ℓ − } ) contains a vertex y , then y ∈ N G ( x j ) \ N G ( { x , . . . , x j − } ), and hence, since σ is E -good, we have x j y ∈ E . Therefore, σ ′′ : x ′ , . . . , x ′ ℓ − , x j is an E -good partial accessibility ordering for I . (cid:3) Corollary 4
For a given bipartite graph G , and a given set E of edges of G , it is possible to checkin polynomial time whether G has a maximum matching M such that M is uniquely restricted and M ⊆ E .Proof: Since G is bipartite, one can determine a maximum independent set I in G in polynomial time.By Lemma 2, G has the desired matching if and only if I has an E -good accessibility ordering. ByLemma 3, this can be checked by starting with the empty partial accessibility ordering for I , whichis trivially E -good, and iteratively extending E -good partial accessibility orderings for I in a greedyway. (cid:3)
3e now invoke the famous Gallai-Edmonds Structure Theorem [6], which will be of central importancefor this and the next section.For a graph G , • let D ( G ) be the set of all vertices of G that are not covered by some maximum matching in G , • let A ( G ) be the set of vertices in V ( G ) \ D ( G ) that have a neighbor in D ( G ), and • let C ( G ) = V ( G ) \ ( A ( G ) ∪ D ( G )).Let G B be the bipartite graph obtained from G by deleting all vertices in C ( G ) and all edges betweenvertices in A ( G ), and by contracting each component H of G [ D ( G )] to a single vertex also denoted H . Note that for a given graph G , the set D ( G ), and hence also A ( G ) as well as C ( G ), can bedetermined in polynomial time [6]. Theorem 5 (Gallai-Edmonds Structure Theorem [6])
Let G be a graph.If D ( G ) , A ( G ) , C ( G ) , and G B are as above, then the following statements hold.(i) Every component of G [ D ( G )] is factor-critical.(ii) Every component of G [ C ( G )] has a perfect matching.(iii) A matching in G is maximum if and only if it is the union of(a) a near-perfect matching in each component of G [ D ( G )] ,(b) a perfect matching in each component of G [ C ( G )] , and(c) a matching with | A ( G ) | edges that matches the vertices in A ( G ) with vertices in differentcomponents of G [ D ( G )] . We proceed to the main result in this section.
Theorem 6
Let G be a graph. Let D ( G ) , A ( G ) , C ( G ) , and G B be as above. Let E be the set of edges aH of G B , where a ∈ A ( G ) and H is a component of G [ D ( G )] , such that the vertex a has a uniqueneighbor, say h , in V ( H ) , and H − h has a unique perfect matching.Some maximum matching in G is uniquely restricted if and only if the following conditions hold.(i) Every component of G [ C ( G )] has a unique perfect matching.(ii) G B has a maximum matching M B such that(a) M B is uniquely restricted and(b) M B ⊆ E (iii) Every component H of G [ D ( G )] has a vertex h such that H − h has a unique perfect matching.Proof: We first prove the necessity. Therefore, let M be a maximum matching in G that is uniquelyrestricted. Theorem 5(iii)(b) implies (i). Let M B be the matching in G B such that M B contains theedge aH , where a ∈ A and H is a component of G [ D ( G )], if and only if M contains an edge betweenthe vertex a and a vertex of H . We will show that M B is as in (ii). Theorem 5(iii)(c) implies that M B is a maximum matching of G B . If M B is not uniquely restricted, then Theorem 5(i) and (iii)4mply that G has a maximum matching M ′ with V G ( M ′ ) = V G ( M ) such that M ′ B = M B , where M ′ B is defined analogously to M B . This implies M ′ = M , which is a contradiction. Hence, (ii)(a) holds.If some edge aH in M B does not belong to E , then either a has at least two distinct neighbors in V ( H ) or a has a unique neighbor h in V ( H ) but H − h does not have a unique perfect matching. Inboth cases, Theorem 5(i) and (iii) imply that G has a maximum matching M ′ with V G ( M ′ ) = V G ( M )that differs from M within H , which is a contradiction. Hence, (ii)(b) holds. If some component H of G [ D ( G )] has no vertex h such that H − h has a unique perfect matching, then Theorem 5(iii)(a)implies that G has a maximum matching M ′ with V G ( M ′ ) = V G ( M ) that differs from M within H ,which is a contradiction. Hence, (iii) holds.Now we prove the sufficiency. Let M be the unique perfect matching in G [ C ( G )]. Let M B beas in (ii). Let M be a matching in G such that for every a ∈ A , the matching M contains an edge ah , where h ∈ V ( H ) and H is a component of G [ D ( G )], if and only if M B contains the edge aH . ByTheorem 5(iii)(c), M covers all of A ( G ). By (ii)(b), M is uniquely determined. For every component H of G [ D ( G )] such that M contains an edge ah with h ∈ V ( H ), (ii)(b) implies that H − h has aunique perfect matching M H . For every component H of G [ D ( G )] such that M does not contain anedge ah with h ∈ V ( H ), (iii) implies that H has a vertex h such that H − h has a unique perfectmatching M H . Let M = [ H : H is a component of G [ D ( G )] M H and M = M ∪ M ∪ M . By Theorem 5(iii), M is a maximum matching in G . We will show that M is uniquely restricted. For a contradiction, we assume that M ′ is a maximum matching in G with M ′ = M and V G ( M ′ ) = V G ( M ). By (i) and Theorem 5(iii)(b), M ′ contains M . By (ii)(a) and(b), M ′ contains M . By (ii)(b) and (iii), M ′ contains M . Altogether, M ⊆ M ′ , which implies thecontradiction M = M ′ . (cid:3) Corollary 7
For a given graph G , it is possible to check in polynomial time whether some maximummatching in G is uniquely restricted.Proof: If some graph H has a perfect matching M , then M is uniquely restricted if and only if H − e has no perfect matching for every e ∈ M . Therefore, the conditions (i) and (iii) from Theorem 6can be checked in polynomial time. By Corollary 4, condition (ii) from Theorem 6 can be checked inpolynomial time. Now, Theorem 6 implies the desired statement. (cid:3) Note that the constructive proofs of Lemma 2, Corollary 4, and Theorem 6 also lead to an efficientalgorithm that determines a maximum matching in a given graph G that is uniquely restricted, if sucha matching exists. It is convenient to split this section into two subsections, one about bipartite graphs, and one aboutnot necessarily bipartite graphs.
Throughout this subsection, let G be a bipartite graph with partite sets A and B .5or a matching M in G , let D ( M ) be the digraph with vertex set V ( G ) and arc set { ( a, b ) : a ∈ A, b ∈ B, and ab ∈ E ( G ) \ M } ∪ { ( b, a ) : a ∈ A, b ∈ B, and ab ∈ M } . Note that M -alternating paths and cycles in G correspond to directed paths and cycles in D .Let A ( M ) = n x ∈ A : d − D ( M ) ( x ) = 0 o and B ( M ) = n x ∈ B : d + D ( M ) ( x ) = 0 o . Note that A ( M ) = A \ V G ( M ) and d − D ( M ) ( a ) = 1 for every a ∈ A \ A ( M ) ,B ( M ) = B \ V G ( M ) and d + D ( M ) ( b ) = 1 for every b ∈ B \ B ( M ).Let V + ( M ) = [ a ∈ A ( M ) V + D ( M ) ( a ) and V − ( M ) = [ b ∈ B ( M ) V − D ( M ) ( b ) , that is, V + ( M ) is the set of vertices of G that are reachable from a vertex in A ( M ) on an M -alternating path, and V − ( M ) is the set of vertices of G that can reach a vertex in B ( M ) on an M -alternating path.K˝onig’s classical method [4] of finding a maximum matching in a bipartite graph relies on the followingresult (cf. Section 16.3 of [7]). Theorem 8 (K˝onig [4])
A matching M in a bipartite graph G is maximum if and only if G containsno M -alternating path between a vertex in A ( M ) and a vertex in B ( M ) , that is, if and only if V + ( M ) ∩ V − ( M ) = ∅ . In view of the correspondence between M -alternating cycles in G and directed cycles in D ( M ),Golumbic, Hirst, and Lewenstein’s [3] characterization of a uniquely restricted matching in a bipartitegraph can be rephrased as follows. Theorem 9 (Golumbic, Hirst, and Lewenstein [3])
A matching M in a bipartite graph G isuniquely restricted if and only if D ( M ) is acyclic. Our main result in this subsection is the following.
Theorem 10
Let M be a maximum matching in a bipartite graph G .Every maximum matching in G is uniquely restricted if and only if D ( M ) is acyclic, and the twosubgraphs G [ V + ( M )] and G [ V − ( M )] of G induced by V + ( M ) and V − ( M ) , respectively, are forests. The rest of this subsection is devoted to the proof of Theorem 10.
Lemma 11
Let M be a maximum matching in a bipartite graph G .If M ′ is a maximum matching in G , then V + ( M ′ ) = V + ( M ) and V − ( M ′ ) = V − ( M ) .Proof: Since the non-trivial components of ( V ( G ) , M ∆ M ′ ) are M - M ′ -alternating cycles and M - M ′ -alternating paths of even length, it suffices, by an inductive argument, to show that V + ( M ′ ) =6 + ( M ) and V − ( M ′ ) = V − ( M ) if either M ′ = M ∆ E ( C ), where C is an M -alternating cycle, or M ′ = M ∆ E ( P ), where P is an M -alternating path between some vertex a in A ( M ) and some vertex a ′ in A \ A ( M ). In the first case, D ( M ′ ) arises from D ( M ) by inverting the orientation of the edges of C , A ( M ′ ) = A ( M ), and B ( M ′ ) = B ( M ), which easily implies V + ( M ′ ) = V + ( M ) and V − ( M ′ ) = V − ( M ). Now, let M ′ = M ∆ E ( P ), where P is as above. D ( M ) contains a directed path ~P from a to a ′ such that P is the underlying undirected path of ~P . Furthermore, D ( M ′ ) arises by inverting theorientation of the arcs of ~P . Since M = M ′ ∆ E ( P ), a ′ ∈ A ( M ′ ), and a ∈ A \ A ( M ′ ), in order tocomplete the proof, it suffices, by symmetry, to show V + ( M ′ ) ⊆ V + ( M ) and V − ( M ′ ) ⊆ V − ( M ).If x ∈ V + ( M ) \ V + ( M ′ ), then some directed path in D ( M ) from a vertex in A ( M ) to x intersects ~P , which implies that D ( M ′ ) contains a directed path from a ′ to x , that is, x ∈ V + D ( M ′ ) ( a ′ ) ⊆ V + ( M ′ ),which is a contradiction. Hence, V + ( M ′ ) ⊆ V + ( M ). Similarly, if x ∈ V − ( M ) \ V − ( M ′ ), then somedirected path in D ( M ) from x to a vertex b in B ( M ) intersects ~P , which implies that D ( M ) containsa directed path from a ∈ A ( M ) to b ∈ B ( M ). By Theorem 8, M is not maximum, which is acontradiction. (cid:3) Lemma 12
Let M be a maximum matching in a bipartite graph G .If every maximum matching in G is uniquely restricted, then the two subgraphs G [ V + ( M )] and G [ V − ( M )] of G induced by V + ( M ) and V − ( M ) , respectively, are forests.Proof: For a contradiction, we may assume, by symmetry, that G [ V + ( M )] is not a forest. For a cycle C in G [ V + ( M )] and a maximum matching M ′ in G , let ~C ( M ′ ) be the subdigraph of D ( M ′ ) such that C is the underlying undirected graph of ~C ( M ′ ). Since M ′ is uniquely restricted, Theorem 9 impliesthat ~C ( M ′ ) is not a directed cycle in D ( M ′ ). Therefore, the set S ( C, M ′ ) = n x ∈ V ( C ) : d − ~C ( M ′ ) ( x ) = 0 o is not empty. Note that (cid:12)(cid:12)(cid:12)n x ∈ V ( C ) : d + ~C ( M ′ ) ( x ) = 0 o(cid:12)(cid:12)(cid:12) = | S ( C, M ′ ) | , that is, ~C ( M ′ ) contains equallymany sink vertices as source vertices.We assume that C and M ′ are chosen such that | S ( C, M ′ ) | is minimum.Let x ∈ S ( C, M ′ ). Since d + ~C ( M ′ ) ( x ) = 2, we have x ∈ A . Since x ∈ V ( C ) ⊆ V + ( M ), Lemma 11implies x ∈ V + ( M ′ ). Hence, there is a directed path ~P in D ( M ′ ) from some vertex a in A ( M ′ ) to x . First, we assume that ~P and ~C ( M ′ ) only share the vertex x . Let ~Q be a directed path in ~C ( M ′ )from x to some vertex y ∈ V ( C ) with d + ~C ( M ′ ) ( y ) = 0. Since d − ~C ( M ′ ) ( y ) = 2, we have y ∈ B . Since y ∈ V + D ( M ′ ) ( a ) ⊆ V + ( M ′ ), Theorem 8 implies y ∈ B \ B ( M ′ ). This implies that there is some vertex a ′ such that a ′ y ∈ M ′ . If ~R is the concatenation of ~P , ~Q , and the arc ( y, a ′ ), and M ′′ = M ′ ∆ E ( R ),then | S ( C, M ′′ ) | is strictly smaller than | S ( C, M ′ ) | , which is a contradiction. Hence, ~P and ~C ( M ′ )share a vertex different from x . This implies that ~P contains a directed subpath ~P ′ from a vertex y in V ( C ) \ { x } to x such that ~P ′ is internally disjoint from ~C ( M ′ ). If z is such that ( z, y ) is an arcof ~C ( M ′ ), and Q is the path in C between x and y that contains z , then C ′ = P ′ ∪ Q is a cycle in G [ V + ( M )] such that | S ( C ′ , M ′ ) | is strictly smaller than | S ( C, M ′ ) | , which is a contradiction. Hence,we may assume that d − ~C ( M ′ ) ( y ) = 0. Now, if R is one of the two paths in C between x and y , then C ′′ = P ′ ∪ R is a cycle in G [ V + ( M )] such that | S ( C ′′ , M ′ ) | is strictly smaller than | S ( C, M ′ ) | , whichis a contradiction. (cid:3) If M is a maximum matching in G , and a ∈ A ( M ) and a ′ b ′ ∈ M are such that b ′ is a neighbor of a , then M ′ = ( M \ { a ′ b ′ } ) ∪ { ab ′ } is a maximum matching in G , and we say that M ′ arises from M
7y an edge exchange . Similarly, if b ∈ B ( M ) and a ′ b ′ ∈ M are such that a ′ is a neighbor of b , then M ′′ = ( M \ { a ′ b ′ } ) ∪ { a ′ b } is a maximum matching in G , and also in this case, we say that M ′′ arisesfrom M by an edge exchange. Lemma 13
Let M be a maximum matching in a bipartite graph G .If D ( M ) is acyclic, then every maximum matching in G arises from M by a sequence of edgeexchanges.Proof: If M ′ is any maximum matching in G , then, since D is acyclic, the non-trivial components of( V ( G ) , M ∆ M ′ ) are M - M ′ -alternating paths P , . . . , P k , each starting with an edge in M and endingwith an edge in M ′ . Clearly, M ′ = M ∆ E ( P )∆ · · · ∆ E ( P k ). Since the maximum matching M ∆ E ( P )arises from M by a sequence of edge exchanges, the statement follows easily by an inductive argument. (cid:3) Lemma 14
Let M be a maximum matching in a bipartite graph G . Let D ( M ) be acyclic, and letthe two subgraphs G [ V + ( M )] and G [ V − ( M )] of G induced by V + ( M ) and V − ( M ) , respectively, beforests.If M ′ arises from M by an edge exchange, then D ( M ′ ) is acyclic.Proof: By symmetry, we may assume that a ∈ A ( M ) and a ′ b ′ ∈ M are such that b ′ is a neighborof a , and that M ′ = ( M \ { a ′ b ′ } ) ∪ { ab ′ } . Note that A ( M ′ ) = ( A ( M ) \ { a } ) ∪ { a ′ } , d − D ( M ) ( a ) = 0,and d − D ( M ′ ) ( a ′ ) = 0. If ~C is a directed cycle in D ( M ′ ), then, since D ( M ) is acyclic, ~C contains thearc ( b ′ , a ) of D ( M ′ ). This implies that G [ V + D ( M ) ( a )], and hence, also G [ V + ( M )] contains the cycle C ,which is a contradiction. Hence, D ( M ′ ) is acyclic. (cid:3) We are now in a position to prove Theorem 10.
Proof of Theorem 10:
The necessity follows from Theorem 9 and Lemma 12. For the sufficiency, let M ′ be any maximum matching of G . By Lemma 13, M ′ arises from M by a sequence of edge exchanges.By Lemma 11 and Lemma 14, it follows by induction on the number of these edge exchanges that D ( M ′ ) is acyclic. Therefore, by Theorem 9, M ′ is uniquely restricted. (cid:3) In order to extend Theorem 10 to graphs that are not necessarily bipartite, we again rely on theGallai-Edmonds Structure Theorem.
Theorem 15
Let G be a graph. Let D ( G ) , A ( G ) , C ( G ) , and G B be as above.Every maximum matching in G is uniquely restricted if and only if the following conditions hold.(i) Every component of G [ C ( G )] has a unique perfect matching.(ii) For every component H of G [ D ( G )] , every near-perfect matching in H is uniquely restricted.(iii) Every maximum matching of G B is uniquely restricted.(iv) If an edge aH of G B , where a ∈ A ( G ) and H is a component of G [ D ( G )] , is contained in somemaximum matching of G B , then the vertex a has a unique neighbor in V ( H ) . roof: In view of Theorem 5(iii), the proof of the necessity is straightforward; in fact, it can be doneusing very similar arguments as the proof of the necessity in Theorem 6. Therefore, we proceed toshow the sufficiency. Let M be a maximum matching in G . By Theorem 5(iii)(b), (i) implies that M ∩ E ( G [ C ( G )]) is uniquely determined. By Theorem 5(iii)(a) and (c), (iii) and (iv) imply that M ∩ { uv ∈ E ( G ) : u ∈ A ( G ) and v ∈ D ( G ) } is uniquely determined, which also implies that for everycomponent H of G [ D ( G )], the unique vertex of H that is not covered by an edge in M ∩ E ( G [ D ( G )])is uniquely determined. Now, by Theorem 5(iii)(a), (ii) implies that M ∩ E ( G [ D ( G )]) is uniquelydetermined, which completes the proof. (cid:3) Note that the factor-critical graphs in which every near-perfect matching is uniquely restricted (cf.Theorem 15(ii)) are exactly the factor-critical graphs G with the minimum possible number | V ( G ) | ofdistinct near-perfect matchings. In [1] it is shown that these are exactly the connected graphs whoseblocks are odd cycles. Corollary 16