Local Maximum Stable Sets Greedoids Stemmed from Very Well-Covered Graphs

Lo cal Maxim um Stable Sets Greedoids Stemmed fro m V ery W ell-Co v ered Graphs V adim E. Levit Ariel Univ ersit y Cen ter of Samaria, ISRAEL E-mail: levitv@ariel.ac.il Eugen Mandrescu Holon Institute of T ec hnology , ISRAEL E-mail: eugen m@hit.ac.il Abstract A maximum stable s et i n a graph G is a stable set of maximum cardinali ty . S is called a lo c al maxim um stable set of G , and w e write S ∈ Ψ( G ), if S is a maximum stable set of th e subgraph ind uced by the closed neighborho o d of S . A greedoid ( V , F ) is called a lo c al maximum stable set gr e e doid if there exists a graph G = ( V , E ) such that F = Ψ( G ). Nemhauser and T rotter Jr. [27], prov ed that any S ∈ Ψ( G ) is a subset of a ma ximum stable set of G . In [15] w e ha ve sho wn that th e family Ψ ( T ) of a forest T fo rms a greedoid on its vertex set. The cas es where G is bipartite, tri angle-free, w ell-cov ered, while Ψ( G ) is a g reedoid, w ere analyzed in [17], [19], [21], respectively . In this p ap er w e demonstrate that i f G is a very well -cov ered graph, then the family Ψ( G ) is a greedoid if and only if G has a unique p erfect matching. Keywords: ver y w ell-cov ered graph , local maximum stable set, p erfect matching, greedoid, K¨ onig-Egerv´ ary graph. 1 In tro duction Throughout this pap er G = ( V , E ) is a s imple (i.e., a finite, undirec ted, lo opless and without m ultiple edges) graph with v ertex set V = V ( G ) and edge set E = E ( G ). If X ⊂ V , then G [ X ] is the subg r aph of G spanned b y X . If A, B ⊂ V and A ∩ B = ∅ , then ( A, B ) stands for the set { e = ab : a ∈ A, b ∈ B , e ∈ E } . The neighb orho o d of a vertex v ∈ V is the set N ( v ) = { u : u ∈ V and vu ∈ E } . F or A ⊂ V , we denote N G ( A ) = { v ∈ V − A : N ( v ) ∩ A 6 = ∅} and N G [ A ] = A ∪ N ( A ), or s hortly , N ( A ) and N [ A ]. If N ( v ) = { u } , then v is a p endant vertex a nd u v is a p en dant e dge o f G . K n , C n , P n denote r esp ectively , the complete graph o n n ≥ 1 vertices, the chordless cycle on n ≥ 3 v ertices, a nd the chordless path on n ≥ 2 v ertices. A matching in a g r aph G = ( V , E ) is a se t M ⊆ E such tha t no tw o edges of M share a common vertex. A maximum matching is a matching of maximum cardinalit y . By µ ( G ) is denoted the size of a maximum matching. A matc hing is p erfe ct if it saturates all the vertices of the gra ph. 1 If f or every tw o incident edg es of a cycle C exactly o ne of them b elongs to a ma tchin g M , then C is called an M -alternating cycle [9]. It is clear that an M -alternating cycle should b e of even leng th. A matc hing M in G is called a lternating cycle-fr e e if G has no M -alterna ting cycle. Alternating cycle-free matchings for bipartite gr a phs were first defined in [9 ]. F or example, the matching { ab, cd, ef } of the graph G from Fig ure 1 is alter nating cycle-free. ✇ ✇ ✇ ✇ ✇ ✇ ✇    a b c d e f g G ✇ ✇ ✇ ✇ ✇ ✇ u v t w y x H Figure 1: The unique cycle of H is alterna ting with r esp ect to the matching { y v , tx } . A matching M = { a i b i : a i , b i ∈ V ( G ) , 1 ≤ i ≤ k } o f g raph G is called a uniquely re stricte d matching if M is the unique per fect matching of G [ { a i , b i : 1 ≤ i ≤ k } ] [7]. Theorem 1.1 [7] A matching M in a gr aph G is uniquely r estricte d i f and only if G c ontains no alternating cycle with re sp e ct to M , i .e., M is alternating cy cle-fr e e. F or instanc e , all the maximum matchings of the graph G in Figur e 1 a re uniquely restricted, while the g raph H from the same figure ha s b oth uniquely restricted maximum matchings (e.g., { uv , xw } ) and non- uniquely restr icted maximum matc hings (e.g., { xy , tv } ). A stable set in G is a set of pair wise non-adjacent v er tices. A stable set of maximum size will b e referred to as a maximum stable set of G , and the stability n umb er of G , denoted by α ( G ), is the cardinalit y of a maxim um stable set in G . Let Ω( G ) stand for the s et of all maximum stable sets o f G . In genera l, α ( G ) ≤ α ( G − e ) and µ ( G − e ) ≤ µ ( G ) holds for any edge e of a gra ph G . An edge e of G is α -critic al ( µ -critic al ) if α ( G ) < α ( G − e ) ( µ ( G ) > µ ( G − e ), resp ectively). It is w orth observing that there is no general connection betw een the α -cr itical and µ -critical edges of a g r aph. Recall that G is ca lled a K¨ onig-Egerv´ ary gr aph provided α ( G ) + µ ( G ) = | V ( G ) | [3], [32]. As a well-kno wn example, every bipartite gr aph is a K¨ onig -Eger v´ ar y graph [4], [11]. Theorem 1.2 If G i s a K¨ oni g-Egerv´ ary g r aph, t hen the fol lowing a ssertions hold: (i) [16] every maximum matching is c ontaine d in ( S, V ( G ) − S ) , for e ach S ∈ Ω ( G ) ; (ii) [18] the α -cri tic al e dges ar e a lso µ - critic al, and the y c oincide in a bip artite gr aph. A set A ⊆ V ( G ) is a lo c al maximum stabl e set of G if A ∈ Ω( G [ N [ A ]]) [15]; by Ψ ( G ) we denote the set of all lo c al maximum stable sets of the gra ph G . ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c d e f g G Figure 2: { a } , { e, d } , { a, d, f } ∈ Ψ ( G ) , while { b } , { a, e } , { c, f } a re not in Ψ ( G ). The following theor em concer ning maximum stable sets in g eneral gr a phs, due to Nemhause r and T r otter J r. [27], s hows that for a sp ecia l subgra ph H o f a graph G , some maximum stable set o f H ca n b e e nla rged to a maximum stable s e t of G . 2 Theorem 1.3 [27] Every lo c al maximum stable set of a gr aph is a su bset of a maximum stable set. Let us notice that the conv erse of Theor em 1.3 is not generally true. F or instance, C n has no prop er lo ca l max im um stable set, for any n ≥ 4. The graph G in Figure 2 shows another coun ter e xample: an y S ∈ Ω( G ) con tains s o me lo ca l maximum sta ble set, but th ese lo cal maxim um s ta ble sets a re o f differen t ca rdinalities. As examples, { a, d, f } ∈ Ω( G ) and { a } , { d, f } ∈ Ψ( G ), while for { b, e , g } ∈ Ω( G ) only { e, g } ∈ Ψ( G ) . Definition 1.4 [1], [10] A gr e e doid is a p air ( V , F ) , wher e F ⊆ 2 V is a non-empty set system satisfying the f ol lowing c onditions: Accessibility: for every non-empty X ∈ F ther e is an x ∈ X such tha t X − { x } ∈ F ; Exchange: for X , Y ∈ F , | X | = | Y | + 1 , ther e i s an x ∈ X − Y s u ch that Y ∪ { x } ∈ F . Definition 1.5 [23] A gr e e doid ( V , F ) is c al le d a lo c al maximum stable set gr e e doid if the r e exists a gr aph G = ( V , E ) such that F = Ψ ( G ) . In fac t, the fo llowing theo rem says that, in th e case of lo ca l ma ximum stable set gree doids, it is enough to chec k only the accessibility proper ty . Theorem 1.6 [23] If the family Ψ( G ) of a gr aph G s atisfi es the ac c essibility pr op erty, t hen ( V ( G ) , Ψ( G )) is a gr e e doid. In the sequel, we use F instead of ( V , F ), as the ground set V will b e, usually , the vertex set o f some gr a ph. Theorem 1.7 [15] The f amily of lo c al maximum stable sets of a for est forms a gr e e doid on its vertex s et . The conclusion of Theorem 1.7 is not sp ecific for forests. F or instance, the f amily Ψ( G ) o f the gr aph G in Figure 3 is a greedo id. ✇ ✇ ✇ ✇ ✇ ✇ a b c d e f G ✇ ✇ ✇ ✇ ✇ ✇ u v t w y x H Figure 3: Both G and H ar e bipartite, but only Ψ( G ) for ms a g reedoid. Notice that Ψ( H ) is not a g reedoid, where H is from Figure 3, because the accessibility prop erty is not satisfied; e.g., { y , t } ∈ Ψ ( H ), while { y } , { t } / ∈ Ψ( H ). In addition, o ne can see that all the maximum matc hings of the g raph G in Fig ure 3 ar e uniquely restricted, while the g raph H from the same figure has b oth uniquely restricted max imum matchings (e.g., { uv , xw } ) and non-uniquely r estricted maxim um match ings (e.g., { xy , tv } ). It turns o ut that this is the rea son that Ψ ( H ) is not a g reedoid, while Ψ( G ) is a g reedoid. Theorem 1.8 [17] F or a bip artite gr aph G, Ψ( G ) is a gre e doid on its vertex set if and only if al l its ma ximum matchings ar e uniquely r estricte d. 3 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ f a b c d e g G ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇    H Figure 4: Ψ( G ) is not a gr eedoid, Ψ( H ) is a gre e doid. The case of bipartite graphs owning a unique cycle, whose f amily of loca l maximum sta ble sets forms a g reedoid is ana lyzed in [14]. The g r aphs from Figure 4 are non-bipar tite K¨ o nig-Eger v´ ar y graphs, and all their maximum matchings are uniq ue ly restricted. Let us remark that b oth gr aphs are also triangle-free, but only Ψ( H ) is a greedoid. It is clear that { b, c } ∈ Ψ( G ), while G [ N [ { b , c } ]] is no t a K¨ onig- Egerv´ ary graph. As one can see fro m the following theorem, this observ atio n is the rea l reas on for Ψ( G ) not to b e a greedoid. Theorem 1.9 [19] If G is a t riangle-fr e e gr aph, then Ψ( G ) is a gr e e doid if and only if al l maximum matchings of G ar e uniquely r estricte d and the close d n eighb orho o d of every lo c al maximum stable set of G induc es a K ¨ onig-Egerv´ ary g r aph. Let X b e a graph with V ( X ) = { v i : 1 ≤ i ≤ n } , and { H i : 1 ≤ i ≤ n } be a family of graphs. Joining ea ch v i ∈ V ( X ) to all the v er tice s of H i , w e obtain a new graph, calle d the c or ona of X and { H i : 1 ≤ i ≤ n } and denoted b y G = X ◦ { H 1 , H 2 , ..., H n } . F or instance, see Fig ure 5. If H 1 = H 2 = ... = H n = H , we write G = X ◦ H , and in this c ase, G is called the c or ona o f X and H . ✇ ✇ ✇ ✇ v 1 v 2 v 3 v 4 X ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁                ❅ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ ✟ ✟ ✟ ✟ ✟ ✟ x y z u t v 1 v 2 v 3 v 4 K 3 K 2 P 3 C 4 G Figure 5: The co rona G = X ◦ { K 3 , K 2 , P 3 , C 4 } . If each H i is a complete graph, then X ◦ { H 1 , H 2 , ..., H n } is called the clique c or ona of X and { H 1 , H 2 , ..., H n } ; notice that the clique corona is well-cov er e d (and very well-co vered, whenever H i = K 1 , 1 ≤ i ≤ n ). Reca ll that G is well-c over e d if all its maximal stable sets hav e the sa me cardinalit y [28], and G is very wel l-c over e d if, in addition, it has no isola ted vertices and | V ( G ) | = 2 α ( G ) [5]. A num be r of class e s of well-cov e r ed gra phs w ere completely described (see, for instance, the following referenc e s : [5], [6], [8], [12], [29], [3 0], [31]. Theorem 1.10 (i) [ 6] L et G b e a c onne cte d gr aph of girth ≥ 6 , wh ich is isomorp hic to neither C 7 nor K 1 . Then G is wel l-c over e d if and only if G = H ◦ K 1 , for some gr aph H of girth ≥ 6 . (ii) [2], [20] L et G b e a gr aph having girth ≥ 5 . Then G is very wel l-c over e d if and only if G = H ◦ K 1 , for so me gr aph H of gi rth ≥ 5 . (iii) [12] G i s very wel l-c over e d if and only i f G is a well-c over e d K¨ onig-Egerv´ ary gr aph. (iv) [33] G = X ◦ { H 1 , H 2 , ..., H n } is wel l-c over e d if and only if al l H i ar e c omplete. 4 It is easy to pro ve that ev e r y gr aph having a perfect matc hing consisting of pendant edges is very well-cov er ed. The con verse is not genera lly true (see, for instance, th e graphs depicted in Figur e 6). Moreover, there are well-co vered graphs without p erfect matchings; e.g., K 3 . ✇ ✇ ✇ ✇ C 4 ✇ ✇ ✇ ✇ ✇ ✇ G Figure 6: V ery well-cov er ed gr aphs with no pe rfect ma thc hing cons is ting of only p endant edges. Theorem 1.11 [5] F or a g r aph G without i solate d ve rtic es the fol lowing ar e e quivalent: (i) G is very wel l-c over e d; (ii) ther e exists a p erfe ct matching M in G that satisfies pr op erty P , i.e., “ N ( x ) ∩ N ( y ) = ∅ , and e ach v ∈ N ( x ) − { y } is ad jac ent to al l vertic es of N ( y ) − { x } ” hold for e very e dge xy ∈ M ; (iii) ther e exists at le ast o ne p erfe ct match ing in G and every p erfe ct matching in G satisfies pr op erty P . V ario us cases of w ell-covered gra phs generating loca l maxim um s table set greedoids, w ere treated in [21], [2 2], [24], [25]. Theorem 1.12 L et G = X ◦ { H 1 , H 2 , ..., H n } , wher e H 1 , H 2 , ..., H n ar e non- empty gr aphs. (i) [24] if G = P n and al l H i , 1 ≤ i ≤ n , a r e c omplete gr aphs, then Ψ ( G ) is a gr e e doid; (ii) [21] if H i = K 1 , 1 ≤ i ≤ n , then Ψ ( G ) is a gr e e doid; (iii) [22] if all H 1 , H 2 , ..., H n ar e c omplete gr aphs, then Ψ ( G ) is a gr e e doid; (iv) [25] Ψ( G ) i s a gr e e doid if a nd o nly if every Ψ( H i ) , i = 1 , 2 , ..., n , is a gr e e doid. It turns out that the prop erty of having a unique m aximum matc hing is of crucial imp or- tance for very-well co vered graphs to g e ne r ate lo cal max im um stable set gre edoids. Theorem 1.13 [26] L et G b e a very wel l-c over e d gr aph of girth at le ast 4 . Then Ψ( G ) i s a gr e e doid if and o nly i f G has a unique maximum matching. In this pap er we completely characterize very well-cov er ed graphs who se fa milies o f lo cal maximum stable sets a r e gre e doids. 2 V ery w ell-co v ered graphs p ro du cing greedoids Notice that S 1 = { a , b } and S 2 = { c, d } ar e stable sets in the gr aph G 1 from Fig ure 7, S 1 ∈ Ψ( G 1 ), a nd both G 1 [ N [ S 1 ]] and G 1 [ N [ S 2 ]] are K¨ onig -Eger v´ ary gr aphs. On the other hand, S 2 = { x, y } ∈ Ψ( G 2 ), where G 2 is fro m Fig ur e 7, but G [ N [ S 3 ]] is not a K¨ onig-Eger v´ ary graph. Theorem 2.1 [20] L et G b e a very wel l-c over e d gr aph. Then G [ N [ S ]] is a K¨ onig-Egerv´ ary gr aph, for eve ry S ∈ Ψ( G ) . 5 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ ✁ ✁ ✁ ✁ ✁ ✁       ❳ ❳ ❳ ❳ ❳ ❳ ❆ ❆ ❆ ❆ ❆ ❆    a b c d e G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇    x y G 2 Figure 7: Non-bipartite well-cov e red graphs. Mor eov er , G 1 is very well-co vered. Concerning the gra ph G 1 from Figure 7, let us re ma rk t hat { b, d } , { b, e } are stable sets, |{ b, d } | < | N ( { b, d } ) | and |{ b, e } | = | N ( { b, e } ) | , but only { b, e } ∈ Ψ( G 1 ). Lemma 2.2 If S is a stable set in a very well-c over e d gr aph G , then S ∈ Ψ( G ) if and only if | S | = | N ( S ) | . Pro of. Accor ding to Theor ems 1.1 0 (iii ) and 1 .11, G is a K¨ onig - Egerv ´ ary graph having a per fect matching, say M . If S is a stable set in G , there must b e some A ∈ Ω( G ), such that S ⊆ A , b eca us e G is well-co vered. By Theo rem 1.2 (i) , we ha ve that M ⊆ ( A, V ( G ) − A ). Since M is a p erfect matching, it follows that S is matched into N ( S ), and further, | S | = | M ( S ) | ≤ | N ( S ) | , where M ( S ) = { y ∈ V : xy ∈ M , x ∈ S } . Let S ∈ Ψ( G ). According to Theore m 2.1, G [ N [ S ]] is a K ¨ onig-E gerv´ ary graph, and consequently , we get that | S | = | M ( S ) | ≥ | N ( S ) | . Hence, we infer that | S | = | N ( S ) | . Conv er sely , let S be a stable set in G satisfying | S | = | N ( S ) | . Since S is matched by M int o N ( S ), we infer that the restriction of M to G [ N [ S ]] is a perfect matching. Therefore, | S | = α ( G [ N [ S ]]), and this implies S ∈ Ψ( G ). ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇       x y v G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅    a b c d e f u v w G 2 Figure 8: G 1 and G 2 are not very w ell-c overed graphs. Only G 1 is well-cov e red. Notice that the ab ove lemma can fail in a non-very well-co vered graph. F or instance, S = { x, y } ∈ Ψ( G 1 ), while | S | < | N ( S ) | , where G 1 is from Figure 8 and it is well-cov ered. F urther, the sets S 1 = { a, c } , S 2 = { e, d } and S 3 = { v , w } belong to Ψ( G 2 ), where G 2 is from Figure 8, and they satisfy: | S 1 | = | N ( S 1 ) | , | S 2 | < | N ( S 2 ) | , a nd | S 3 | > | N ( S 3 ) | . Concerning the very well-cov ered gr aph G 1 from Figure 7 , we see that A, B ∈ Ψ( G 1 ), where B = { a } , A = B ∪ { a } , a nd | N ( A ) | = | N ( B ) | + 1. The following lemma shows that in a v er y w ell-covered graph the existence of an acce ssibility chain is equiv alent to the fact that one c a n have a c hain of stable sets , where e a ch additional v e rtex a dded to a stable s et increases the size of its op en neighbor ho o d by exactly one element. 6 Lemma 2.3 If A = B ∪ { v } is a stable set in a very wel l- c over e d gr aph G , and B ∈ Ψ( G ) , then A ∈ Ψ ( G ) if and only if | N ( A ) | = | N ( B ) | + 1 . Pro of. Assume that A = B ∪ { v } ∈ Ψ ( G ). Since G is v er y w ell-covered, by Lemma 2 .2, it follows that | N ( A ) | − | N ( B ) | = | A | − | B | = 1 . Conv er sely , since | N ( A ) | = | N ( B ) | + 1 = | B | + 1 = | A | , Lemma 2.2 implies immediately that A ∈ Ψ( G ). Lemma 2.3 fails for gr a phs that a re not v ery w ell-covered, for instance, B 1 = { x, y } and A 1 = B 1 ∪ { v } belo ng to Ψ ( G 1 ), but | N ( A 1 ) | = | N ( B 1 ) | + 2 , where G 1 is from Figure 8, and also B 2 = { v } and A 2 = B 2 ∪ { w } b elong to Ψ ( G 2 ), but | N ( A 2 ) | = | N ( B 2 ) | , wher e G 2 is from Figure 8. Let us no tice that the g raphs G 1 , G 2 and G 3 from Figure 9 are very well-cov e red; by Theorem 1.8 or 1.9, neither Ψ( G 2 ) nor Ψ( G 3 ) is a g r eedoid. How ever, Ψ( G 1 ) is a g r eedoid. ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✘ ✘ ✘ ✘ ✘ ✘ ❳ ❳ ❳ ❳ ❳ ❳ G 1 ✇ ✇ ✇ ✇ G 2 ✇ ✇ ✇ ✇ ✇ ✇ ✟ ✟ ✟ ✟ ✟ ✟ G 3 Figure 9: G 1 , G 2 , G 3 are very well-co vered gr aphs. G 1 has a unique p erfect matching. Theorem 2.4 L et G b e a very we l l- c over e d gr aph. Then Ψ( G ) forms a gr e e doid if and only if G h as a u nique maximum ma tching. Pro of. Supp ose that Ψ( G ) forms a greedoid. B y Theorem 1.1 1, G has at le ast one p erfect matching, sa y M . Since Ψ( G ) is a greedoid, every S ∈ Ω( G ) has an a ccessibility c hain { x 1 } ⊂ { x 1 , x 2 } ⊂ ... ⊂ { x 1 , x 2 , ..., x α − 1 } ⊂ { x 1 , x 2 , ..., x α } = S. Let us denote S i = { x 1 , x 2 , ..., x i } , 1 ≤ i ≤ α , and S 0 = ∅ . Since S i − 1 ∈ Ψ ( G ) , S i = S i − 1 ∪ { x i } ∈ Ψ( G ) and G is w ell-cov ered, Lemma 2 .3 implies that | N ( x i ) − N [ S i − 1 ] | = 1 , b e c ause | N ( x i ) − N [ S i − 1 ] | = | N ( S i ) − N ( S i − 1 ) | = | N ( S i ) | − | N ( S i − 1 ) | . Let { y i } = N ( x i ) − N [ S i − 1 ] , 1 ≤ i ≤ α . Hence, M = { x i y i : 1 ≤ i ≤ α } is a maximum matching in G . Let us v alidate that M is a uniquely res tricted maximum matc hing in G . W e induct on k = | S k | in o rder to s how that the restriction of M to H k = G [ N [ S k ]], which we denote by M k , is a uniquely res tricted maximum matc hing in H k . F or k = 1 , S 1 = { x 1 } ∈ Ψ( G ) and this implies that N ( x 1 ) = { y 1 } , unless x 1 is an iso lated vertex. In this case, M 1 = { x 1 y 1 } is a uniquely re s tricted maxim um matching in H 1 . If x 1 is an isola ted vertex, then M 1 = ∅ is a uniquely r estricted maximum matching in H 1 . Suppo se that the ass ertion is true for a ll j ≤ k − 1. Let us notice that N [ S k ] = N [ S k − 1 ] ∪ { x k } ∪ { y k } . 7 As we kno w, N ( x k ) − N [ S k − 1 ] = { y k } . Since H k is K ¨ onig-E gerv´ ary gr aph, M k is a maximum matching in H k . The edg e x k y k is α -critical in H k , b ecause { y k } = N ( x k ) − N [ S k − 1 ], and hence, x k y k is als o µ - critical in H k , according to Theor em 1.2 (ii) . Therefore, any maxim um matching of H k contains t he edge x k y k . Since M k = M k − 1 ∪ { x k y k } and M k − 1 is a uniquely restricted maximum ma tc hing in H k − 1 = H k − { x k , y k } , it follows that M k is a uniquely restricted maximum ma tchin g in H k . Conv er sely , assume that G ha s a unique p erfect matching, say M . W e show that Ψ ( G ) sa tisfies the a ccessibility pro p erty , i.e., for every non-empty X ∈ Ψ( G ) there is an x ∈ X such that X − { x } ∈ Ψ( G ). Let S ∈ Ψ( G ). Accor ding to Theorem 1 .3, there is so me A ∈ Ω( G ), such that S ⊆ A . By Theorem 1.2 (i) , M ⊆ ( A, V ( G ) − A ) a nd this implies that S is ma tched in to N ( S ), and further, | S | = | M ( S ) | ≤ | N ( S ) | , where M ( S ) = { y ∈ V : xy ∈ M , x ∈ S } . According to Theor em 2.1, G [ N [ S ]] is a K¨ onig -Eger v ´ ary graph, and this fact ensures that | S | = | M ( S ) | ≥ | N ( S ) | . Hence, we infer that N ( S ) = M ( S ). Suppo se that S does not satisfies t he accessibility prop er t y , i.e., S − { x } / ∈ Ψ( G ) for ev er y x ∈ S . This implies that N ( S − { x } ) = N ( S ), for every x ∈ S . Conse quently , each vertex in N ( S ) has at least tw o neighbo rs in S . W e show that there is an even cycle C in G [ N [ S ]], such that half of its edge s are in M . Let x 1 y 1 ∈ M and x 1 ∈ S . Since | N ( y 1 ) ∩ S | ≥ 2, ther e is a vertex, s ay x 2 , b elonging to N ( y 1 ) ∩ S . Let x 2 y 2 ∈ M ; such an edge exists, b ecause M matc hes S into M ( S ). Now, since | N ( y 2 ) ∩ S | ≥ 2, there is a vertex, say x 3 , b elonging to N ( y 2 ) ∩ S . If x 3 = x 1 , then the cycle C spanned b y { x 1 , y 1 , x 2 , y 2 } has half o f its e dg es in M . If x 3 6 = x 1 , then we consider the edge in M that saturates x 3 , say x 3 y 3 ∈ M . Since G [ N [ S ]] is finite, after a n umber of steps, w e find some v ertex in N ( S ), say y k , that is joined by an e dge to some x j for j < k . Clearly , the cycle C , with V ( C ) = { x i , y i : j ≤ i ≤ k } and E ( C ) = { x i , y i : j ≤ i ≤ k } ∪ { y i x i +1 : 1 ≤ i ≤ k − 1 } ∪ { x j y k } is even and has half o f its edges in M . Ther efore, M ′ = ( M − E ( C )) ∪ ( E ( C ) − M ) is a perfect matching in G and M 6 = M ′ , in contradiction with the uniq ueness of M in G . Consequently , Ψ( G ) satisfies the accessibility prop er t y , and, a ccording to Theorem 1.6, Ψ( G ) is a g r eedoid. Let us remark t hat the v ery w ell-cov ered g r aph G 1 in Figure 9 has a C 3 and a C 4 ; one edge o f C 4 belo ngs t o the unique p erfect matc hing M of G 1 , bu t none of the edges of C 3 is included in M . Lemma 2.5 No e dge of some C q , for q = 3 or q ≥ 5 , b elongs to a p erfe ct matching in a very wel l-c over e d gr aph . Pro of. If the graph G is very well-cov ered, then by Theor em 1 .11, G has a p erfect matching, say M , and each per fect matching satisfies Prop erty P . Let xy ∈ M . Then, Pr op erty P implies tha t N ( x ) ∩ N ( y ) = ∅ , i.e., xy b elong s to no C 3 in G . F urther, if v ∈ N ( x ) − { y } a nd u ∈ N ( y ) − { x } , Prope rty P ass ur es that v u ∈ E ( G ), i.e., xy belong s to no C q , for q ≥ 5. 8 The very well-co vered gra phs G 1 , G 2 , and G 3 from Figure 1 0 have chordless alterna ting cycles of length 4. In addition, G 3 has an alternating cycle of length 6 , namely , { e 1 , e 2 , ..., e 6 } is alterna ting with r e sp ect to the p erfect matching { e 1 , e 3 , e 5 } . ✇ ✇ ✇ ✇ G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✟ ✟ ✟ ✟ ✟ ✟ G 2 ✇ ✇ ✇ ✇ ✇ ✇ e 1 e 2 e 3 e 4 e 5 e 6 G 3 Figure 10 : V er y well-cov ered graphs, each ha ving more than one p er fect matching. Lemma 2.6 L et G b e a very wel l-c over e d gr aph and M b e one of its maximum m atchings. Ther e exists an alternating cycle with r esp e ct t o M if and only if ther e is an alternating chor d less cycle of length fo ur with r esp e ct t o M . Pro of. Acco rding to Theorem 1 .11, every maximum matching of G is p er fect. Supp o se C 1 is an alternating cycle wit h resp ect to a p er fect matching M = { a i b i : 1 ≤ i ≤ | V ( G ) | / 2 } . Without loss of g enerality , assume that C 1 = { a 1 , b 1 , a 2 , b 2 , a 3 , ..., a k − 1 , b k − 1 , a k , b k } is a cycle on 2 k > 4 vertices with edg es E ( C 1 ) = { a 1 b 1 , b 1 a 2 , a 2 b 2 , b 2 a 3 , ..., a k − 1 b k − 1 , b k − 1 a k , a k b k , b k a 1 } . Since a 1 b 1 ∈ M , b k ∈ N ( a 1 ) − { b 1 } , a 2 ∈ N ( b 1 ) − { a 1 } , Pr op erty P implies that a 2 is adjacent to b k . Thus the cycle on 2 k − 2 vertices C 2 = { a 2 , b 2 , a 3 , ..., a k − 1 , b k − 1 , a k , b k } with edges E ( C 2 ) = { a 2 b 2 , b 2 a 3 , ..., a k − 1 b k − 1 , b k − 1 a k , a k b k , b k a 2 } is still alternating with resp ect to M . It is clear that reducing the size of the cycle in this wa y one ca n easily r each C k − 1 of size 2 k = 4. Accor ding to Lemma 2.5, C k − 1 is a n induced cycle of length four. The conv erse is evident. The conclusion of Lemma 2.6 ca n b e tr ue for no n-well-co vered graphs; e.g., the p erfect matching { e 1 , e 2 , e 3 } of the g raph G 1 from Figure 11 admits alternating cycles of length six and chordless of length fo ur . On the other hand, Le mma 2.6 can fail for well-co vered graphs; e.g., the perfect matc hing { e 1 , e 2 , e 3 } of the graph G 2 from Figure 11 admits a unique alternating cycle of length s ix, while the perfect ma tching { e 1 , e 2 , e 3 , e 4 } of the graph G 3 from Figure 11 admits an alterna ting cycle of length four that ha s chords. ✇ ✇ ✇ ✇ ✇ ✇ e 1 e 2 e 3 G 1 ✇ ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅ e 1 e 2 e 3 G 2 ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇       ❅ ❅ ❅ e 1 e 2 e 3 e 4 G 3 Figure 11 : G 1 is not well-co verd. G 2 , G 3 are well-co vered, but not very well-co vered gr aphs. 9 Theorem 2.7 L et G b e a very well- c over e d gr aph. Then the f ol lowing ar e true: (i) Ψ( G ) is a gr e e doid; (ii) G has a u niquely r estricte d maximum matching; (iii) G has a n alternating cyc le-fr e e maximum matching; (iv) G has a n alternating C 4 -fr e e maximu m matching; (v) every maximum matching in G is alternating cycle-fr e e; (vi) every maximum matching in G is alternating C 4 -fr e e; (vii) al l maximu m match ings of G ar e uniquely r estricte d. Pro of. Fir s tly , Theorem 1 .11 implies that ea ch max imum matching of G is p erfect. (i) = ⇒ (ii) Theorem 2.4 claims that G m ust hav e a unique p erfect matching, say M . Clearly , M is a uniquely r estricted max im um matching. (ii) = ⇒ (iii) It is true, by Theorem 1.1. (iii) = ⇒ (iv) Clear. (iv) = ⇒ (v) In fact, G has a p erfect matc hing, say M , which is alter nating C 4 -free. Hence, by Lemma 2.6, M is alternating cycle-free. Consequently , by Theor em 1.1, G has no other maximum matc hing s , and th us the assertion (v) is true. (v) = ⇒ (vi) Clear. (vi) = ⇒ (vii) By Lemma 2.6 and Theor em 1.1, it follows that every maximum ma tch ing of G is uniquely restr ic ted. (vii) = ⇒ (i) Since all maximum matchings of G ar e both p erfect and uniquely restric ted, it f ollows that G has a unique perfect matc hing. Conse q uent ly , Ψ( G ) is a greedoid, according to Theorem 2.4. Corollary 2.8 Each very wel l-c over e d C 4 -fr e e gr aph G has a unique maximum matching, and, c onse quently, pr o duc es a lo c al maximum stable sets gr e e doid. Pro of. Combining Theorem 1 .11 and Lemma 2.6, w e infer tha t G has a unique p erfect matching. Hence, Theore m 2.4 ensur es that Ψ( G ) is a greedo id. 3 Conclusions In this pap er w e hav e prov e d that a v ery well-cov er ed gr aph pro duce s a lo cal maxim um stable set g r eedoid if and only if it has a uniq ue p erfect matching. ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅    e 1 e 2 e 3 e 4 e 5 x y G Figure 12: The w ell-cov ered graph G has { e 1 , e 2 , e 3 , e 4 , e 5 } as its unique perfect matching. G is not very w ell- c ov er ed, since α ( G ) < 5 = | V ( G ) | / 2 . Nevertheless, the ass ertion is not tr ue for every well-co vered gra phs with a unique p erfect matching; e.g ., Ψ ( G ) is no t a greedo id, where G is the w ell-cov ered graph from Figure 1 2, bec ause { x, y } ∈ Ψ( G ), while { x } , { y } / ∈ Ψ( G ). Theore m 1.12 p oints out to a num b er o f ex- amples of well-cov ered gr aphs whos e families of local maxim um stable gr aphs for m greedoids. F or general well-cov er ed graphs w e pr op ose the fo llowing. Problem 3.1 Char acterize well- c over e d gr aphs pr o ducing lo c al max imu m stable set gr e e doids. 10 References [1] A. Bj¨ or ne r , G. M. Ziegler, Int ro duct ion to gr e e doids , in N. White (ed.), Matr oid Applic ations , 284 -357, Cambridge Univ e rsity Press, 1992 . [2] N. Dean, J. Zito, W el l-c over e d gr aphs and extendability , Discr ete Mathema tics 12 6 (199 4 ) 67-80 . [3] R. W. Deming, Indep endenc e numb ers of gr aphs - an extension of the K¨ onig–Egerv´ a ry the or em , Discrete Mathema tics 27 (19 7 9) 23– 33. [4] E . Egerv a ry , On c ombinatori al pr op erties of matric es , Mat. Lap ok 38 (193 1) 16- 28. [5] O. F av aron, V ery wel l- c over e d gr aphs , Discrete Mathematics 42 (198 2) 177-1 8 7. [6] A. Finbow, B. Hartnell, R. J. Nowak owski, A ch ar acterization of we l l- c over e d gr aphs of girth 5 or g r e ater , Jour nal of Co m binatorial Theo r y , B 57 (19 93) 44-6 8 . [7] M. C. Go lumbic, T . Hirst, M. Lewenstein, Uniquely r estricte d matchi ngs , Algorithmica 31 (200 1) 139-15 4. [8] B. Ha r tnell, M. D. P lummer, On 4 -c onne cte d claw-fr e e wel l-c over e d gr aphs , Discrete Applied Mathematics 64 (199 6) 57- 6 5. [9] S. Krogdahl, The dep endenc e gr aph for b ases in m atro ids , Discrete Ma thematics 19 (1977) 47- 59. [10] B . Korte, L. Lov´ asz, R. Schrader, Gr e e doids , Spr ing er-V erlag, Berlin, 19 91. [11] D. K¨ onig, Gr aphen un d Matrizen , Ma t. Lap ok 38 (19 3 1) 116-1 19. [12] V. E. Levit, E. Mandrescu, Wel l-c over e d and K¨ onig-Egerv´ ary gr aphs , Congres sus Nu- merantium 130 (1998) 20 9 -218 . [13] V. E. Levit, E. Mandr escu, On the structure of α -stable gr aphs , Discrete Mathematics 236 (200 1) 227-2 43. [14] V. E . Levit, E . Mandrescu, Unicycle bip artite gr aphs with only uniquely r est ricte d max- imum matchi ngs , Proc eedings of the Third International C o nference on Com bina to rics, Computability a nd Lo gic, (DMTCS’1), Springer, (C. S. Calude, M. J . Dinneen and S. Sburlan eds.) (2001 ) 151-158 . [15] V. E. Levit, E. Mandrescu, A new gr e e doid: the family of lo c al m aximum stable sets of a for est , Discr ete Applied Mathematics 1 24 (200 2) 91-101 . [16] V. E. Levit and E. Mandresc u, On α + -stable K¨ onig–Egerv´ ary gr aphs , Discrete Mathe- matics 263 (2003) 179 –190 . [17] V. E . Lev it, E. Mandresc u, L o c al maximum stable sets in bip artite gr aph s with u niquely r estricte d maximu m matchings , Discrete Applied Mathematics 132 (200 4) 163 -174. [18] V. E. Levit, E. Mandres cu, On α - critic al e dges in K¨ onig-Egervary gr aph s , Discrete Math- ematics 306 (2006) 168 4-16 9 3. 11 [19] V. E . Levit, E . Mandrescu, T riangle-fr e e gr aphs with uniquely r estricte d maximum match- ings and t heir c orr esp onding gr e e doids , Discrete Applied Mathema tics 155 (2007) 2414– 2425. [20] V. E. Le vit, E. Mandrescu, S ome struct ur al pr op erties of very well- c over e d gr aphs , Co n- gressus Numerantium 186 (200 7) 97–10 6. [21] V. E. Levit, E . Mandrescu, Wel l-c over e d gr aphs and gr e e doids , Pro cee dings of the 1 4 th Computing: The Australa sian Theory Symp osium (CA TS20 0 8), W ollongo ng, NSW, Con- ferences in Research and Practice in Information T echnology V olume 77 (20 08) 89-9 4. [22] V. E. Levit, E. Mandre scu, The clique c or ona op er ation and gr e e doids , Combinatorial Optimization a nd Applicatio ns, Second International Confere nce, COCOA 2008, Lecture Notes in Computer Science 5165 (200 8) 384 -392. [23] V. E. Levit, E . Mandr escu, In t erval gr e e doids and lo c al maximum stable sets in gr aphs , (2008) arXiv:0 811.4 089 v1 [ma th.CO] 13 pp. [24] V. E. Levit, E. Mandrescu, On lo c al maximum stable sets of the c or ona of a p ath with c omplete gr aphs , Pr o ceedings of the 6 th Congress of Romanian Mathematicians (2009) 565-5 69. [25] V. E. Levit, E. Mandres cu, Gr aph op er ations that ar e go o d for gr e e doids , Discrete Applied Mathematics 158 (2010) 14 18-14 23. [26] V. E. Levit, E. Mandrescu, V ery wel l-c over e d gr aphs of girth at le ast four and lo c al m ax - imum stable set gr e e doids , Discr e te Mathema tics, Algo rithms a nd Applicatio ns (2010) (accepted). a rXiv:1008 .2897 v1[cs.DM] 7 pp. [27] G. L. Nemhauser, L. E. T ro tter, Jr., V ertex p ackings: structu r al pr op erties and algo- rithms , Mathematica l Pro gramming 8 (19 75) 232- 248. [28] M. D . Plummer , Some c overing c onc epts in gr aphs , Jo urnal of Com bina torial Theo ry 8 (1970) 91- 98. [29] E . P risner, J. T opp, P . D. V estergaard, Wel l- c over e d simplicial, chor dal, and cir cu lar ar c gr aphs , J ournal of Gr a ph Theory 21 (1996) 1 13-11 9. [30] G. Ravindra, Wel l-c over e d gr aphs , J. Co mbin. Inform. System Sci. 2 (19 77) 20-21 . [31] J . A. Staples , On some su b-classes of wel l-c over e d gr aphs , Ph. D. Thesis, V anderbilt Univ ersity , 1975. [32] F. Sterb oul, A char acterization of t he gr aphs in which the tr ansversal n u mb er e qu als t he matching numb er , Journal of Combinatorial Theory Series B 2 7 (1979 ) 228–22 9 . [33] J . T opp, L. V o lkmann, On the wel l c over e dness of p r o ducts o f gr aphs , Ar s Combinatoria 33 (199 2) 199-21 5. 12