On upper bounds for parameters related to construction of special maximum matchings

On upp er b ounds for param eters related to construction of sp ecial maximum matchings Artur Kho jabagh y an a ∗ and V ahan V. Mkrtc h y a n a b † a Departmen t of Informatics a nd Applied Mathematics, Y erev an State Univ ersit y , Y erev an, 0025 , Armenia b Institute for Informat ics a nd Automation Problems, National Academ y of Scien ces of R epublic of Armenia, 00 1 4, Arme nia F or a graph G let L ( G ) and l ( G ) denote the siz e of the largest and smalles t maxim um matc hing of a graph obtained from G b y remov ing a maxim um matc hing of G . W e sho w that L ( G ) ≤ 2 l ( G ) , and L ( G ) ≤ 3 2 l ( G ) pro vided that G con tains a p erfect matching. W e also c haracterize the class of graphs for whic h L ( G ) = 2 l ( G ). Our c haracterization implies the existence of a p olynomial a lgorithm for testing the prop erty L ( G ) = 2 l ( G ). Finally w e s ho w that it is N P -complete to test whether a graph G con taining a perf ect matc hing satisfies L ( G ) = 3 2 l ( G ) . 1. In tro duction In the paper graphs are assume d to b e finite, undirected, without lo ops o r mu ltiple edges. Let V ( G ) and E ( G ) denote the sets o f v ertices a nd edges of a gra ph G , respectiv ely . If v ∈ V ( G ) and e ∈ E ( G ), then e is said to co v er v if e is inciden t to v . F or V ′ ⊆ V ( G ) and E ′ ⊆ E ( G ) let G \ V ′ and G \ E ′ denote the g r a phs obtained from G b y remo ving V ′ and E ′ , respectiv ely . Mor eov er, let V ( E ′ ) denote the set of v ertices of G that are co v ered b y an edge from E ′ . A subgraph H of G is said to b e spanning fo r G , if V ( E ( H )) = V ( G ). The length of a path (cycle) is the n umber of its edges. A k -path ( k -cycle) is a path (cycle) of length k . A 3-cycle is called a triangle. A set V ′ ⊆ V ( G ) ( E ′ ⊆ E ( G ) ) is said to b e indep enden t, if V ′ ( E ′ ) con tains no adjacen t v ertices (edges). An indep enden t set of edges is called matching. A matc hing of G is called p erfect, if it co v ers all v ertices of G . Let ν ( G ) denote the cardinalit y of a la r gest matc hing of G . A matchin g of G is maxim um, if it contains ν ( G ) edges. F or a p ositiv e in teger k and a match ing M of G , a (2 k − 1)-path P is called M - augmen ting, if the 2 nd , 4 th , 6 th ,..., (2 k − 2) th edges o f P b elong to M , while the endv ertices of P are not co v ered b y an edge of M . The f ollo wing theorem of Berg e giv es a sufficien t and necess ary condition for a matchin g to b e maxim um: Theorem 1 (Ber ge [ 2]) A matching M of G is max imum, if G c ontains no M -augmen ting p ath. ∗ email: arturkho jaba gh yan@gmail.co m † email: v ahanmkr tc hy a n2002@ { y su.am, ipia.sc i.am, yaho o.com } 1 2 Artur Kho jabaghy an, V ahan V. Mkrtc hy a n F or t wo matc hings M a nd M ′ of G consider t he subgraph H of G , where V ( H ) = V ( M △ M ′ ) and E ( H ) = M △ M ′ . The connected components of H are called M △ M ′ - alternating comp onen t s. Note that M △ M ′ alternating comp onen ts are alw ay s paths or cycles of ev en length. F or a graph G define: L ( G ) ≡ max { ν ( G \ F ) : F is a maxim um matc hing of G } , l ( G ) ≡ min { ν ( G \ F ) : F is a maximum matc hing of G } . It is kno wn that L ( G ) a nd l ( G ) are N P -hard calculable ev en for connected bipartite graphs G with maxim um de gree three [ 4], though there are p olynomial algorithms whic h construct a max im um matc hing F of a tree G suc h that ν ( G \ F ) = L ( G ) and ν ( G \ F ) = l ( G ) (to b e presen ted in [ 5]). In the same pap er [ 5] it is sho wn that L ( G ) ≤ 2 l ( G ) . In the presen t pap er w e r e- pro ve this equalit y , and also show that L ( G ) ≤ 3 2 l ( G ) provided that G con tains a perfect matc hing. A naturally arising question is the c haracterization o f graphs G with L ( G ) = 2 l ( G ) a nd the graphs G with a p erfect matching that satisfy L ( G ) = 3 2 l ( G ) . In this pap er w e solv e these problems b y giving a c haracterization of graphs G with L ( G ) = 2 l ( G ) that implies the existence of a p olynomial algorit hm for testing this prop erty , a nd b y s ho wing that it is N P -complete to test whether a bridgeless cubic g r aph G satisfies L ( G ) = 3 2 l ( G ) . Recall that b y P etersen theorem an y bridgeless cubic graph con ta ins a perfect matc hing (see, for example, theorem 3.4.1 of [ 6]). T erms and concepts that w e do not define can b e found in [ 1, 2, 6, 8]. 2. Some auxiliarly results W e w ill need the follow ing : Theorem 2 L et G b e a gr aph. Then: (a) for any two maximum matchings F , F ′ of G , we have ν ( G \ F ′ ) ≤ 2 ν ( G \ F ) ; (b) L ( G ) ≤ 2 l ( G ) ; (c) If L ( G ) = 2 l ( G ) , F L , F l ar e two maximum matching s of the gr aph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) , an d H L is any maximum matching of the gr aph G \ F L , then: (c1) F l \ F L ⊂ H L ; (c2) H L \ F l is a maximum matching of G \ F l ; (c3) F L \ F l is a maximum matching of G \ F l ; (d) if G c ontains a p erfe ct matching, then L ( G ) ≤ 3 2 l ( G ) . Pro of. (a)Let H ′ b e an y ma ximum matc hing in the graph G \ F ′ . Then: ν ( G \ F ′ ) = | H ′ | = | H ′ ∩ F | + | H ′ \ F | ≤ | F \ F ′ | + ν ( G \ F ) = | F ′ \ F | + ν ( G \ F ) ≤ 2 ν ( G \ F ) . (b) follows from (a). On upp er b ounds fo r parameters related to construction of special maxim um matc hing s 3 (c) Consider the pro of of (a) and tak e F ′ = F L , H ′ = H L and F = F l . Since L ( G ) = 2 l ( G ), w e mus t hav e equalities throughout, thus prop erties (c1)-(c3) should b e true. (d) L et F L , F l b e t wo p erfect matc hings of the gra ph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) , a nd assu me H L to b e a maxim um matching of the graph G \ F L . Define: X = { e = ( u , v ) ∈ F L : u and v are inciden t to an edge from H L ∩ F l } , x = | X | , k = | H L ∩ F l | ; Clearly , ( H L \ F l ) ∪ X is a matc hing of the graph G \ F l , therefore, taking into accoun t that ( H L \ F l ) ∩ X = ∅ , w e deduce l ( G ) = ν ( G \ F l ) ≥ | H L \ F l | + | X | = | H L | − | H L ∩ F l | + | X | = L ( G ) − k + x. Since F L is a p erfect matc hing, it co v ers the set V ( H L ∩ F l ) \ V ( X ), whic h con tains | V ( H L ∩ F l ) \ V ( X ) | = 2 | ( H L ∩ F l ) | − 2 | X | = 2 k − 2 x v ertices. D efine the set E F L as follo ws: E F L = { e ∈ F L : e co v ers a v ertex from V ( H L ∩ F l ) \ V ( X ) } . Clearly , E F L is a matc hing of G \ F l , to o, and therefore l ( G ) = ν ( G \ F l ) ≥ | E F L | = 2 k − 2 x. Let us sho w that max { L ( G ) − k + x, 2 k − 2 x } ≥ 2 L ( G ) 3 . Note that if x ≥ k − L ( G ) 3 then L ( G ) − k + x ≥ L ( G ) − k + k − L ( G ) 3 = 2 L ( G ) 3 ; if x ≤ k − L ( G ) 3 then 2 k − 2 x ≥ 2 L ( G ) 3 , th us in both cases we ha ve l ( G ) ≥ 2 L ( G ) 3 or L ( G ) l ( G ) ≤ 3 2 . The pro of of the theorem 2 is completed .  Lemma 1 (L em ma 2.20, 2.41 of [ 5]) L et G b e a gr ap h, and ass ume that u and v ar e vertic es of de gr e e one sharing a neigh b our w ∈ V ( G ) . Then: L ( G ) = L ( G \{ u, v , w } ) + 1 , l ( G ) = l ( G \{ u, v , w } ) + 1 . Pro of. The pro ofs of thes e t w o equalities are similar, thus w e will stop only on the proof of the first one. Our proof is base d on the ideas of [ 5]. First of all, observ e that ν ( G ) = ν ( G \{ u, v , w } ) + 1. Let us show that L ( G ) ≥ L ( G \{ u, v , w } )+1 . T ake any maxim um matc hing F of G \{ u, v , w } with ν (( G \{ u, v , w } ) \ F ) = L ( G \{ u, v , w } ), and let H b e any maxim um matc hing of ( G \{ u, v , w } ) \ F . Define: F ′ = F ∪ { ( u, w ) } , H ′ = H ∪ { ( v , w ) } . 4 Artur Kho jabaghy an, V ahan V. Mkrtc hy a n Observ e that F ′ is a maxim um matc hing o f G , and H ′ is a matc hing of G \ F ′ . Th us, L ( G ) ≥ ν ( G \ F ′ ) ≥ | H ′ | = 1 + | H | = 1 + ν (( G \{ u, v , w } ) \ F ) = 1 + L ( G \{ u, v , w } ) . T o complete the pro of of the first equalit y , it suffices to show that L ( G ) ≤ L ( G \{ u, v , w } )+ 1. First of all, let us sho w that there is a maximum matc hing F ′ of G with ν ( G \ F ′ ) = L ( G ), suc h that F ′ con tains one of the edges ( u, w ) and ( v , w ). T ake an y maximum matc hing F ′ of G with ν ( G \ F ′ ) = L ( G ), and assume that F ′ ∩ { ( u, w ) , ( v , w ) } = ∅ . Morev er, let H ′ b e a maxim um ma t ching of G \ F ′ . Since F ′ is a maxim um matc hing of G , F ′ m ust con tain an edge ( w, z ), where z 6 = u, v . Define: F ′′ =  ( F ′ \{ ( w , z ) } ) ∪ { ( w , u ) } , if ( w , u ) / ∈ H ; ( F ′ \{ ( w , z ) } ) ∪ { ( w , v ) } , if ( w , u ) ∈ H. Observ e that F ′′ is a maxim um matc hing o f G , and H ′ is a matc hing of G \ F ′′ . Th us, ν ( G \ F ′′ ) ≥ | H ′ | = ν ( G \ F ′ ) = L ( G ) . The last inequality implies that ν ( G \ F ′′ ) = L ( G ). Moreo ve r , F ′′ ∩ { ( u, w ) , ( v , w ) } 6 = ∅ . Th us, initially w e can assume that F ′ is a maxim um matc hing of G with ν ( G \ F ′ ) = L ( G ), suc h tha t F ′ con tains one of the edges ( u, w ) and ( v , w ). Without loss of generalit y , w e can also assum e that this edge is ( u, w ). No w, w e claim that there is a maxim um matc hing H ′ of G \ F ′ that con tains the edge ( v , w ). T ake any maxim um matc hing H ′ of G \ F ′ and supp ose that ( v , w ) / ∈ H ′ . Since H ′ is a maxim um matc hing of G \ F ′ , there m ust ex ist an edge ( w , z ) ∈ H ′ , where z 6 = u, v . Define: H ′′ = ( H ′ \{ ( w , z ) } ) ∪ { ( w , v ) } . Observ e that H ′′ is a maxim um matc hing of G \ F ′ , sinc e | H ′′ | = | H ′ | = ν ( G \ F ′ ) = L ( G ). Moreo ve r , it contains the edge ( w , v ). Th us, initially w e can assume tha t H ′ is a maxim um matc hing of G \ F ′ that contains the edge ( v , w ). W e are ready to sho w that L ( G ) ≤ L ( G \{ u, v , w } ) + 1. Since ν ( G ) = ν ( G \{ u, v , w } ) + 1, w e hav e that F ′ \{ ( u, w ) } is a maxim um matc hing of G \{ u, v , w } . T aking in to accoun t that H ′ \{ ( v , w ) } is a matc hing of ( G \{ u, v , w } ) \ ( F ′ \{ ( u, w ) } ), w e deduce: L ( G ) = ν ( G \ F ′ ) = | H ′ | = 1 + | H ′ \{ ( v , w ) }| ≤ 1 + ν (( G \{ u , v , w } ) \ ( F ′ \{ ( u, w ) } )) ≤ 1 + L ( G \{ u , v , w } ) .  Corollary 1 L et G b e a gr aph with L ( G ) = 2 l ( G ) . Then ther e ar e no vertic es u, v o f de gr e e one, that ar e adjac ent to the same vertex w . Pro of. Suppo se not. Then lemma 1 and (b) of theorem 2 imply L ( G ) = 1 + L ( G − { u, v , w } ) ≤ 1 + 2 l ( G − { u, v , w } ) = 1 + 2( l ( G ) − 1) < 2 l ( G ) a contradiction.  On upp er b ounds fo r parameters related to construction of special maxim um matc hing s 5 3. Characterization of graphs G satisfying L ( G ) = 2 l ( G ) Let T b e the set o f a ll triangles of G that con tain at least t w o vertice s of degree t w o. Note that an y v ertex of degree tw o lies in at most o ne triangle from T . F r o m each triangle t ∈ T c ho ose a v ertex v t of degree tw o, and define V 1 ( G ) as follows : V 1 ( G ) = { v : d G ( v ) = 1 } ∪ { v t : t ∈ T } Theorem 3 L et G b e a c onne cte d gr aph with | V ( G ) | ≥ 3 . Then L ( G ) = 2 l ( G ) if and only if (1) G \ V 1 ( G ) is a bip artite gr aph with a bip artition ( X, Y ) ; (2) | V 1 ( G ) | = | Y | and any y ∈ Y has exactly one neighb our in V 1 ( G ) ; (3) t he gr aph G \ V 1 ( G ) c ontains | X | vertex disjoint 2 -p aths. Pro of. Suffic iency . Let G b e a connected graph with | V ( G ) | ≥ 3 satisfying the conditions (1)-(3). Let us show that L ( G ) = 2 l ( G ). F or each vertex v with d ( v ) = 1 tak e the edge inciden t to it and define F 1 as the unio n of all these edges. F or eac h v ertex v t ∈ V 1 ( G ) tak e t he edge that connects v t to a v ertex of degree t wo, and define F 2 as the union o f all those edges. Set: F = F 1 ∪ F 2 . Note that F is a matching with | F | = | V 1 ( G ) | = | Y | . Moreo v er, since G is bipartite and | V 1 ( G ) | = | Y | , the definitions of F 1 and F 2 imply that there is no F -augmen ting path in G . Th us, b y Berge theorem, F is a maxim um matchin g of G , and ν ( G ) = | F | = | V 1 ( G ) | = | Y | . Observ e tha t t he graph G \ F is a bipartite graph with ν ( G \ F ) ≤ | X | , th us l ( G ) ≤ ν ( G \ F ) ≤ | X | . No w, consider the | X | v ertex disjoin t 2-paths o f the graph G \ V 1 ( G ) guaran teed by (3). (2) implies that thes e 2-paths to g ether w ith the | F | = | V 1 ( G ) | = | Y | edges of F fo rm | X | v ertex disjoint 4-paths of the g r aph G . Consider matc hings M 1 and M 2 of G obtained from these 4-paths b y a dding the first and the third, the second and the fourth edges of these 4-pat hs to M 1 and M 2 , r esp ectiv ely . Define: F ′ = ( F \ M 2 ) ∪ ( M 1 \ F ) . Note that F ′ is a matc hing of G and | F ′ | = | F | , thus F ′ is a maxim um matc hing of G . Since F ′ ∩ M 2 = ∅ , w e ha v e L ( G ) ≥ ν ( G \ F ′ ) ≥ | M 2 | = 2 | X | ≥ 2 l ( G ) . (b) of theorem 2 implies that L ( G ) = 2 l ( G ). Necessit y . Now , a ssum e that G is a connected graph with | V ( G ) | ≥ 3 and L ( G ) = 2 l ( G ). By pro ving a series of claims, w e sho w that G \ V 1 ( G ) satisfies the conditions (1)- (3) of the theorem. 6 Artur Kho jabaghy an, V ahan V. Mkrtc hy a n Claim 1 F or a ny maximum matchings F L , F l of the gr aph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) , F L ∪ F l induc es a sp anning sub gr aph, that is V ( F L ) ∪ V ( F l ) = V ( G ) . Pro of. Suppo se t ha t there is a v ertex v ∈ V ( G ) that is co v ered neither b y F L nor b y F l . Since F L and F l are maximum matc hing s of G , fo r eac h edge e = ( u, v ) the v ertex u is inciden t to an edge from F L and to an edge from F l . Case 1: there is a n edge e = ( u, v ) such that u is inciden t to an edge from F L ∩ F l . Note that { e } ∪ ( F L \ F l ) is a matc hing of G \ F l whic h con tradicts (c3) of the theorem 2 . Case 2 : for eac h edge e = ( u, v ) u is inciden t to an edge f L ∈ F L \ F l and to an edge f l ∈ F l \ F L . Let H L b e any maxim um matc hing of G \ F L . Due to (c1) of theorem 2 f l ∈ H L . D efine: H ′ L = ( H L \{ f l } ) ∪ { e } . Note that H ′ L is a maxim um matching of G \ F L suc h that F l \ F L is not a subset of H ′ L con tradicting (c1) of theorem 2.  Claim 2 F or a ny maximum matchings F L , F l of the gr aph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) , the alternating c omp onents F L △ F l ar e 2 -p aths. Pro of. It suffice s to sho w that there is no edge f L ∈ F L that is adjacen t to tw o edges from F l . Supp ose that some edge f L ∈ F L is adjacent to e dges f ′ l and f ′′ l from F l . Let H L b e an y maxim um matc hing of G \ F L . Due to (c1) of theorem 2 f ′ l , f ′′ l ∈ H L . This implies that { f L } ∪ ( H L \ F l ) is a matching of G \ F l whic h c on tradicts (c2) of theorem 2.  Claim 3 F or a ny maximum matchings F L , F l of the gr aph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) (a) if u ∈ V ( F l ) \ V ( F L ) then d ( u ) = 1 or d ( u ) = 2 . Mor e over, in the l a tter c ase, if v a n d w den ote the two neighb ours of u , wher e ( u, w ) ∈ F l , then d ( w ) = 2 and ( v , w ) ∈ F L . (b) if u ∈ V ( F L ) \ V ( F l ) then d ( u ) ≥ 2 . Pro of. (a) Assume that u is cov ered b y an edge e l ∈ F l and u / ∈ V ( F L ). Supp ose that d ( u ) ≥ 2, and there is an edge e = ( u, v ) suc h tha t e / ∈ F l . T aking in to accoun t the claim 1, w e nee d only to consider the fo llo wing four cases: Case 1: v ∈ V ( F l ) \ V ( F L ). This is imp ossible, since F L is a maxim um matc hing. Case 2: v is cov ered b y an edge f ∈ F L ∩ F l ; Let H L b e an y maxim um matc hing of G \ F L . Due to (c1) of theorem 2 e l ∈ H L , th us e / ∈ H L . Define: F ′ L = ( F L \{ f } ) ∪ { e } . Note that F ′ L is a maxim um matching, and H L is a matc hing of G \ F ′ L . Moreo ve r , ν ( G \ F ′ L ) ≥ | H L | = ν ( G \ F L ) = L ( G ) , On upp er b ounds fo r parameters related to construction of special maxim um matc hing s 7 th us H L is a maxim um matc hing of G \ F ′ L and ν ( G \ F ′ L ) = L ( G ). This is a c o n tradiction b ecause F ′ L △ F l con tains a component whic h is not a 2-path con tradicting claim 2. Case 3: v is inciden t to an edge f L ∈ F L , f l ∈ F l and f L 6 = f l . Let H L b e a n y maxim um matc hing of G \ F L . Due to (c1) of theorem 2, e l , f l ∈ H L . Define: F ′ L = ( F L \{ f L } ) ∪ { e } . Note that F ′ L is a maxim um matching, and H L is a matc hing of G \ F ′ L . Moreo ve r , ν ( G \ F ′ L ) ≥ | H L | = ν ( G \ F L ) = L ( G ) , th us H L is a maxim um matc hing of G \ F ′ L and ν ( G \ F ′ L ) = L ( G ). This is a c o n tradiction b ecause F ′ L △ F l con tains a component whic h is not a 2-path con tradicting claim 2. Case 4: v is cov ered b y an edge e L ∈ F L and v / ∈ V ( F l ) . Note that if e L is not adjacen t to e l then claim 2 implies that the edges e, e L and the edge ˜ e ∈ F l \ F L that is adjacen t to e L w ould form an augmen ting 3-path with respect to F L , whic h w ould con tr adict the maximalit y of F L . Th us it remains to consider the case when e L is adjacen t to e l and d ( u ) = 2. Let w b e t he v ertex adjacent to b oth e l and e L . Let us show t hat d ( w ) = 2. Let H L b e a n y maxim um matc hing of G \ F L . Due to (c1) of theorem 2, e l ∈ H L . Define: F ′ L = ( F L \{ e L } ) ∪ { e } . Note that F ′ L is a maxim um matching, and H L is a matc hing of G \ F ′ L . Moreo ve r , ν ( G \ F ′ L ) ≥ | H L | = ν ( G \ F L ) = L ( G ) , th us H L is a maxim um matc hing of G \ F ′ L and ν ( G \ F ′ L ) = L ( G ). If d ( w ) ≥ 3 there is a v ertex w ′ 6 = u, v suc h that ( w , w ′ ) ∈ E ( G ) and w ′ satisfies one o f the conditions of c ases 1,2 and 3 with resp ect t o F ′ L and F l . A con t r a diction. Th us d ( w ) = 2. Clearly , ( v , w ) = e L ∈ F L . (b) This follows from (a) of claim 3 a nd corollary 1.  Claim 4 L et F L , F l b e any maximum matchings of the gr a p h G w i th ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) . Then for a ny ma ximum matchin g H L of the gr aph G \ F L ther e is no e dge of F L ∩ F l which is ad j a c ent to two e dges fr om H L . Pro of. Due to (c3) of theorem 2 an y edge from H L that is inciden t to a v ertex co ve red b y an edge of F L ∩ F l is also inciden t to a v ertex from V ( F L ) \ V ( F l ). If there we r e an edge e ∈ F L ∩ F l whic h is adjacen t to tw o edges h L , h ′ L ∈ H L , then the edges h L , e and h ′ L w ould form an augmen ting 3-path with respect to F l , whic h w ould con tradict the maximality of F l .  Claim 5 (1) for a ny maximum matchings F L , F l of the gr aph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) , we have ( V ( F L ) \ V ( F l )) ∩ V 1 ( G ) = ∅ ; 8 Artur Kho jabaghy an, V ahan V. Mkrtc hy a n (2) ther e is a maximum matching F l of G with ν ( G \ F l ) = l ( G ) and a maximum match- ing F L of the gr ap h G with ν ( G \ F L ) = L ( G ) , such that V 1 ( G ) ⊆ V ( F L ∩ F l ) ∪ ( V ( F l ) \ V ( F L )) . Pro of. (1) On the opp osite assumption, consider a v ertex x ∈ ( V ( F L ) \ V ( F l )) ∩ V 1 ( G ). Since x ∈ V 1 ( G ) then d ( x ) ≤ 2. On the other hand, (b) of claim 3 implies tha t d ( x ) ≥ 2, th us d ( x ) = 2. Then there ar e v ertices y , z suc h that ( x, z ) ∈ F L , ( z , y ) ∈ F l . Note that due to (a) of claim 3, w e ha ve d ( y ) ≤ 2 . Let us sho w that d ( y ) = 1. Supp ose that d ( y ) = 2. Then due to (a) of claim 3, w e ha v e that d ( z ) = 2, th us G is the triangle, whic h is a contradiction, since G do es not satisfy L ( G ) = 2 l ( G ). Th us d ( y ) = 1. Since x ∈ V 1 ( G ), w e imply that there is a v ertex w with d ( w ) = 2 suc h that w, x, z f o rm a triangle. Note t hat w is co v ered neither b y F L nor b y F l , which con tradicts claim 1. (2) Let e t b e an edge of a tria ngle t ∈ T connecting the v ertex v t ∈ V 1 ( G ) to a v ertex of degree t w o. Let us show that there is a maximum matching F l of G with ν ( G \ F l ) = l ( G ) suc h that e t ∈ F l for eac h t ∈ T . Cho ose a maxim um matching F l of G with ν ( G \ F l ) = l ( G ) t ha t con tains as man y edges e t as p ossible. Let us sho w that F l con tains a ll edges e t . S upp ose that there is t 0 ∈ T suc h that e t 0 / ∈ F l . Define: F ′ l = ( F l \{ e } ) ∪ { e t 0 } , where e is the e dge of F l that is adja cen t to e t 0 . Note tha t ν ( G \ F ′ l ) ≤ ν ( G \ F l ) = l ( G ) , th us F ′ l is a maxim um matc hing of G with ν ( G \ F l ) = l ( G ). Note that F ′ l con tains more edges e t than do es F l whic h c on tradicts the c hoice o f F l . Th us, there is a maxim um match ing F l of G with ν ( G \ F l ) = l ( G ) suc h that e t ∈ F l for all t ∈ T . Now, fo r this maximum matc hing F l of G choose a maxim um matc hing F L of the graph G with ν ( G \ F L ) = L ( G ) , suc h that V ( F L ∩ F l ) ∪ ( V ( F l ) \ V ( F L )) co ve r s maxim um n umber of v ertices from V 1 ( G ). Let us show that V 1 ( G ) ⊆ V ( F L ∩ F l ) ∪ ( V ( F l ) \ V ( F L )). Supp ose that there is a v ertex x ∈ V 1 ( G ) suc h that x / ∈ V ( F L ∩ F l ) ∪ ( V ( F l ) \ V ( F L )). Note that due to claim 1 a nd (b) of claim 3, any v ertex of degree one is either inciden t to an edge fr o m F L ∩ F l or to an edge V ( F l ) \ V ( F L ). Th us due to definition of V 1 ( G ), d ( x ) = 2 and if y and z denote the t wo neigh b ors of x , then d ( y ) = 2 a nd ( y , z ) ∈ E ( G ). Since x / ∈ V ( F L ∩ F l ), w e ha ve that ( x, y ) / ∈ F L , and since x / ∈ ( V ( F l ) \ V ( F L )), we ha ve that ( y , z ) / ∈ F L , thus ( x, z ) ∈ F L , as F L is a maxim um matching. Let H L b e an y maxim um matc hing of G \ F L . As L ( G ) = 2 l ( G ), w e ha v e ( x, y ) ∈ H L ((c1) of theorem 2). Define: F ′ L = ( F L \{ ( x, z ) } ) ∪ { ( y , z ) } . Note that F ′ L is a maxim um matching of G , H L is a matc hing of G \ F L , th us ν ( G \ F ′ L ) ≥ | H L | = ν ( G \ F L ) = L ( G ) . Therefore F ′ L is a maxim um matc hing of G with ν ( G \ F ′ L ) = L ( G ). No w, observ e that V ( F ′ L ∩ F l ) ∪ ( V ( F l ) \ V ( F ′ L )) co v ers more v ertices than does V ( F L ∩ F l ) ∪ ( V ( F l ) \ V ( F L )) whic h con tradicts the choice of F L . The pro of of the c laim 5 is completed.  On upp er b ounds fo r parameters related to construction of special maxim um matc hing s 9 Claim 6 F or any maximum matchings F L , F l of the gr aph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) , we have (1) V ( F L ) \ V ( F l ) is an indep endent set; (2) no e dge of G c onne cts t wo vertic es that ar e c over e d by b oth F L \ F l and F l \ F L ; (3) no e dge of G is adjac ent to two differ ent e dges fr om F L ∩ F l ; (4) no e dge of G c onne cts a vertex c over e d by F L ∩ F l to a vertex c over e d by b oth F L \ F l and F l \ F L ; (5) if ( u, v ) ∈ F L ∩ F l then e i ther u ∈ V 1 ( G ) or v ∈ V 1 ( G ) . Pro of. (1)There is no edge of G connecting t w o v ertices from V ( F L ) \ V ( F l ) since F l is a maxim um matc hing. (2) follow s from (c1) and (c2) of theorem 2. (3) follow s from (c3) of theorem 2. (4) Supp ose that there is an edge e = ( y 1 , y 2 ), suc h that y 1 is co v ered b y an edge ( z , y 1 ) ∈ F L ∩ F l and y 2 is cov ered by b oth F L \ F l and F l \ F L . Consider a maxim um matc hing H L of the graph G \ F L . Note that y 1 m ust b e inciden t to an edge from H L , as otherwise w e could replace the edge of H L that is adja cent to e a nd belongs a lso to F l \ F L ((c1) of theorem 2) by the edge e to o bt a in a new maximum matc hing H ′ L of the graph G \ F L whic h w o uld not satisfy (c1) o f t heorem 2. So let y 1 b e inciden t to an edge h L ∈ H L , which connects y 1 with a v ertex x ∈ V ( F L ) \ V ( F l ). Note that due to claim 4 , z is not inciden t to an edge from H L . No w, let x 1 b e a v ertex such tha t ( x, x 1 ) ∈ F L \ F l (suc h a ve r tex exists since x ∈ V ( F L ) \ V ( F l )). As F l is a maxim um matc hing, x 1 is inciden t to an edge ( x 1 , x 2 ) ∈ F l \ F L . By (c1) of theorem 2, ( x 1 , x 2 ) ∈ H L . Moreov er, b y claim 2, x 2 is not adjacen t to an edge f rom F L . Th us the edges ( z , y 1 ) , ( y 1 , x ) , ( x, x 1 ) and ( x 1 , x 2 ) form an F L − H L alternating 4-path P . Define: F ′ L = ( F L \ E ( P )) ∪ ( H L ∩ E ( P )) , H ′ L = ( H L \ E ( P )) ∪ ( F L ∩ E ( P )) . Note tha t F ′ L is a maxim um matching of G , H ′ L is a matc hing of G \ F ′ L of cardinality | H L | , and ν ( G \ F ′ L ) ≥ | H ′ L | = | H L | = ν ( G \ F L ) = L ( G ) , th us H ′ L is a maxim um matc hing of G \ F ′ L and ν ( G \ F ′ L ) = L ( G ). This is a c o n tradiction since the edge e connects tw o v ertices whic h are co ve red b y F ′ L \ F l and F l \ F ′ L ((2) of claim 6). (5)Supp ose that e = ( u, v ) ∈ F L ∩ F l . Since G is connected and | V | ≥ 3, we , without loss of generality , ma y assume that d ( v ) ≥ 2, and t here is w ∈ V ( G ) , w 6 = u suc h that ( w , v ) ∈ E ( G ) . Consider a maxim um matc hing H L of the graph G \ F L . Note that, without loss of generalit y , w e can assume that v is inciden t to an edge f r om H L , as otherwise w e could replace the edge of H L that is inciden t to w ( H L is a maxim um matc hing of G \ F L ) 10 Artur Kho jabaghy an, V ahan V. Mkrtc hy a n b y the edge ( w , v ) to obtain a new maxim um matc hing H ′ L of the graph G \ F L suc h that v is inciden t to an edge from H ′ L . So w e can ass ume that there is an edge ( v , q ) ∈ H L , q 6 = u . Note that due to claim 4, u is not inciden t to a n edge from H L . (c3) o f t heorem 2 implies that q is inciden t to an edge from ( q , q 1 ) ∈ F L \ F l . As F l is a maxim um mat ching, q 1 is inciden t to a n edge ( q 1 , q 2 ) ∈ F l \ F L . By (c1) of theorem 2, ( q 1 , q 2 ) ∈ H L . Moreov er, by claim 2, q 2 is not adjacen t to an edge from F L . Th us the edges ( u, v ) , ( v , q ) , ( q , q 1 ) and ( q 1 , q 2 ) form an F L − H L alternating 4- path P . Define: F ′ L = ( F L \ E ( P )) ∪ ( H L ∩ E ( P )) , H ′ L = ( H L \ E ( P )) ∪ ( F L ∩ E ( P )) . Note that F ′ L is a maxim um matching of G , H ′ L is a matc hing of G \ F ′ L of cardinality | H L | , and ν ( G \ F ′ L ) ≥ | H ′ L | = | H L | = ν ( G \ F L ) = L ( G ) , th us H ′ L is a maxim um matc hing of G \ F ′ L and ν ( G \ F ′ L ) = L ( G ). Since u ∈ V ( F l ) \ V ( F ′ L ) (a) of claim 3 implies that either d ( u ) = 1 and therefore u ∈ V 1 ( G ), or d ( u ) = d ( v ) = 2 and therefore either u ∈ V 1 ( G ) or v ∈ V 1 ( G ). Pro of of the claim 6 is comple t ed.  W e are ready to complete the proof of the theorem. T ak e an y maxim um matc hings F L , F l of the g raph G guaranteed by the (2) of claim 5 and consider the follo wing part it ion of V ( G \ V 1 ( G )) = V ( G ) \ V 1 ( G ): X = X ( F L , F l ) = V ( F L ) \ V ( F l ) , Y = Y ( F L , F l ) = V ( G ) \ ( V 1 ( G ) ∪ X ) . Claim 6 implies that X and Y are indep endent sets of v ertices of G \ V 1 ( G ), thus G \ V 1 ( G ) is a bipartite graph with a bipartition ( X , Y ). The c hoice of maxim um matchings F L , F l , ( a ) of claim 3, (5) of claim 6 and the definition of the set Y imply (2) of t he theorem 3. Let us sho w that it satisfies (3), to o. Consider the alternating 2-paths of ( H L \ F l ) △ ( F L \ F l ) . (c2), (c3) of theorem 2 and the definition of the set X imply t ha t there are | X | suc h 2-paths. Moreo ver, these 2-paths are in fact 2- paths of the graph G \ V 1 ( G ). Thu s G satisfies (3) of t he theorem. The pro of of the the orem 3 is completed.  Corollary 2 The pr op erty of a gr aph L ( G ) = 2 l ( G ) c a n b e teste d in p olynomial time. Pro of. First of all note that the prop ert y L ( G ) = 2 l ( G ) is additiv e, tha t is, a graph satis- fies this prop erty if and only if all its connected componen ts do. Th us w e can concen t rate only on connected graphs. All connec t ed graphs with | V ( G ) | ≤ 2 satisfy the equalit y L ( G ) = 2 l ( G ), th us w e can assume that | V ( G ) | ≥ 3 . On upp er b ounds for parameters r elated to construction of sp ecial maxim um matc hings 11 Next, w e construct a set V 1 ( G ), whic h can b e done in linear time. Now, w e need to c hec k whethe r the graph G \ V 1 ( G ) satisfies the conditions (1)-(3) of the theorem 3. It is w ell-known that the prop erties (1) and (2) can be che ck ed in p olynomial time, so w e will consider only the testing of (3). F rom a graph G \ V 1 ( G ) with a bipartition ( X , Y ) w e construc t a net w ork ~ G with new v ertices s and t . The arcs of ~ G are defined as follo ws: • connect s to eve ry ve rtex of X with an arc of capacity 2; • connect ev ery v ertex of Y to t b y an arc of capacity 1; • for e v ery edge ( x, y ) ∈ E ( G ) , x ∈ X , y ∈ Y add an arc connecting the v ertex x to the v ertex y whic h has capacit y 1. Note that • the v alue of the maxim um s − t flow in ~ G is no more than 2 | X | (the capacit y of the cut ( S, ¯ S ), where S = { s } , ¯ S = V ( ~ G ) \ S , is 2 | X | ); • the v alue of the maxim um s − t flo w in ~ G is 2 | X | if and only if the graph G \ V 1 ( G ) con tains | X | v ertex disjoin t 2-pat hs, th us (3) also can b e tested in p olynomial time.  Remark 1 R e c ently Monnot an d T oulouse in [ 7] pr ove d that 2 -p ath p artition pr oblem r emains N P -c omple te even for bip artite gr ap hs of maximum d e gr e e thr e e. F ortunately, in the or em 3 w e ar e de a l i n g with a sp e c i a l c ase of this pr oblem whi c h en ables us to pr esent a p olynomial algorithm in c or ol lary 2. 4. N P -completeness of testing L ( G ) = 3 2 l ( G ) in the class of bridgeless cubic graphs The reader ma y think that a result analogous to corollary 2 can b e pro ved fo r the prop- ert y L ( G ) = 3 2 l ( G ) in the class of graphs containing a p erfect matc hing. Unfortunately this fails already in the class of bridgeless cubic g r aphs, whic h b y the w ell-know n theorem of P etersen ar e kno wn to p ossess a perfect matching (see theorem 3.4.1 of [ 6]). Theorem 4 It is N P -c omplete to test the pr op erty L ( G ) = 3 2 l ( G ) i n the c lass of bridgeless cubic gr aphs. Pro of. Cle a rly , the pro blem of testing the prop ert y L ( G ) = 3 2 l ( G ) f or graphs con taining a perfect match ing is in N P , since if w e a re given p erfect matc hing s F L , F l of the g r a ph G with ν ( G \ F L ) = L ( G ) , ν ( G \ F l ) = l ( G ) then w e can calculate L ( G ) and l ( G ) in p olynomial time. W e will use the w ell-kno wn 3-edge-coloring problem ([ 3]) to establish the NP-completeness of our problem. 12 Artur Kho jabaghy an, V ahan V. Mkrtc hy a n Let G b e a bridgeless cubic graph. Consider a bridgeless cubic graph G △ obtained from G b y replacing ev ery v ertex of G b y a triangle. W e claim that G is 3-edge-colora ble if and only if L ( G △ ) = 3 2 l ( G △ ). Supp ose that G is 3-edge-color a ble. Then G △ is also 3-edge-colora ble, whic h means that G △ con tains t w o ed ge disjoin t p erfect matc hings F and F ′ . This implies that L ( G △ ) ≥ ν ( G △ \ F ) ≥ | F ′ | = | V ( G △ ) | 2 , On the other hand, the set E ( G ) forms a perfect matching of G △ , and l ( G △ ) ≤ ν ( G △ \ E ( G )) = | V ( G △ ) | 3 , since ev ery compo nen t of G △ \ E ( G ) is a triangle. Th us: L ( G △ ) l ( G △ ) ≥ 3 2 , (d) of theorem 2 implies that L ( G △ ) l ( G △ ) = 3 2 . No w a ssume that L ( G △ ) l ( G △ ) = 3 2 . Note that for ev ery p erfect matc hing F of the graph G △ the graph G △ \ F is a 2-factor , therefore L ( G △ ) = | V ( G △ ) | − w ( G △ ) 2 , l ( G △ ) = | V ( G △ ) | − W ( G △ ) 2 where w ( G △ ) and W ( G △ ) denote the minim um and maxim um n um b er of o dd cyc les in a 2-f a ctor o f G △ , resp ectiv ely . Since L ( G △ ) l ( G △ ) = 3 2 w e ha v e W ( G △ ) = | V ( G △ ) | + 2 w ( G △ ) 3 . T aking in to accoun t that W ( G △ ) ≤ | V ( G △ ) | 3 , w e ha v e: W ( G △ ) = | V ( G △ ) | 3 , w ( G △ ) = 0 . 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