Characterization Of A Class Of Graphs Related To Pairs Of Disjoint Matchings

Characterization Of A Class Of Graphs R elated T o P airs Of Dis join t Matc hings ∗ An ush Tseruny an Departmen t of Informatics and Applied Mathemat ics, Y erev an State Unive rsit y , Y erev an , 0025, Armenia Email: anush@math.ucla.edu, an ush tseruny an@ya ho o.com The wo rk on this paper wa s supp orted by a g rant of Armenian Nationa l Science and Educational F und. De dic ate d to m y mother, f a ther a nd sister Ar evik Abstract F or a given graph consider a pair of disjoint matc hings the union of wh ic h contai ns as many edges as p ossible. F urthermore, consider the ratio of the cardinalities of a maxim um matc h in g and the largest m atc hin g in those pairs. It is kno wn th at for any graph 5 4 is the tigh t upp er b ound f or this ratio. W e c h aracterize the class of graphs for whic h it is precisely 5 4 . Our c haracterization implies that these graph s con tain a spanning subgraph , ev ery connected comp onent of w hic h is the min imal graph of th is class. Keyw ords : matc hing, pair of disjoin t m atc hin gs, maxim um matc hin g. 1 In tr o du ction In this pap er w e consider finite undirected graphs without multiple edges, lo ops, or isolated v ertices. Let V ( G ) and E ( G ) b e the sets of v ertices and edges of a graph G , resp ectiv ely . W e denote by β ( G ) the cardinality of a maximum matchin g of G . Let B 2 ( G ) b e the set of pairs of disjoint matchings of G . Set: λ ( G ) . = max {| H | + | H ′ | : ( H , H ′ ) ∈ B 2 ( G ) } . ∗ The current work is the main part o f the a uthor’s Mas ter’s thesis defended in May 2007 . 1 F urthermore, let us introduce a no ther parameter: α ( G ) . = max {| H | , | H ′ | : ( H , H ′ ) ∈ B 2 ( G ) and | H | + | H ′ | = λ ( G ) } , and define a set: M 2 ( G ) . = { ( H , H ′ ) ∈ B 2 ( G ) : | H | + | H ′ | = λ ( G ) and | H | = α ( G ) } . While w orking on the problems o f constructing a maxim um matc hing F of a graph G suc h tha t β ( G \ F ) is maximized or minimized, Kama lia n a nd Mkrtc h y an designed p olynomial algorithms for solving these problems for t r ees [5 ]. Unfortunat ely , the problems turned out to b e NP-hard already f or connected bipartite graphs with maxim um degree three [6], th us there is no hop e for t he p olynomial time calculation o f β 1 ( G ) ev en for bipartite graphs G , where β 1 ( G ) = max { β ( G \ F ) : F is a maximum matching of G } . Note that fo r any graph G λ ( G ) = β ( G ) + β 1 ( G ) if and only if α ( G ) = β ( G ) . Th us, β 1 ( G ) can b e efficien tly calculated for bipartite graphs G with β ( G ) = α ( G ) since λ ( G ) can b e calculated for that graphs by using a standard algorithm of finding a maxim um flo w in a net work. Let us also note that the calculation of λ ( G ) is NP-hard ev en fo r the class of cubic graphs since the c hromatic class of a cubic graph G is three if and only if λ ( G ) = | V ( G ) | (see [4 ]) . Being intereste d in the classification of graphs G , fo r whic h β ( G ) = α ( G ), Mkrtc h yan in [8] prov ed a sufficien t condition, whic h due to [2, 3], can b e form ulated as: if G is a matc hing co v ered tree then β ( G ) = α ( G ). Not e that a graph is said to be matc hing co v ered (see [9]) if its ev ery edge b elongs to a maxim um matc hing (not necessarily a p erfect matc hing as it is usually defined, see e.g. [7]). In contrast with the theory of 2-matc hings, where ev ery graph G admits a maxim um 2-matc hing that includes a maxim um matching [7 ], there are graphs (ev en t r ees) that do not ha v e a “ maxim um” pair of disjoint matc hings (a pair from M 2 ( G )) tha t includes a maxim um matc hing. The follo wing is the b est result that can b e stat ed ab out the ratio β ( G ) α ( G ) for a ny graph G (see [10]): 1 ≤ β ( G ) α ( G ) ≤ 5 4 . The aim o f the pap er is the c hara cterization of the class of g raphs G , for whic h the ra tio β ( G ) α ( G ) obtains its upp er b ound, i.e. the equalit y β ( G ) α ( G ) = 5 4 holds. 2 Figure 1: Spanner Our characterization theorem is form ulated in terms o f a special graph called spanner (figure 1), whic h is the minimal graph f or whic h β 6 = α (what is remark able is that the equalit y β α = 5 4 also holds for spanner). T his kind of theorems is common in graph theory: see [2] for c haracterization of planar o r line graphs. Another example may b e T utte’s Conjecture (no w a b eautiful theorem thanks to Rob ertson, Sanders, Seymour and T omas) ab o ut the c hro matic index of bridgeless cubic graphs, whic h do not contain P etersen graph a s a minor. On the other hand, let us note that in con trast with the examples given ab ov e, our theorem do es not provide a forbidden/excluded gra ph characterization. Quite the con trary , the theorem implies that eve ry graph satisfying the men tioned equalit y admits a spanning subgraph every connected comp onen t of whic h is a spanner. 2 Main Notati o ns and D e finitio n s Let G b e a graph and d G ( v ) b e the degree of a v ertex v o f G . Definition 2.1 A subset of E ( G ) is c al le d a m atching if it do es not c ontain adjac e nt e dge s . Definition 2.2 A matchin g of G with maximum numb er of e dges is c a l le d maxi m um. Definition 2.3 A vertex v of G is c over e d (misse d) by a matching H of G , if H c ontains (do es not c ontain) an e dg e inciden t to v . Definition 2.4 A se quenc e v 0 , e 1 , v 1 , ..., v n − 1 , e n , v n is c a l le d a tr ail in G if v i ∈ V ( G ) , e j ∈ E ( G ) , e j = ( v j − 1 , v j ) , and e j 6 = e k if j 6 = k , for 0 ≤ i ≤ n , 1 ≤ j, k ≤ n . The n um b er of edges, n , is called the length of a trail v 0 , e 1 , v 1 , ..., v n − 1 , e n , v n . T rail is called eve n (o dd) if its length is eve n (o dd). T rails v 0 , e 1 , v 1 , ..., v n − 1 , e n , v n and v n , e n , v n − 1 , ..., v 1 , e 1 , v 0 are considered equal. T rail T is also considered a s a subgraph of G , a nd thus , V ( T ) a nd E ( T ) are used to denote the sets of vertic es and edges of T , resp ectiv ely . 3 Definition 2.5 A tr ail v 0 , e 1 , v 1 , ..., v n − 1 , e n , v n is c a l le d a cycle if v 0 = v n . Similarly , cycles v 0 , e 1 , v 1 , ..., v n − 1 , e n , v 0 and v i , e i +1 , ..., e n , v 0 , e 1 , ..., e i , v i are considered equal fo r any 0 ≤ i ≤ n − 1. If T : v 0 , e 1 , v 1 , ..., v n − 1 , e n , v n is a trail that is not a cycle then v 0 , v n and e 1 , e n are called the end-vertice s and end-edges of T , resp ectiv ely . Definition 2.6 A tr ail v 0 , e 1 , v 1 , ..., v n − 1 , e n , v n is c a l le d a p ath if v i 6 = v j for 0 ≤ i < j ≤ n . Definition 2.7 A cycle v 0 , e 1 , v 1 , ..., v n − 1 , e n , v 0 is c al le d simple if v 0 , e 1 , v 1 , ..., v n − 2 , e n − 1 , v n − 1 is a p ath. Belo w we omit v i -s a nd write e 1 , e 2 , ..., e n instead of v 0 , e 1 , v 1 , ..., v n − 1 , e n , v n when denoting a trail. Definition 2.8 F or a tr ail T : e 1 , e 2 , ..., e n of G and i ≥ 1 , defi n e sets E b i ( T ) , E e i ( T ) , E i ( T ) and V 0 ( T ) , V b i ( T ) , V e i ( T ) , V i ( T ) as fol lows: E b i ( T ) . = { e j : 1 ≤ j ≤ min { n, i }} , E e i ( T ) . = { e j : max { 1 , n − i + 1 } ≤ j ≤ n } , E i ( T ) . = E b i ( T ) ∪ E e i ( T ) , and V 0 ( T ) . = { v ∈ V ( G ) : v is an end-vertex of T } , V b i ( T ) . = { v ∈ V ( G ) : v is incident to an e dge fr om E b i ( T ) } , V e i ( T ) . = { v ∈ V ( G ) : v is incident to an e dge fr om E e i ( T ) } , V i ( T ) . = V b i ( T ) ∪ V e i ( T ) . The same notations are used for sets of trails. F or example, for a set of t rails D , V 0 ( D ) denotes the set of end-v ertices of a ll trails from D , that is: V 0 ( D ) . = [ T ∈ D V 0 ( T ) . Let A a nd B b e sets of edges of G . Definition 2.9 A tr ail e 1 , e 2 , ..., e n is c al le d A - B alternating if the e d ges with o dd indic es b elong to A \ B and others to B \ A , or vic e-versa . 4 If X is an A - B alternating tr a il then X A ( X B ) denotes the gr aph induced b y the set of edges of X that b elong to A ( B ). The set o f A - B alternat ing t rails o f G that are not cycles is denoted b y T ( A, B ). The subsets of T ( A, B ) con taining only ev en and o dd trails a re denoted b y T e ( A, B ) a nd T o ( A, B ), resp ectiv ely . W e use the notat io n C instead of T do denote the corresp onding sets of A - B alternating cycles ( e.g. C e ( A, B ) is the set of A - B alternating ev en cycles ). The set of the trails from T o ( A, B ) start ing with a n edge from A ( B ) is denoted b y T A o ( A, B ) ( T B o ( A, B )). No w, let A and B b e matc hings of G ( no t necessarily disjoin t). Note that A - B a lt ernat ing trail is either a path, or an ev en simple cycle. Definition 2.10 A n A - B alternating p ath P is c a l le d maximal if ther e i s no other A - B alternating tr ail (a p ath or an even simple cycle) that c ontains P as a pr o p er subtr ail. W e use the notat io n M P instead of T to denote the corresp onding sets of maximal A - B alternating paths ( e.g. M P B o ( A, B ) is t he subset o f M P ( A, B ) containing only those maximal A - B alternating paths whose length is o dd and whic h start (and also end) with an edge fro m B ). T erms and concepts that w e do not define can b e found in [2, 7, 11]. 3 General Prop er t ies and S tructur al Lemmas Let G b e a graph, and A and B b e (not necessarily disjoin t) matc hings of it. The following are prop erties of A - B alternating cycles a nd maximal pat hs. First note that a ll cycles from C e ( A, B ) are simple as A and B are match ings. Prop erty 3.1 I f the c onne cte d c omp onen ts of G ar e p aths or even simple cycles , and ( H , H ′ ) ∈ M 2 ( G ) , then H ∪ H ′ = E ( G ) . Prop erty 3.2 I f C ∈ C e ( A, B ) and v ∈ V ( C ) then d C A ( v ) = d C B ( v ) . Prop erty 3.3 Eve ry e dge e ∈ A △ B 1 lies either on a cycle fr om C e ( A, B ) o r on a p ath fr om M P ( A, B ) . Prop erty 3.4 (1) if F ∈ C e ( A, B ) ∪ T e ( A, B ) then A and B h ave the same numb er of e dges that li e on F , 1 A △ B denotes the symmetric difference of A and B , i.e . A △ B = ( A \ B ) ∪ ( B \ A ) . 5 (2) if T ∈ T A o ( A, B ) then the numb er of e dges fr om A lying on T is one mor e than the numb er of ones fr om B . These observ ations imply: Prop erty 3.5 | A | − | B | = | M P A o ( A, B ) | − | M P B o ( A, B ) | . Berge’s w ell-kno wn theorem stat es that a mat ching M of a g r a ph G is maximum if and only if G do es not con tain an M -aug men ting path [2, 7, 11]. This theorem immediately implies: Prop erty 3.6 I f M is a maximum matching and H is a m a tching of a gr aph G then M P H o ( M , H ) = ∅ , and ther ef o r e, | M | − | H | = | M P M o ( M , H ) | . The pro of of the follo wing pro p ert y is similar to t he one of prop erty 3.6: Prop erty 3.7 I f ( H , H ′ ) ∈ M 2 ( G ) then M P H ′ o ( H , H ′ ) = ∅ . Prop erty 3.8 I f λ ( G ) = 2 α ( G ) and ( H , H ′ ) ∈ B 2 ( G ) for wh i c h | H | + | H ′ | = λ ( G ) , then M P o ( H , H ′ ) = ∅ and ( H , H ′ ) ∈ M 2 ( G ) . Pro of. Assume that M P o ( H , H ′ ) 6 = ∅ . D enote by O and E the sets of edges lying on the paths f r om M P o ( H , H ′ ) with o dd and ev en indices, resp ectiv ely (indices start with 1). Set H 1 = ( H \ E ) ∪ O , and H ′ 1 = ( H ′ \ O ) ∪ E . Note t ha t ( H 1 , H ′ 1 ) ∈ B 2 ( G ) and | H 1 | + | H ′ 1 | = | H | + | H ′ | = λ ( G ) (as H 1 ∪ H ′ 1 = H ∪ H ′ ). Also note that M P H 1 o ( H 1 , H ′ 1 ) = M P o ( H , H ′ ) and M P H ′ 1 o ( H 1 , H ′ 1 ) = ∅ . Due to prop ert y 3.5, | H 1 | − | H ′ 1 | = | M P H 1 o ( H 1 , H ′ 1 ) | = | M P o ( H , H ′ ) | > 0, i.e. | H 1 | > | H ′ 1 | , a nd therefore | H 1 | > λ ( G ) 2 = α ( G ) , whic h con tradicts the definition of α ( G ). Thu s, M P o ( H , H ′ ) = ∅ , and, due to prop ert y 3.5, | H | = | H ′ | = λ ( G ) 2 = α ( G ), whic h means that ( H , H ′ ) ∈ M 2 ( G ). No w let M b e a fixed maxim um matc hing of G . Over all ( H , H ′ ) ∈ M 2 ( G ), consider the pairs (( H , H ′ ) , M ) f or whic h | M ∩ ( H ∪ H ′ ) | is maximized. Denote t he set of those pairs b y M 2 ( G, M ): M 2 ( G, M ) . = { ( H , H ′ ) ∈ M 2 ( G ) : | M ∩ ( H ∪ H ′ ) | is maxim um } . Let ( H , H ′ ) b e an arbitrarily c hosen pair from M 2 ( G, M ). 6 Lemma 3.9 F or every p ath P : m 1 , h 1 , ..., m l − 1 , h l − 1 , m l fr om M P M o ( M , H ) (1) m 1 , m l ∈ H ′ ; (2) l ≥ 3 . Pro of. Let us sho w that m 1 , m l ∈ H ′ . If l = 1 then P = m 1 , m 1 ∈ M \ H , and m 1 is not adjacen t to a n edge from H as P is maximal. Th us, m 1 ∈ H ′ as otherwise w e could enlarge H b y adding m 1 to it whic h w ould contradict ( H , H ′ ) ∈ M 2 ( G ). Thus , supp ose that l ≥ 2. Let us show that m 1 ∈ H ′ . If m 1 / ∈ H ′ then define H 1 . = ( H \{ h 1 } ) ∪ { m 1 } . Clearly , H 1 is a matc hing, and H 1 ∩ H ′ = ∅ , | H 1 | = | H | , whic h means that ( H 1 , H ′ ) ∈ M 2 ( G ). But | M ∩ ( H 1 ∪ H ′ ) | > | M ∩ ( H ∪ H ′ ) | , whic h contradicts ( H , H ′ ) ∈ M 2 ( G, M ). Th us m 1 ∈ H ′ . Similarly , it can b e show n that m l ∈ H ′ . No w let us show that l ≥ 3. D ue to prop erty 3.7, M P H ′ o ( H , H ′ ) = ∅ , thus there is i, 1 ≤ i ≤ l , suc h that m i ∈ M \ ( H ∪ H ′ ), since { m 1 , m l } ⊆ H ′ , and w e hav e l ≥ 3. Lemma 3.10 Each ve rtex lying on a p ath fr om M P M o ( M , H ) is incid ent to an e dge fr om H ′ . Pro of. Assume the contrary , and let v b e a v ertex lying on a path P from M P M o ( M , H ) , whic h is not inciden t to an edge from H ′ . Clearly , v is inciden t to an edge e = ( u, v ) ∈ M \ ( H ∪ H ′ ) lying on P . If u is not inciden t to an edge from H ′ to o, then H and H ′ ∪ { e } are disjoin t matc hings and | H | + | H ′ ∪ { e }| > | H | + | H ′ | = λ 2 ( G ) , whic h con tra dicts ( H , H ′ ) ∈ M 2 ( G ). On the other hand, if u is incident to an edge f ∈ H ′ , then consider the pair ( H , H ′′ ), where H ′′ . = ( H ′ \{ f } ) ∪ { e } . Note that H and H ′′ are disjoint matc hings and | H ′′ | = | H ′ | , whic h means that ( H, H ′′ ) ∈ M 2 ( G ). But | M ∩ ( H ∪ H ′′ ) | > | M ∩ ( H ∪ H ′ ) | contradicting ( H , H ′ ) ∈ M 2 ( G, M ). F or a path P ∈ M P M o ( M , H ) , consider one of its end-edges f ∈ E 1 ( P ). D ue to statemen t (1) o f lemma 3.9, f ∈ M ∩ H ′ . By maximality of P , f is adjacen t to only one edge from H , th us it is an end-edge of a path P f from M P e ( H , H ′ ) ∪ M P H ′ o ( H , H ′ ). Moreo ver, P f ∈ M P e ( H , H ′ ) according to pro p ert y 3.7 . D efine a set Y ⊆ M P e ( H , H ′ ) as follows: Y ( M , H, H ′ ) . = { P f : P ∈ M P M o ( M , H ) , f ∈ E 1 ( P ) } . Lemma 3.11 (1) Th e end-e dges o f p a ths of M P M o ( M , H ) lie on diff e r ent p aths of Y ( M , H, H ′ ) ; 7 (2) | Y ( M , H, H ′ ) | = 2 | M P M o ( M , H ) | = 2( β ( G ) − α ( G ) ) . (3) F or eve ry P ∈ Y ( M , H, H ′ ) , P : h ′ 1 , h 1 , ..., h ′ n , h n , wher e h ′ i ∈ H ′ , h i ∈ H , 1 ≤ i ≤ n , (a) h ′ 1 and h 1 lie on a p ath fr om M P M o ( M , H ) , but h ′ n and h n do not lie on any p ath fr om M P M o ( M , H ) ; (b) n ≥ 2 . Pro of. (1) is true as otherwise w e would hav e a path from Y ( M , H, H ′ ) with b oth end-edges from H ′ con tradicting Y ( M , H, H ′ ) ⊆ M P e ( H , H ′ ). F urthermore, (1) to g ether with prop erty 3.6 imply ( 2 ). No w, let us prov e (3a). By the definition of Y ( M , H, H ′ ), h ′ 1 is an end-edge of a path P 1 from M P M o ( M , H ) , and therefore h 1 lies on P 1 to o. h n do es no t lie on a n y path from M P M o ( M , H ) as otherwise, due to lemma 3 .1 0, b ot h v ertices inciden t to h n w o uld b e inciden t to edges from H ′ , whic h contradicts the maximality of P . Note t ha t h n is not inciden t to an inner v ertex (not an end-v ertex) of a path P 1 from M P M o ( M , H ) as an y suc h v ertex is inciden t to an edge fr o m H lying on P 1 , and therefore differen t from h n . h n is inciden t neither to an end-v ertex of a path P 1 from M P M o ( M , H ) as it w ould contradict the maximality of P 1 . Th us, h n is not a djacen t to an edge lying on a path from M P M o ( M , H ) , and therefore h ′ n do es not lie on an y path from M P M o ( M , H ) . The pro of of (3a) is complete. Statemen t (3b) immediately follo ws from (3a). T aking in to account that | H | = α ( G ), | H ′ | = λ ( G ) − α ( G ), and | H | ≥ | H ′ | , w e get the follo wing result (also obtained in [10]) as a corollary from the statemen ts (2 ) a nd (3b) of lemma 3.1 1: Corollary 3.12 α ( G ) ≥ λ ( G ) − α ( G ) ≥ 4 ( β ( G ) − α ( G )) , i.e. β ( G ) α ( G ) ≤ 5 4 . 4 Spanner, S-F ores t and S-G r aph The gra ph on figure 1 is called spanner. A ve rtex v of spanner S is called i - v ertex, 1 ≤ i ≤ 3, if d S ( v ) = i . The 3-v ertex closest to a vertex v of spanner is referred as the base of v . The t w o paths of the spanner of length four connecting 1-v ertices are called sides . F or spanner S define sets U ( S ) and L ( S ) as follow s: U ( S ) . = { e ∈ E ( S ) : e is inciden t to a 1-v ertex } , L ( S ) . = { e ∈ E ( S ) \ U ( S ) : e is inciden t to a 2-v ertex } . 8 Note that for spanner S , and for ev ery ( H , H ′ ) ∈ M 2 ( S ), the edge connecting the 3- v ertices do es no t b elong to H ∪ H ′ , hence λ ( S ) = 8, α ( S ) = λ ( S ) − α ( S ) = 4, and β ( S ) α ( S ) = 5 4 , as β ( S ) = 5. The pair ( H , H ′ ) sho wn on figure 1 b elongs to M 2 ( S ). It can b e implied fro m the lemma 3.11 t ha t spanner is the minimal graph for whic h the parameters α and β are not equal. Prop erty 4.1 F or sp anner S and ( H , H ′ ) ∈ M 2 ( S ) , 2 -vertic es and 3 -vertic es of S ar e c ov- er e d by b oth H an d H ′ . Prop erty 4.2 F or every 1 -vertex v of sp anner S ther e is ( H, H ′ ) ∈ M 2 ( S ) such that v is misse d by H ( H ′ ). Definition 4.3 S -for es t is a for est whose c onne cte d c omp onen ts ar e sp a n ners. An i -vertex of a connected comp onen t (spanner) of an S - forest F is referred simply as an i -ve rtex of F . If S 1 , S 2 , ..., S k are connected comp onen t s of S -forest F then define sets U ( F ) and L ( F ) as f ollo ws: U ( F ) . = k [ i =1 U ( S i ); L ( F ) . = k [ i =1 L ( S i ) . Prop erty 4.4 I f the numb e r of c onne cte d c omp one n ts (sp a n ners) of a n S -for est F is k , then λ ( F ) = 2 α ( F ) = 8 k , and β ( F ) = 5 k , thus β ( F ) α ( F ) = 5 4 . Prop erty 4.5 I f F is an S -fo r est and ( H , H ′ ) ∈ M 2 ( F ) then H ∪ H ′ = U ( F ) ∪ L ( F ) . Definition 4.6 S -gr aph is a gr aph c ontaining an S -for est as a sp anning sub gr aph (b elow, we wil l r efe r to it as a sp anning S -for e st of an S -gr aph). Note that, spanning S -forest of an S -graph is not unique in general. It is easy t o see that spanner, S - forests, and S - graphs con tain a p erfect matc hing, a nd for S -forest it is unique. Let G b e an S -graph with a spanning S -forest F . 9 Prop erty 4.7 I f F has k c on n e cte d c omp onents (sp ann ers) then β ( G ) = β ( F ) = 5 k . Let us define an i - j edge of F as an edge connecting an i -vertex to a j -verte x of F . Also define: ∆( G, F ) . = { e ∈ E ( G ) : e connects a 1-v ertex of F to its ba se } , B ( G, F ) . = E ( G ) \ ( L ( F ) ∪ U ( F ) ∪ ∆( G, F )) . Prop erty 4.8 F or any L ( F ) - B ( G, F ) alternating even cycle the numb ers of 2 - 2 and 3 - 3 e dges lying on it ar e e qual. Pro of. Consider a n L ( F )- B ( G, F ) alternating even cycle C : ( u 1 , v 1 ) , ( v 1 , u 2 ) , ( u 2 , v 2 ) , ..., ( v n − 1 , u n ) , ( u n , v n ) , ( v n , u 1 ) , where ( u i , v i ) ∈ L ( F ) , i = 1 , 2 , ..., n ; ( v n , u 1 ) ∈ B ( G , F ) , ( v j , u j +1 ) ∈ B ( G , F ) , j = 1 , 2 , ..., n − 1. F or a v ertex w of the cycle C let δ ( w ) b e the frequency of app earance of the v ertex w during the circumference of C (the num b er o f indices i 0 for whic h w = u i 0 or w = v i 0 ). As an y v ertex w lying on C is inciden t to an edge from L ( F ) tha t lies on C b efore or after v during the circumference, and as edges from L ( F ) are 2-3 edges, w e g et: X w i s a 2-vertex lying on C δ ( w ) = X w i s a 3-ve rtex lying on C δ ( w ) = n. On the ot her hand, denote b y m 22 , m 33 , m 23 the num b ers of 2 -2, 3 -3, 2 -3 edges lying on C , resp ectiv ely . As f or eac h v ertex w lying on C , 2 δ ( w ) is the num b er o f edges that lie on C and are inciden t to w , implies: X w i s a 2-ve rtex lying on C 2 δ ( w ) = 2 m 22 + m 23 , X w i s a 3-ve rtex lying on C 2 δ ( w ) = 2 m 33 + m 23 , where the left sides of the equalities represen t the n um b ers of edges lying on C a nd inciden t to 2- v ertices and 3-vertice s of C , resp ective ly . Th us, m 22 = m 33 . 10 5 Main Theo rem Theorem F or a gr a ph G ( G do es not c on tain isolate d vertic es ), the e q uali ty β ( G ) α ( G ) = 5 4 holds, if and only if G is an S -g r aph with a sp ann ing S -for est F , s a tisfying the fol lowing c onditions : (a) 1 - vertic es of F ar e not incident to any e d ge fr om B ( G, F ) ; (b) if a 1 -vertex u of F is incident to an e dge fr o m ∆( G, F ) , then the 2 2 -vertex of F adjac ent to u is not incident to any e dge fr om B ( G, F ) ; (c) for every L ( F ) - B ( G, F ) alternating even cycle C of G c ontaining a 2 - 2 e dge, the gr aph C B ( G,F ) is not bip artite. The pro of of the theorem is lo ng, so it is divided into subse ctions: Necessit y and Suffi- ciency , whic h, in their turn, are split in to num b ers of lemmas. 5.1 Necessit y In this subsection, we assume that β ( G ) α ( G ) = 5 4 , and prov e that G is an S - graph. Then, on the con trary assumptions w e prov e consequen tly that the conditions (a), (b) and (c) are satisfied for an arbitrary spanning S - f orest of G . As one can see, w e prov e a statemen t stronger than the Necessit y of the theorem. Let G b e a graph, M b e a fixed maxim um matching of it, and ( H , H ′ ) b e an arbitrarily c ho sen pair fro m M 2 ( G, M ). Supp ose that for the g raph G the equalit y β ( G ) α ( G ) = 5 4 holds. Due to coro lla ry 3 .12, we hav e: α ( G ) = λ ( G ) − α ( G ) = 4( β ( G ) − α ( G )) . ( ‡ ) Lemma 5.1.1 Each p ath fr om Y ( M , H , H ′ ) is of length fo ur, e ach p ath fr om M P M o ( M , H ) is of length five, and every e dge fr om H lies on a p ath fr om Y ( M , H, H ′ ) . Pro of. Due to equality ( ‡ ) and statemen t (2) of lemma 3.11 , w e get: | H | = 2 | Y ( M , H, H ′ ) | . Therefore, as there are at least t w o edges from H lying on eac h path of Y ( M , H , H ′ ) (state- men t (3b) of t he lemma 3.11), the length of each pa th from Y ( M , H , H ′ ) is precisely four, 2 W e wr ite “the” here a s if the c ondition (a) is s atisfied then there is only one 2 -vertex adjacent to u (the 2-vertex connected to u via the edge fr om U ( F )). 11 and ev ery edge from H lies on a path from Y ( M , H , H ′ ). Moreov er, the length of ev ery path from M P M o ( M , H ) is precisely fiv e (due to statemen t (2) of the lemma 3.9 it is at least fiv e for an y g r aph), a s otherwise we would hav e either an edge fro m H not lying on any path from Y ( M , H, H ′ ), or a path from Y ( M , H , H ′ ) with length g r eat er than four. This lemma implies t hat each path P from M P M o ( M , H ) together with the tw o paths from Y ( M , H, H ′ ) starting fr o m the end-edges o f P form a spanner ( fig ure 2 ). Figure 2: Since there a r e β ( G ) − α ( G ) paths in M P M o ( M , H ) (prop ert y 3.6), we get: Corollary 5.1.2 Ther e is a sub gr aph F of the gr aph G that is an S -for est c ontaining β ( G ) − α ( G ) sp anners as its c onne cte d c o mp onents. No w, let F b e a n S -fo r est arbitrarily c hosen among the ones describ ed in the corolla ry 5.1.2. Due to prop erty 4.4, α ( F ) = λ ( F ) − α ( F ) = 4( β ( G ) − α ( G )), therefore due to equality ( ‡ ), α ( F ) = λ ( F ) − α ( F ) = α ( G ) = λ ( G ) − α ( G ). This means that M 2 ( F ) ⊆ M 2 ( G ). Let ( H , H ′ ) b e an ar bitr arily ch osen pair from M 2 ( F ) ⊆ M 2 ( G ). Not e that the c hoice of ( H , H ′ ) differs from t he one ab ov e (w e k eep this nota tion as the reader may ha v e already got used with a pair from M 2 ( G ) denoted by ( H , H ′ )). Lemma 5.1.3 If ( u, v ) ∈ E ( G ) \ E ( F ) , and u / ∈ V ( F ) or u is a 1 -vertex of F , then v is a 3 -vertex of F . Pro of. D ue t o prop erty 4.2, without loss o f generality , w e ma y assume t ha t u is misse d by H ′ . Clearly , v ∈ V ( F ) as otherwise v w ould also b e missed by H ′ and w e could “enlarge” H ′ b y “adding” ( u , v ) to it, whic h con tra dicts ( H, H ′ ) ∈ M 2 ( G ). No w, let us sho w tha t v is neither a 1-v ertex nor a 2-v ertex. Supp ose for con tradiction that it is, and let S 0 b e the spanner (connected comp onent) of F con taining v . Define 12 Figure 3: S 0 matc hings H 1 , H ′ 1 as follows (figure 3): H 1 . = ( H \ E ( S 0 )) ∪ M 0 , where M 0 is the p erfect matching of S 0 ; H ′ 1 . = ( H ′ \ E ( S 0 )) ∪ J 0 , where J 0 is a matc hing of cardinality three satisfying J 0 ⊆ L ( S 0 ) ∪ { ( u, v ) } (it alw ays exists). Clearly , H 1 ∩ H ′ 1 = ∅ , and, since | H ∩ E ( S 0 ) | = | H ′ ∩ E ( S 0 ) | = 4, | M 0 | = 5 a nd | J 0 | = 3, w e ha v e | H 1 | + | H ′ 1 | = ( | H | − 4 + 5) + ( | H ′ | − 4 + 3) = λ ( G ) and | H 1 | > | H | . Th is contradicts ( H , H ′ ) ∈ M 2 ( G ), concluding t he pro of o f the lemma. Lemma 5.1.4 If ( u, v ) ∈ E ( G ) \ E ( F ) , then u ∈ V ( F ) and i f u is a 1 -vertex of F then v i s its b ase. Pro of. Assume the con trary . Let ( u, v ) ∈ E ( G ) \ E ( F ), where u either b elongs to V ( G ) \ V ( F ), or is a 1-v ertex whose ba se is not v . As ( u , v ) satisfies the conditions of the lemma 5.1.3, implies that v is a 3-v ertex o f F . Let W 0 b e the side of the spanner S 0 (connected comp onent of F ) con taining v . It is easy to notice that u do es not b elong to V ( W 0 ) as otherwise it w o uld b e a 1- v ertex of S 0 whose base is v . Due t o prop erty 4.2, without loss of generalit y , w e ma y assume that u is missed b y H ′ . Define matchings H 1 and H ′ 1 as f ollo ws (figure 4): H 1 . = ( H \ E ( W 0 )) ∪ { ( u , v ) } ∪ ( U ( S 0 ) ∩ E ( W 0 )) , H ′ 1 . = ( H ′ \ E ( W 0 )) ∪ { e } , where e is an edge from L ( S 0 ∩ E ( W 0 ). 13 Figure 4: W 0 Clearly , H 1 and H ′ 1 are disjoin t matchin gs. Moreo ve r, since | H ∩ E ( W 0 ) | = | H ′ ∩ E ( W 0 ) | = 2, | H 1 | + | H ′ 1 | = ( | H | − 2 + 1 + 2) + ( | H ′ | − 2 + 1) = λ ( G ) and | H 1 | > | H | , whic h contradicts ( H , H ′ ) ∈ M 2 ( G ) concluding t he pro of o f the lemma. Lemma 5.1.5 G is an S -gr aph with sp anning S -for e st F . Pro of. Lemma 5.1.4 asserts that there is no edge inciden t to a v ertex from V ( G ) \ V ( F ), i.e. all v ertices fr o m V ( G ) \ V ( F ) are isolated. This is a contradiction as w e assume that G has no isolated v ertices (see the b eginning of In tro duction). Th us, V ( G ) \ V ( F ) = ∅ and F is a spanning S -forest of G , whic h means that G is an S -graph. Due to this lemma, F (an arbitrarily chose n S - f orest with β ( G ) − α ( G ) spanners) is spanning. Ob viously , the conv erse is a lso true. So, w e ma y say that F is an arbitrary spanning S -forest of G . Lemma 5.1.6 The gr aph G with its sp anning S -for est F satisfies the c ondition (a) of the the or em. Pro of. Let u b e a 1-v ertex of F . Lemma 5.1.4 asserts that if e = ( u , v ) is an edge from E ( G ) \ E ( F ) then v is the base of u , thu s e ∈ ∆( G, F ). This means that the condition ( a ) is satisfied. Lemma 5.1.7 The gr aph G w ith its sp a nning S -for est F satisfi e s the c o ndition (b) of the the or em. Pro of. Assume that u is a 1- vertex of F inciden t to a n edge ( u, w ) from ∆( G, F ) ( w is t he base of u ), and v is the 2-ve rtex of F adjacent to u (this 2-v ertex is unique as, due to lemma 5.1.6, u can b e incident only to edges from U ( F ) a nd ∆( G, F )). On the opp osite assumption v is inciden t to an edge ( v , w ′ ) from B ( G, F ) (figure 5a). Let us construct a subgraph F ′ of G by remo ving ( v , w ) fro m F and adding ( w , u ): F ′ . = ( F \{ ( v , w ) } ) ∪ { ( w , u ) } . 14 Figure 5: Note that F ′ is a spanning S - forest o f G , f or which v is a 1-v ertex (figure 5b), and B ( G, F ′ ) = B ( G, F ). Thus , ( v , w ′ ) ∈ B ( G , F ′ ). On the o ther hand, the g r a ph G with its spanning S - forest F ′ satisfies the conditio n (a) of the theorem (lemma 5.1.6). Th us, v cannot b e inciden t to an edge from B ( G, F ′ ), and we hav e a contradiction. Lemma 5.1.8 The gr aph G w ith its sp a nning S -for est F satisfi e s the c o ndition (c) of the the or em. Pro of. Supp ose for contradiction that there exists an L ( F )- B ( G, F ) alternating eve n cycle C of the graph G containing a 2- 2 edge e 0 and the g raph C B ( G,F ) is bipart it e. The definition o f L ( F ) implies that for each v ∈ V ( C ), 1 ≤ d C L ( F ) ( v ) ≤ 2. Therefore, due to prop ert y 3.2, 1 ≤ d C B ( G,F ) ( v ) ≤ 2, and if v is a 2-vertex , then d C B ( G,F ) ( v ) = 1. Hence, the connected comp onen ts of the bipa r tite graph C B ( G,F ) are paths or cycles of ev en length. Cho ose ( H 1 , H ′ 1 ) ∈ M 2 ( C B ( G,F ) ). D ue to prop ert y 3.1 , H 1 ∪ H ′ 1 = E ( C B ( G,F ) ). Let F 0 b e the subgraph of F whose connected comp onen ts are those sides of the spanners (connected comp onen ts) S 1 , S 2 , ..., S β ( G ) − α ( G ) of F , whic h do not contain an y edge from C . Note that none of the edges of F 0 is adjacen t to an edge lying on C since otherwise one of the edges from L ( F ) ∩ E ( F 0 ) would lie on C a s w ell con tradicting t he definition of F 0 . Let ( H 2 , H ′ 2 ) ∈ M 2 ( F 0 ). As the sides of spanners are pa ths, a gain due to pro p ert y 3.1, H 2 ∪ H ′ 2 = E ( F 0 ). Define the sets U ′ , L ′ , U ′′ as f ollo ws: U ′ . = { e ∈ U ( F ) : e is adjacen t to a n edge e ′ ∈ E ( C B ( G,F ) ) } , L ′ . = { e ∈ L ( F ) : e / ∈ C , e is adja cent to an edge e ′ ∈ E ( C B ( G,F ) ) } , U ′′ . = { e ∈ U ( F ) : e is adjacen t to an edge e ′ ∈ L ′ } . Define a pair of disjoin t matc hings ( H 3 , H ′ 3 ) as follo ws: H 3 . = { e ∈ U ′ ∪ L ′ : the edge e ′ ∈ E ( C B ( G,F ) ) adjacen t to e do es not b elong to H 1 } , 15 H ′ 3 . = { e ∈ U ′ ∪ L ′ : the edge e ′ ∈ E ( C B ( G,F ) ) adjacen t to e do es not b elong to H ′ 1 } , Note that H 3 ∪ H ′ 3 = U ′ ∪ L ′ . Define a pair of disjoin t matc hings ( H 4 , H ′ 4 ) as follows: H 4 . = { e ∈ U ′′ : the edge e ′ ∈ L ′ adjacen t to e do es not b elong to H 3 } , H ′ 4 . = { e ∈ U ′′ : the edge e ′ ∈ L ′ adjacen t to e do es not b elong to H ′ 3 } , Note that H 4 ∪ H ′ 4 = U ′′ . Finally , define a pair of disjoint matc hings ( H 0 , H ′ 0 ) as follo ws: H 0 . = 4 [ i =1 H i , H ′ 0 . = 4 [ i =1 H ′ i . Since U ( F ) = ( E ( F 0 ) ∩ U ( F )) ∪ U ′ ∪ U ′′ and L ( F ) = ( E ( F 0 ) ∩ L ( F )) ∪ L ′ ∪ E ( C L ( F ) ), w e ha v e H 0 ∪ H ′ 0 = U ( F ) ∪ ( L ( F ) \ E ( C L ( F ) )) ∪ E ( C B ( G,F ) ) . Hence, | H 0 | + | H ′ 0 | = | H 0 ∪ H ′ 0 | = | U ( F ) | + | L ( F ) | − | E ( C L ( F ) ) | + | E ( C B ( G,F ) ) | = | U ( F ) | + | L ( F ) | since | E ( C L ( F ) ) | = | E ( C B ( G,F ) ) | . F rom the equalit y ( ‡ ) w e hav e that λ ( G ) = 8( β ( G ) − α ( G )). Therefore, as | U ( F ) | + | L ( F ) | = 8( β ( G ) − α ( G ) ), we get: | H 0 | + | H ′ 0 | = λ ( G ) . As λ ( G ) = 2 α ( G ) (equalit y ( ‡ )), due to prop ert y 3.8, there is no maximal H 0 - H ′ 0 alternating o dd path ( M P o ( H 0 , H ′ 0 ) = ∅ ). Let us show that there is one and get a contradiction. Let e 1 , e 2 b e the edges from U ( F ) adja cen t to e 0 . Clearly e 1 , e 2 ∈ U ′ as e 0 ∈ E ( C B ( G,F ) ). The construction of H 0 and H ′ 0 (or r ather H 3 and H ′ 3 ) implies that the path e 1 , e 0 , e 2 is an H 0 - H ′ 0 alternating o dd path. Let u b e the 1-v ertex inciden t to e 1 . Lemmas 5.1.6 and 5.1.7 imply that u is not inciden t to an edge other than e 1 . The same can b e sho wn for e 2 . Therefore, the path e 1 , e 0 , e 2 is a maximal H 0 - H ′ 0 alternating path of o dd length ( b elongs to M P o ( H 0 , H ′ 0 )). This contradiction concludes the pro of of the lemma. Lemmas 5.1 .5, 5 .1.6, 5.1.7 and 5.1.8 imply the follow ing statemen t, whic h is stronger than the Necess it y of the theorem: Statemen t 1 If for a gr aph G the e quality β ( G ) α ( G ) = 5 4 holds, then G is an S -gr aph any sp anning S -for est F of which sa tisfies the c onditions (a), (b) and ( c) of the the o r em. 16 5.2 Sufficiency The structure of the pro of of the Sufficiency is the follo wing: for an S -graph G with a spanning S -forest F satisfying the conditions (a) and (b) of the theorem we show that if β ( G ) α ( G ) 6 = 5 4 then the condition (c) is not satisfied. The pro of of the Sufficiency is more complicated than the one o f the Necessit y . Therefore, w e first presen t the idea o f the pro of briefly . Let G b e an S -g r a ph with a spanning S -forest F satisfying the condition (a) o f the theorem. Let us mak e also an additiona l assumption: Assumption 1 Ther e is a p air ( S, S ′ ) ∈ M 2 ( G ) \ M 2 ( F ) such that ( S ∪ S ′ ) ∩ ∆( G, F ) = ∅ . Cho ose an arbitrary pair ( H , H ′ ) from M 2 ( F ). Let T U , T L , T B b e the sets of edges from S ′ \ H ′ that b elong to U ( F ) , L ( F ) , B ( G, F ) re- sp ectiv ely . Note t hat T U , T L , T B are pairwise disjoin t, and, due to assumption 1, w e ha v e: S ′ \ H ′ = T U ∪ T L ∪ T B . Define Q, Q U , Q L , Q B as the sets of all vertice s that are inciden t to edges f r o m S ′ \ H ′ , T U , T L , T B , resp ectiv ely . Note that Q U , Q L , Q B are pairwise disjoin t, | Q | = 2 | S ′ \ H ′ | , and Q = Q U ∪ Q L ∪ Q B . Using pa t hs from M P ( S, H ) we construct a set of trails A ′′ ⊆ T L ( F ) o ( L ( F ) , B ( G, F )) such that a ll edges lying on trails from A ′′ that b elong to B ( G, F ) are from S , and V 0 ( A ′′ ) = Q B . Note that the trails from A ′′ are connected with edges from T B (as sho wn in figure 10) making L ( F )- B ( G, F ) alternating ev en cycles, i.e. eac h edge from T B lies on one suc h cycle. Also note that for any suc h cycle C , the gr a ph C B ( G,F ) do es not contain o dd cycles as E ( C B ( G,F ) ) ⊆ S ∪ S ′ , therefore C B ( G,F ) is bipart it e. After this, assuming that the condition (b) is a lso satisfied and β ( G ) α ( G ) 6 = 5 4 (it is shown that these assumptions t o gether are stronger than assumption 1), w e pro v e that T B con tains a 2-2 edge. Therefore, at least o ne of the L ( F )- B ( G , F ) alternating ev en cycles describ ed ab ov e contains a 2-2 edge con tradicting the condition (c) of the theorem. The construction of A ′′ is a step-b y-step pro cess. First, fro m M P ( S, H ) w e construct a set of paths A for whic h Q U ∪ Q L ∪ Q B ⊆ V 2 ( A ). Then, A is transformed to a set of trails A ′ for which V 0 ( A ′ ) = Q L ∪ Q B . And finally , A ′ is tra nsformed to A ′′ men tio ned ab o v e. No w, let us start the pro of. As mentioned a b o v e, assume that G is an S -graph with a spanning S - forest F satisfying the condition (a) of the theorem, and the assumption 1 holds. Cho ose ( S, S ′ ) and ( H , H ′ ) as describ ed ab o v e. 17 In o rder t o characterize the set Q define t he sets R 1 , R 2 , R 3 , R as follow s: R 1 . = { v ∈ V ( G ) : v is inciden t to an edge fr o m H ′ \ S ′ } , R 2 . = { v ∈ R 1 : v is a 1-v ertex of F } , R 3 . = { v ∈ V ( G ) : v is a 1- v ertex of F incid en t to an edge from H \ S } , R . = ( R 1 \ R 2 ) ∪ R 3 . W e claim that R is a set of v ertices “ p oten tially” inciden t to edges from S ′ \ H ′ . F ormally: Lemma 5.2.1 Q ⊆ R . Pro of. Assume that v ∈ Q and e is the edge from S ′ \ H ′ inciden t to v . If v is not a 1-v ertex then, due to pr o p ert y 4.1, v is inciden t to an edge from H ′ \ S ′ , and therefore, b elongs to R 1 \ R 2 . On the other hand, if v is a 1-v ertex then, due to condition (a ) and assumption 1, e ∈ U ( F ). Moreo v er, e ∈ H \ S as e b elongs to S ′ and do es not b elong to H ′ . Th us, v ∈ R 3 . F urther, w e sho w tha t in fact Q = R . W e intro duce R a s its definition is muc h easier to w o r k with. Consider the paths from M P ( S, H ) (they are the main w orking to o ls throughout the whole pro of ). F rom condition (a) of the theorem a nd assumption 1 we get the follo wing coro lla ries: Corollary 5.2.2 A ny 1 - vertex of F is incident to at most o ne e d g e fr om S △ H . Corollary 5.2.3 (1) I f a 1 -vertex (a n e dge fr om U ( F ) ) lies on a p ath fr om M P ( S, H ) , then it is an end - vertex (end-e dge) of the latter. (2) No 1 -vertex (e d g e fr om U ( F ) ) lies on a cycle fr o m C e ( S, H ) . Lemma 5.2.4 If e ∈ S is an end-e dg e of a p ath P ∈ M P ( S, H ) an d v is an end-v ertex of P incid ent to e , then e ∈ U ( F ) and v is a 1 -vertex of F . Pro of. Let e 1 ∈ S b e a n end-edge of a path P : e 1 , e 2 , ..., e n ∈ M P ( S, H ), a nd v b e an end-v ertex of P inciden t to e 1 . If v is not a 1- v ertex then, due to prop erty 4.1, v is inciden t to an edge e ∈ H . Due to the definition of alternating path, e 6 = e 1 , therefore the path e, e 1 , e 2 , ..., e n b elongs to P ( S, H ), whic h contradicts the maximalit y of P (see the definition 2.10 o f a maximal path). As a coro llary from lemma 5.2.4 w e get: 18 Corollary 5.2.5 Every p ath fr om M P S o ( S, H ) is of le n gth at le ast five. Lemma 5.2.6 | ( S \ H ) ∩ U ( F ) | = 2 | M P S o ( S, H ) | + | M P e ( S, H ) | . Pro of. D ue to prop ert y 3.3, ev ery edge from S \ H lies either o n a path from M P ( S, H ) or a cycle f r o m C e ( S, H ). Moreov er, corollary 5.2 .3 implies that ev ery edge from ( S \ H ) ∩ U ( F ) is an end-edge of a path from M P ( S, H ), hence | ( S \ H ) ∩ U ( F ) | ≤ 2 | M P S o ( S, H ) | + | M P e ( S, H ) | . Corollary 5.2.5 implies t ha t eac h path from M P S o ( S, H ) has t wo different end-edges. Th us, due to lemma 5.2.4, | ( S \ H ) ∩ U ( F ) | ≥ 2 | M P S o ( S, H ) | + | M P e ( S, H ) | , which completes the pro of of the lemma. The follo wing t w o lemmas provide us with three inequalities, t he b oundary cases (equali- ties) o f which are related to a num b er o f useful prop erties. Those inequalities together imply that the equalities are indeed the case. Lemma 5.2.7 2 | H ′ \ S ′ | − | R | ≤ 2 | H ′ \ S ′ | − | Q | ≤ 2( | M P S o ( S, H ) | − M P H o ( S, H ) | ) , and the e quality c ases in the first and se c ond i n e qualities hold if and only i f Q = R and λ ( F ) = λ ( G ) , r esp e ctively. Pro of. By the definition of λ , λ ( F ) = | H | + | H ′ | ≤ | S | + | S ′ | = λ ( G ). Due to prop ert y 3.5 , this is equiv alen t to | H ′ | − | S ′ | ≤ | S | − | H | = | M P S o ( S, H ) | − M P H o ( S, H ) | , or | H ′ \ S ′ | − | S ′ \ H ′ | ≤ | M P S o ( S, H ) | − M P H o ( S, H ) | . T aking into account that 2 | S ′ \ H ′ | = | Q | w e get: 2 | H ′ \ S ′ | − | Q | ≤ 2( | M P S o ( S, H ) | − M P H o ( S, H ) | ) , and the equality holds if and only if λ ( F ) = λ ( G ). F urthermore, lemma 5.2.1 implies that | Q | ≤ | R | , and we get 2 | H ′ \ S ′ | − | R | ≤ 2 | H ′ \ S ′ | − | Q | , and the equalit y holds if and only if | Q | = | R | , whic h means that Q = R , due to lemma 5.2.1. Lemma 5.2.8 2 | H ′ \ S ′ | − | R | ≥ | ( H ′ \ ( S ∪ S ′ )) ∩ U ( F ) | + 2( | M P S o ( S, H ) | − M P H o ( S, H ) | ) , and the e quality c ase holds i f a n d only if for any p ath fr o m M P ( S, H ) , if its end-e dg e e ∈ H \ S then e ∈ U ( F ) . 19 Pro of. First note that | R 1 | = 2 | H ′ \ S ′ | . Clearly , an y v ertex from R 2 is inciden t to one edge from ( H ′ \ S ′ ) ∩ U ( F ) and vice vers a. F urthermore, any edge from ( H ′ \ S ′ ) ∩ U ( F ) either b elongs to ( H ′ \ ( S ∪ S ′ )) ∩ U ( F ), or b elongs to ( S \ H ) ∩ U ( F ). Th us, due to lemma 5.2 .6, w e get: | R 2 | = | ( H ′ \ ( S ∪ S ′ )) ∩ U ( F ) | + | ( S \ H ) ∩ U ( F ) | = = | ( H ′ \ ( S ∪ S ′ )) ∩ U ( F ) | + 2 | M P S o ( S, H ) | + | M P e ( S, H ) | . Moreo v er, prop erty 3.3 together with corollary 5 .2 .3 implies that ev ery v ertex v ∈ R 3 is an end-v ertex of a path fro m M P ( S, H ), and the end-edge of that pat h inciden t to v is from H \ S . Th us, | R 3 | ≤ 2 | M P H o ( S, H ) | + | M P e ( S, H ) | , and the equality holds if and only if the vice v ersa is also true, i.e. any end-edge e ∈ H \ S of a path from M P ( S, H ) is fro m U ( F ). Hence we hav e | R | = | R 1 | − | R 2 | + | R 3 | ≤ 2 | H ′ \ S ′ | − ( | ( H ′ \ ( S ∪ S ′ )) ∩ U ( F ) | + 2 | M P S o ( S, H ) | + | M P e ( S, H ) | )+ +2 | M P H o ( S, H ) | + | M P e ( S, H ) | , or 2 | H ′ \ S ′ | − | R | ≥ | ( H ′ \ ( S ∪ S ′ )) ∩ U ( F ) | + 2( | M P S o ( S, H ) | − M P H o ( S, H ) | ) . Lemmas 5.2.7 and 5.2.8 together imply the following: Corollary 5.2.9 (1) 2 | H ′ \ S ′ | − | R | = 2( | M P S o ( S, H ) | − M P H o ( S, H ) | ); (2) Q = R ; (3) λ ( F ) = | H | + | H ′ | = | S | + | S ′ | = λ ( G ); (4) I f e ∈ H \ S is an end- e dge of a p ath fr om M P ( S, H ) and v is an en d -vertex incide nt to e then e ∈ U ( F ) and v is a 1 -ve rtex of F ; (5) ( H ′ \ ( S ∪ S ′ )) ∩ U ( F ) = ∅ . 20 Statemen t (5) of corollary 5.2.9 and prop ert y 4.5 imply: Corollary 5.2.10 (1) H ′ \ ( S ∪ S ′ ) ⊆ L ( F ); (2) V ertic es incid ent to e dges fr om H ′ \ ( S ∪ S ′ ) b elo ng to R 1 \ R 2 ⊆ R = Q . Statemen t (4) of corollary 5.2.9 and lemma 5.2.4 imply the following: Corollary 5.2.11 (1) Th e length of an y p ath fr om M P H o ( S, H ) is at le a s t thr e e; (2) Th e length of an y p ath fr om M P e ( S, H ) is at le a s t four; (3) Every p ath of M P ( S, H ) c onne cts 1 -vertic es (end-vertic es ar e 1 -vertic e s). No w w e are able to g ive a c haracterization of the set T U . Lemma 5.2.12 T U = ( H \ S ) ∩ U ( F ) . Pro of. Let e ∈ ( H \ S ) ∩ U ( F ) a nd v b e the 1 -v ertex inciden t to e . The definition of R 3 implies tha t v ∈ R 3 ⊆ R , and due to statemen t (2) of the corollary 5.2.9, v ∈ Q . As v is a 1-v ertex, v ∈ Q U , th us, e ∈ T U . Hence, ( H \ S ) ∩ U ( F ) ⊆ T U . On the other hand, b y definition T U = ( S ′ \ H ′ ) ∩ U ( F ). F or any edge e ∈ ( S ′ \ H ′ ) ∩ U ( F ) w e hav e that e / ∈ S as e ∈ S ′ , and e ∈ H as e ∈ U ( F ) and e / ∈ H ′ (prop erty 4.5). Thu s, e ∈ ( H \ S ) ∩ U ( F ), i.e. T U ⊆ ( H \ S ) ∩ U ( F ) and the pro of of the lemma is complete. The follow ing lemma describ es the placemen t of edges fro m H ′ lying on S - H alt ernating trails (maximal paths o r simple ev en cycles): Lemma 5.2.13 (1) I f e ∈ H ′ lies on a p ath P ∈ M P ( S, H ) then e ∈ E 2 ( P ) ; (2) No e dge fr o m H ′ lies on a cycle fr om C e ( S, H ) . 21 Pro of. Let e ∈ H ′ and note that if e lies on a path from M P ( S, H ) o r o n a cycle from C e ( S, H ) then e ∈ S . (1). Supp ose that e / ∈ E 1 ( P ). Then, e ∈ L ( F ) as otherwise e ∈ U ( F ) (prop ert y 4.5) and, due to statemen t (1) of corollary 5.2.3, e ∈ E 1 ( P ). Therefore, due to maximalit y of P , e is adjacen t to tw o edges from H \ S lying on P , one of whic h (denote it b y h ) b elongs to U ( F ). Due to statemen t (1) of corollary 5.2.3, h ∈ E 1 ( P ), whic h means that e ∈ E 2 ( P ) \ E 1 ( P ). (2). Supp ose fo r con tra diction that e lies on a cycle C ∈ C e ( S, H ). Here again, e ∈ L ( F ) as otherwise e ∈ U ( F ) (prop erty 4.5) contradicting statemen t (2) of corollary 5.2.3. Therefore, there a re tw o edges from H \ S adjacent to e lying on C , one of whic h b elong s to U ( F ), whic h con tradicts statemen t (2) of corollary 5.2.3. No w let us construct the set of paths A men tioned ab o v e: A . = M P ( S, H ) ∪ ( H ′ \ ( S ∪ S ′ )) (the edges f rom H ′ \ ( S ∪ S ′ ) are considered a s paths of length one). The follow ing lemma pro vides t he a b o v e-mentioned prop erty of A , for whic h A is actually constructed. Lemma 5.2.14 Q ⊆ V 2 ( A ) . Pro of. Due to statemen t 2 of corolla r y 5.2.9, Q = R . So, assume that v ∈ R 1 \ R 2 and e is the edge from H ′ \ S ′ inciden t to v . If e / ∈ S , then e ∈ H ′ \ ( S ∪ S ′ ) and w e are done. On the other hand, if e ∈ S , then, as e / ∈ H , e lies on a path P ∈ M P ( S, H ) (due to statement (2) of lemma 5.2.13, e cannot lie o n a cycle fr o m C e ( S, H )). Hence, according t o statemen t (1 ) of lemma 5.2.13 , e ∈ E 2 ( P ), whic h means that v ∈ V 2 ( P ). No w let v ∈ R 3 b e a 1-ve rtex incident to the edge e ∈ ( H \ S ) ∩ U ( F ). Clearly , e lies on a pat h fro m M P ( S, H ) a s it cannot lie on a cycle from C e ( S, H ) (see statemen t (2) of corollary 5.2.3). Moreo v er, v lies on P to o, and, due to statemen t (1) of corollary 5.2.3, is an end-ve rtex of P , concluding the pro o f of the lemma. No w we in tend to t ransform A to a set of tra ils A ′ ⊆ T L ( F ) o ( L ( F ) , B ( G, F )), suc h tha t V 0 ( A ′ ) = Q L ∪ Q B and an y edge from B ( G, F ) lying on a trail from A ′ b elongs to S . F or eac h path P of M P ( S, H ) we transform only the edges of sets E b 2 ( P ) and E e 2 ( P ) (first tw o and last t w o edges of the path). Note that corolla ry 5.2.5 and statemen ts (1) and (2) of corollar y 5.2.1 1 imply that E b 2 ( P ) and E e 2 ( P ) a r e sets of car dina lity 2 whic h do no t coincide though ma y inters ect. F or eac h path P ∈ M P ( S, H ), and for X = E b 2 ( P ) or X = E e 2 ( P ), where X = { e, e ′ } , e ∈ E 1 ( P ) and e ′ ∈ E 2 ( P ) \ E 1 ( P ), do the follo wing: 22 Case 1 e ∈ H , e ′ ∈ B ( G, F ) (figure 6a). Due to statemen t (4 ) of corolla ry 5.2.9, e ∈ U ( F ). Let e ′′ b e the edge f r o m L ( F ) adjacent to e and e ′ (figure 6a). Figure 6: Remo v e e fro m P and add (concatenate) e ′′ instead (figure 6b). Case 2 e ∈ H , e ′ / ∈ B ( G, F ) (hence e ′ ∈ L ( F )) (figure 7a). Figure 7: Here again due to statemen t (4) o f corollary 5.2.9, e ∈ U ( F ). Remo v e e and e ′ from P (figure 7b). Case 3 e ∈ S . As e ∈ U ( F ) (lemma 5.2.4 ) and e ′ ∈ H ( e ∈ S ), e ′ ∈ L ( F ) (figure 8a). Remo v e e fro m P (figure 8b). 23 Figure 8: Note tha t the transformation describ ed ab o v e is defined correctly b ecause of the follow ing: for a path P ∈ M P ( S, H ), E b 2 ( P ) and E e 2 ( P ) may hav e non-empt y in tersection only if P ∈ M P H o ( S, H ) and the length of P is three. In this case b oth E b 2 ( P ) and E e 2 ( P ) are handled b y the case 1, and the edge e ′ ∈ E b 2 ( P ) ∩ E e 2 ( P ) is treated in the same wa y . Define sets of edges Z Q U and Z Q U as f ollo ws: Z Q U . = { e ∈ H ′ \ ( S ∪ S ′ ) / e is inciden t to a v ertex from Q U } , Z Q U . = ( H ′ \ ( S ∪ S ′ )) \ Z Q U , and let D b e the set of trails obta ined from the paths of M P ( S, H ) b y t he transformation described ab ov e (w e do not sa y that D is a set of paths as it is not hard to construct an example of D con taining a trail that is not a path using the f act tha t v ertex v in case 1 may also lie o n P ). Let A ′ b e the follow ing: A ′ . = D ∪ Z Q U . Some pro p erties of A ′ are giv en b elow. Lemma 5.2.15 No e dge fr om S ∩ H ′ lies on a tr a i l fr om A ′ . Pro of. First note that no edge from Z Q U b elongs to S . So w e need to examine only D . Statemen t (1) of lemma 5.2 .1 3 and the transforma t ion describ ed ab o v e imply that the only edges lying on trails from D that b elong to H ′ are edges denoted by e ′′ in the case 1 (figure 6). But tha t edges do not b elong to S , and the pro o f of the lemma is complete. Lemma 5.2.16 If an e dge e lies on a tr ail fr om A ′ and e ∈ B ( G, F ) ∪ S then e ∈ B ( G, F ) ∩ S 3 . 3 In other words, the set of edges fro m B ( G, F ) lying on trails from A ′ and the set o f edges fro m S lying on tra ils fr om A ′ coincide. 24 Pro of. Clearly , all edges from B ( G, F ) lying on trails from A ′ , lie on trails fro m D as Z Q U ⊆ L ( F ) (statemen t (1) of corollary 5.2 .1 0). First note that a ll edges from B ( G, F ) lying on paths from M P ( S, H ) b elong to S as edges f rom H b elong to U ( F ) ∪ L ( F ) (prop erty 4.5). During the transformation of M P ( S, H ) to D , the only edges w e add are edges e ′′ in case 1 (figure 6), whic h are from L ( F ). Th us, still all edges from B ( G , F ) lying on D b elong to S , and the pro of of the first part of the lemma is complete. Let edge e ∈ S lies on a trail T ∈ A ′ . Ob viously , T ∈ D . Due to assumption 1 S ⊆ U ( F ) ∪ L ( F ) ∪ B ( G, F ). As w e do not add a n y edge fro m H ∩ S during the tra nsformation of M P ( S, H ) to D , e do es not b elong to H . D ue to lemma 5.2.15, e do es not b elong to H ′ either. Therefore, as H ∪ H ′ = U ( F ) ∪ L ( F ) (prop erty 4.5), e ∈ B ( G, F ). Lemma 5.2.17 A ′ ⊆ T L ( F ) o ( L ( F ) , B ( G, F )) . Pro of. D ue t o statemen t (1) of corollary 5.2.1 0, Z Q U ⊆ T L ( F ) o ( L ( F ) , B ( G, F )), so w e need to show the same for D only . Corollary 5.2.3 a nd the construction of D imply that no edge from U ( F ) lies on a trail from D , therefore eve ry edge from H lying on a t rail f rom D b elongs to L ( F ) ( prop erty 4.5). On the other hand, due to lemma 5.2.16, all edges f r om S lying on trails from D are from B ( G, F ). The only edges that do not b elong to S ∪ B ( G, F ) are edges e ′′ in the case 1, whic h are fro m L ( F ) and are adjacen t to edges from B ( G, F ) (figure 6b). All these together imply D ⊆ P ( L ( F ) , B ( G , F )). Moreo v er, the construction of D implies that the edges from E 1 ( D ) b elong to L ( F ). Therefore, D ⊆ T L ( F ) o ( L ( F ) , B ( G, F )). Lemma 5.2.18 V 0 ( A ′ ) ⊆ Q L ∪ Q B . Pro of. Let v b e an end-vertex of a trail P ′ from A ′ . If P ′ ∈ Z Q U then, due to statemen t 2 of corollar y 5.2.10 and the definition o f Z Q U , v ∈ Q \ Q U = Q L ∪ Q B . Therefore, assume that P ′ ∈ D , and let P b e the path from M P ( S, H ) corresp onding t o P ′ . Without loss of generalit y , w e may a ssume that v is t he starting vertex of P ′ . Let e, e ′ , e ′′ b e as it is sho wn in t he corresp o nding figure describing eac h case of the transformat io n of P to P ′ . Assume that the tr ansformation of E b 2 ( P ) is handled by cases 1 or 2 (fig ures 6b and figure 7b). Lemma 5.2.12 implies tha t e ∈ T U ⊆ S ′ \ H ′ . Thu s, e ′′ ∈ H ′ \ S ′ and e ′ ∈ H ′ \ S ′ for cases 1 and 2, resp ectiv ely . Therefore, as v is a 3- v ertex (not a 1-vertex ), v ∈ R 1 \ R 2 ⊆ R = Q (statemen t (2) of corollary 5.2.9). Moreo v er, v / ∈ Q U as Q U con tains only 1-v ertices and 2-v ertices. Hence, v ∈ Q \ Q U = Q L ∪ Q B . No w assume that the transformatio n of E b 2 ( P ) is handled b y case 3 (figure 8b). As e ∈ S, e ′ ∈ H , w e ha v e e ∈ H ′ \ S ′ . Since v is a 2- v ertex (not a 1- vertex ), v ∈ R 1 \ R 2 ⊆ R = Q (statemen t (2) o f corollary 5 .2.9). On the other hand, due to lemma 5 .2.12, e / ∈ T U as e ∈ S . Therefore, v / ∈ Q U . Thu s, v ∈ Q \ Q U = Q L ∪ Q B . Lemma 5.2.19 Q L ∪ Q B ⊆ V 0 ( A ′ ) . 25 Pro of. Let v ∈ Q L ∪ Q B = Q \ Q U . The condition (a) of the theorem implies that v is not a 1-v ertex. Therefore, a s Q = R (statemen t (2) o f the corollary 5.2.9), v ∈ R 1 \ R 2 , hence there is an edge e 0 ∈ H ′ \ S ′ suc h that v and e 0 are inciden t. The follow ing cases ar e p ossible: (1) e 0 ∈ S . Then e ∈ S \ H and e 0 ∈ E 2 ( P ), where P is a path fro m M P ( S, H ) (lemma 5.2.13). Without loss of generality we ma y assume that e 0 ∈ E b 2 ( P ). Let P ′ b e t he trail fr o m D corr esp o nding to P . Consider the f o llo wing tw o sub cases: (a) e 0 ∈ U ( F ). Then e 0 is the starting edge of P (statemen t ( 1 ) of corollary 5.2.3). Clearly , the tra nsformation of E b 2 ( P ) is handled by case 3 (see figure 7b, e 0 corre- sp onds to e in the figure). Hence v is the starting v ertex of P ′ , since v is not the 1-v ertex inciden t to e 0 . (b) e 0 / ∈ U ( F ). Hence e 0 ∈ L ( F ) as e 0 ∈ H ′ (prop erty 4 .5). Due t o statemen t (1) of corollary 5 .2.3, e 0 ∈ E b 2 ( P ) \ E b 1 ( P ). Th us, the transformation of E b 2 ( P ) is handled b y case 2. Let e, e ′ b e a s in figure 7b, and note tha t e 0 corresp onds t o e ′ . Lemma 5.2.12 implies t hat e ∈ T U , hence the 2-v ertex inciden t to e ′ = e 0 b elongs to Q U . Since v / ∈ Q U , v is the 3- v ertex inciden t to e ′ = e 0 , a nd therefore, is the starting v ertex of P ′ . (2) e 0 / ∈ S . By the definitions of Z Q U and Z Q U , e 0 ∈ Z Q U ∪ Z Q U . If e 0 ∈ Z Q U , then w e are done. Therefore, assume t ha t e 0 ∈ Z Q U . The definition of Z Q U implies that there is an edge e 1 ∈ T U adjacen t t o e 0 . Due to lemma 5.2.12, e 1 ∈ ( H \ S ) ∩ U ( F ), and therefore, e 1 is an end-edge of some path P ∈ M P ( S, H ) (prop erty 3.3 and corollar y 5 .2.3). Without loss of generalit y , w e ma y a ssume t hat e 1 ∈ E b 1 ( P ). Note that E b 2 ( P ) \ E b 1 ( P ) 6 = ∅ as an y path from M P ( S, H ) has length at least three (corollary 5.2.5 and statemen t s (1) and (2) of corollary 5.2.11). Th us, let e 2 ∈ E b 2 ( P ) \ E b 1 ( P ). As e 1 ∈ H , e 2 ∈ S . Let P ′ b e the trail from D corresp o nding to P . Since e 0 ∈ L ( F ) (see statemen t (1) of corollary 5.2.10 and the definition of Z Q U ) and e 2 6 = e 0 , we ha v e e 2 / ∈ L ( F ) a nd, due to assumption 1, e 2 ∈ B ( G, F ). Hence, the transformatio n of E b 2 ( P ) is handled b y case 1 , and the edges e ′′ , e, e ′ in figure 6b corresp ond to e 0 , e 1 , e 2 , resp ectiv ely . As e = e 1 ∈ T U , the 2-v ertex inciden t to e ′′ = e 0 b elongs to Q U . Th erefore, v is the 3- vertex inciden t to e ′′ = e 0 . Thu s, v is the starting v ertex of P ′ . F rom lammas 5.2.18 and 5.2.19 w e g et the following corollary: Corollary 5.2.20 V 0 ( A ′ ) = Q L ∪ Q B . Let us prov e one auxiliary lemma, whic h is used at t he end of the pro of of Sufficiency . 26 Lemma 5.2.21 The differ enc e of the numb ers of 2 -vertic es and 3 -vertic es in Q B is 2( | M P S o ( S, H ) | − | M P H o ( S, H ) | ) . Pro of. Let us denote the difference of the num b ers of 2-vertice s and 3-vertice s in a set of v ertices N b y D 23 ( N ). As D 23 ( Q L ) = 0 (any edge from L ( F ) is incident to one 2-vertex and one 3-vertex ), D 23 ( Q B ) = D 23 ( Q L ∪ Q B ), and due to corollar y 5.2 .2 0, D 23 ( Q B ) = D 23 ( V 0 ( A ′ )). Note that fo r any trail P ′ ∈ D and its corresp onding path P ∈ M P ( S, H ) • P ′ starts with a 3 - v ertex if and o nly if P starts with an edge from H (cases 1 and 2), • P ′ starts with a 2 - v ertex if and o nly if P starts with an edge from S (case 3). Th us, there are 2 | M P H o ( S, H ) | + | M P e ( S, H ) | 3-v ertices and 2 | M P S o ( S, H ) | + | M P e ( S, H ) | 2-v ertices in V 0 ( D ). As Z Q U ⊆ L ( F ) (statemen t (1) of corollary 5.2.10), D 23 ( Z Q U ) = 0. Therefore, D 23 ( Q B ) = D 23 ( V 0 ( A ′ )) = D 23 ( Z Q U ) + D 23 ( D ) = D 23 ( D ) = 2 | M P S o ( S, H ) | + | M P e ( S, H ) |− − (2 | M P H o ( S, H ) | + | M P e ( S, H ) | ) = 2( | M P S o ( S, H ) | − | M P H o ( S, H ) | ) . Note that lemmas 5 .2.16 and 5.2 .17, and corollary 5.2.20 are the main prop erties of A ′ that we re men tioned ab ov e while describing the idea of the pro of, and a re used f urther. No w, w e a re going to construct a set of tra ils A ′′ men tio ned a b ov e, whic h is the k ey p oint of our pro of. First, let us prov e tw o lemmas necessary for the construction of A ′′ . Lemma 5.2.22 T L ⊆ E 1 ( D ) . Pro of. First let us pro of that T L ⊆ E 1 ( A ′ ). suppo se for contradiction that there exists an edge e = ( u, v ) ∈ T L , such that e is not an end-edge of an y trail fro m A ′ . Sinc e the set of end-v ertices of all trails from A ′ is Q L ∪ Q B (corollary 5.2.20), there are tr a ils T 1 , T 2 ∈ A ′ suc h that u and v are the end-ve rtices of T 1 and T 2 , resp ectiv ely ( T 1 and T 2 ma y coincide). As A ′ ⊆ T L ( F ) o ( L ( F ) , B ( G, F )), e is adjacen t to tw o edges from L ( F ), whic h is imp o ssible b ecause e itself b elongs to L ( F ). Th us, e is an end-edge of some trail T f rom A ′ . Since Z Q U ⊆ H ′ \ ( S ∪ S ′ ) and e ∈ T L ⊆ S ′ \ H ′ , implies T ∈ D . Lemma 5.2.23 No e dge lies on two differ ent tr a ils fr om A ′ . 27 Pro of. Clearly , the statemen t o f the lemma is true for the following set of paths: M P ( S, H ) ∪ Z Q U . Let us prov e that when M P ( S, H ) is tra nsfor med to D it still remains t rue. During the transformat io n of M P ( S, H ) to D , the only edges w e add a re edges e ′′ in the case 1, whic h b elong to H ′ \ S (figure 6). Clearly , eac h suc h edge e ′′ is attac hed to only one end of one path from M P ( S, H ), thus lies on only one tr a il from D . F urthermore, due to lemma 5.2.12 the edge e sho wn in the fig ure 6 b elongs to T U . Th us e ′′ do es not b elong to S ′ , i.e. e ′′ ∈ H ′ \ ( S ∪ S ′ ). Moreov er, e ′′ ∈ Z Q U as the 2- v ertex inciden t to e ′′ b elongs to Q U since is inciden t to e ∈ T U . Thu s, e ′′ / ∈ Z Q U , and the pro of of the lemma is complete. Let us introduce an op eration called J O I N , using whic h w e “get rid of” edges fr om T L “preserving” the main prop erties obtained for A ′ . Assume that l = ( u, v ) ∈ T L , and T 1 : l 1 b 1 l 2 ...l n b n l and T 2 : l ′ 1 b ′ 1 l ′ 2 ...l ′ m b ′ m l ′ m +1 are tra ils from T L ( F ) o ( L ( F ) , B ( G, F )) suc h that E ( T 1 ) ∩ E ( T 2 ) = ∅ , v is the last ve rtex of T 1 inciden t to l and u is the starting vertex o f T 2 inciden t to l ′ 1 . In this case we sa y tha t ( T 1 , T 2 ) is a T L -adjacen t pair of trails corresponding to l (figure 9a). Figure 9: If ( T 1 , T 2 ) is a T L -adjacen t pair ( T 1 , T 2 ∈ T L ( F ) o ( L ( F ) , B ( G, F ))) suc h that T 1 and T 2 do not coincide then the J O I N o f ( T 1 , T 2 ) is a trail defined as follow s (figure 9b): T = l 1 b 1 l 2 ...l n b n l ′ 1 b ′ 1 ...l ′ m b ′ m l ′ m +1 . As T 1 and T 2 do not ha v e common edges, this definition is correct, i.e. T is really a trail. Moreo v er, T is not a cycle, as T 1 and T 2 do not coincide ( T 1 is not the rev erse of T 2 ). Thus , T ∈ T L ( F ) o ( L ( F ) , B ( G, F )). No w assume tha t W ⊆ T L ( F ) o ( L ( F ) , B ( G, F )) suc h that no tw o trails in it share a common edge. W e sa y that t he set W ′ is a T L -reduction of W if W ′ is obtained from W by removin g arbitrarily c hosen T L -adjacen t trails T 1 , T 2 ∈ W corresp o nding to an edge l = ( u , v ) ∈ T L and, if T 1 and T 2 do not coincide, adding their J O I N . The following lemma is obviously true f or W ′ : 28 Lemma 5.2.24 (1) W ′ ⊆ T L ( F ) o ( L ( F ) , B ( G, F )) ; (2) no two tr a i l s in W ′ shar e c ommon e dg e; (3) no e dge is adde d to W ′ , formal ly: [ T ′ ∈ W ′ E ( T ′ ) ⊆ [ T ∈ W E ( T ) ; (4) E 1 ( W ′ ) = E 1 ( W ) \{ l } and V 0 ( W ′ ) = V 0 ( W ) \{ u, v } . Due to lemma 5.2.22, fo r each edge from T L there is a trail T 1 : l 1 b 1 l 2 ...l n b n l , T 1 ∈ A ′ . Let l = ( u, v ) and v is the last v ertex of T 1 . Similarly , as u ∈ Q L , there is a trail T 2 : l ′ 1 b ′ 1 l ′ 2 ...l ′ m b ′ m l m +1 , T 2 ∈ A ′ , suc h that u is the starting v ertex of T 2 and is inciden t to l ′ 1 (corollary 5.2.20). Thus , ( T 1 , T 2 ) is a T L -adjacen t pair corresp onding to l , fo r eac h l ∈ T L . As no t w o trails from A ′ share a common edge (lemma 5.2.23), A ′ is applicable for T L -reduction op eration. Consider a sequence of sets of trails A ′ = A 0 , A 1 , ..., A k , ... , suc h that A i is a T L - reduction of A i − 1 , for eac h i = 1 , 2 , ...k . Note that k ≤ | T L | as T L -reduction op eration can b e applied not more than | T L | times. F urthermore, consider suc h a sequence A ′ = A 0 , A 1 , ..., A k ha ving maxim um length. Due to lemma 5.2.22 and statemen t (4 ) of lemma 5.2.24, k = | T L | . Set A ′′ = A k . Lemma 5.2.25 (1) A ′′ ⊆ T L ( F ) o ( L ( F ) , B ( G, F )) ; (2) I f an e dge e lies on a tr ail fr om A ′′ and e ∈ B ( G, F ) ∪ S then e ∈ B ( G, F ) ∩ S ; (3) V 0 ( A ′′ ) = Q B . Pro of. Statemen t (1 ) immediately follo ws from stat emen t (1) of lemma 5.2.24. Statemen t (2) is implied from lemma 5.2.1 6 and statemen t ( 3) of lemma 5.2 .24. As k = | T L | , all edges from T L are “remo v e” from trails of A ′ , i.e. no edge from T L lies on a trail from A ′′ . Th us no end-v ertex of a trail from A ′′ b elongs to Q L . As V 0 ( A ′ ) = Q L ∪ Q B (lemma 5.2.20), we get statemen t (3). The follo wing is the main lemma of the Sufficiency part. Lemma 5.2.26 A ny e dge f r om T B lies on a cycle C ∈ C e ( L ( F ) , B ( G, F )) such that C B is bip artite. 29 Figure 10: Pro of. Due to statement (3) o f lemma 5.2.25, each v ertex of Q B is an end-vertex of a path from A ′′ . On the ot her hand, it is inciden t to an edge from T B . F urthermore, ev ery end- v ertex of an y pa th from A ′′ b elongs to Q B . Hence, ev ery edge o f T B lies on a ( not ne c es s arily simple ) cycle C : t 1 , T 1 , ..., t r , T r ( r ≥ 1), where t i ∈ T B ⊆ S ′ , T i ∈ A ′′ , i = 1 , 2 , ..., r (figure 10). Due to statemen t (1) of lemma 5.2.25 C is an L ( F )- B ( G, F ) alternating cycle, i.e. C ∈ C e ( L ( F ) , B ( G, F )). The construction of C and statemen t (2) of lemma 5.2 .25 imply t hat E ( C B ( G,F ) ) ⊆ S ∪ S ′ since t i ∈ S ′ and E (( T i ) B ( G,F ) ) ⊆ S , i = 1 , 2 , ..., r . Therefore, the g r a ph C B ( G,F ) cannot con tain an o dd cycle, i.e. is bipartite. Let us note that all lemmas and corollaries ab o v e in this subsection are pro v ed on the assumption that G is an S -gr a ph with spanning S -forest F satisfying the conditions (a ) of the theorem and the assumption 1 holds. The follo wing lemma is prov ed without any of the assumptions made ab o v e. Lemma 5.2.27 If G ′ is an S -gr aph with sp anning S -for est F ′ satisfying the c ondition (b) of the the or em then ther e is a p air ( S, S ′ ) ∈ M 2 ( G ′ ) such that ( S ∪ S ′ ) ∩ ∆( G ′ , F ′ ) = ∅ . Pro of. Let ( S, S ′ ) b e a pair from M 2 ( G ′ ) suc h that | ( S ∪ S ′ ) ∩ ∆( G ′ , F ′ ) | is minimum. Assume for contradiction that | ( S ∪ S ′ ) ∩ ∆( G ′ , F ′ ) | 6 = 0. Let e = ( u, v ) ∈ ( S ∪ S ′ ) ∩ ∆( G, F ), where u is the 1-v ertex inciden t to e and v is its base. Let w b e t he 2-v ertex adja cent to 30 u and v . The condition (b) of the theorem implies tha t d G ′ \ F ′ ( w ) = 0. Without loss o f generalit y , w e ma y assume that e ∈ S . Let e ′ b e one of the tw o edges inciden t to w , whic h do es not b elong to S ′ . Define a matc hing S 1 as follows : S 1 . = ( S \{ e } ) ∪ { e ′ } . Note that ( S 1 , S ′ ) ∈ M 2 ( G ′ ) and | ( S 1 ∪ S ′ ) ∩ ∆( G ′ , F ′ ) | < | ( S ∪ S ′ ) ∩ ∆( G ′ , F ′ ) | , which con tradicts t he choice of ( S, S ′ ). No w assume that G is an S -graph with spanning S -forest F satisfying the conditions (a) and (b) o f the theorem (do not assume that the assumption 1 ho lds). Assume also tha t β ( G ) α ( G ) 6 = 5 4 , and therefore M 2 ( G ) ∩ M 2 ( F ) = ∅ (pro p ert y 4.4). Th us, M 2 ( G ) \ M 2 ( F ) = M 2 ( G ). This, to gether with lemma 5.2.2 7, implies that assumption 1 holds for G . Th us, all of the lemmas and corollar ies pro ved in this section a re t r ue fo r G . Lemma 5.2.28 | M P S o ( S, H ) | > | M P H o ( S, H ) | . Pro of. Coro llary 3.1 2 and prop ert y 4.4 imply that β ( G ) α ( G ) < 5 4 = β ( F ) α ( F ) . As β ( G ) = β ( F ) (prop erty 4.7), α ( G ) > α ( F ), which means tha t | S | > | H | . Th us, due to prop ert y 3.5, w e get: | M P S o ( S, H ) | > | M P H o ( S, H ) | . Lemmas 5.2.21 and 5.2 .28 to gether imply Corollary 5.2.29 Ther e is at le ast one 2 - 2 e dge in T B . Let e b e one of 2- 2 edges fro m T B . Due to lemma 5.2.2 6, e lies on a cycle C ∈ C e ( L ( F ) , B ( G, F )) such that C B is bipartite. Hence, Corollary 5.2.30 The gr aph G do es not satisfy the c ond ition (c) of the the or em. Let us not that corollary 5.2.30 is pro v en on the assumption that G is an S -graph with spanning S -forest F satisfying the conditions (a) and (b) of the theorem, a nd β ( G ) α ( G ) 6 = 5 4 . Clearly , this is equiv a len t to the follo wing: Statemen t 2 If G is an S -gr aph with sp anning S -fo r est F s atisfying the c onditions (a), (b) and (c) of the the or em then β ( G ) α ( G ) = 5 4 . 31 5.3 Remarks Remark 1 Statemen ts (1) and (2) imply that the theorem can b e r efo r m ulated as follows : F or a gr a ph G the e quality β ( G ) α ( G ) = 5 4 holds, if and on l y if G is an S -gr aph, any sp anning S -for e s t of which satisfi e s the c onditions (a), (b) and (c) . Remark 2 Due to prop ert y 4.8, the condition (c) of the t heorem can b e changed to the follo wing: F or every L ( F ) - B ( G, F ) alternating even cycle C of G c ontaining a 2 - 2 or 3 - 3 e dge, the gr aph C B ( G,F ) is not bip artite. Ac kno wledgemen ts I w ould lik e to express m y sincere gratitude to m y sup ervisor D r. V ahan Mkrtch y an, for stating the problem and g iving an o pp ortunity to p erform a researc h in suc h an interes ting area as Match ing theory is, for teach ing me ho w to write art icles and helping me to cope with the curren t one, for directing and correcting m y chaotic ideas, for encouraging me, and finally , fo r his pa tience. During the researc h pro cess I also collab ora t ed with m y friend and colleague V ahe Mu- so y an, who I wan t to thank for his great in v estmen t in this w ork, for helping me to dev elop the main idea of the theorem, and for numerous of counte r-examples that corrected m y statemen ts as w ell as rejected some of my h yp otheses. 32 References [1] E. Balas, In teger and F ractional Matc hings, Studies on Graphs and Discrete Progra m- ming, Ed.: P . Ha nsen, Ann. Discrete Math., 11 , Nort h- Holland, Amsterdam, 198 1, pp.1- 13. [2] F. Harary , Graph Theory , Addison-W esley , Reading, MA, 1969. [3] F. Hara ry , M. D. 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