Graph Operations that are Good for Greedoids

Graph Op erations that are Go o d for G reedoids V adim E. Levit Departmen t of Computer Science and Mathematics Ariel Univ e rsit y Cen ter of Samaria, ISRAEL levitv@ariel.ac.il Eugen Mandrescu Departmen t of Computer Science Holon Institute of T ec hnology , ISRAEL eugen m@hit.ac.il Abstract S is a lo c al maximum stable set of a graph G , and w e write S ∈ Ψ( G ), if the set S is a maxim um stable set of the subgraph induced by S ∪ N ( S ), where N ( S ) is the neighborho od of S . In [4] w e hav e prov ed that Ψ( G ) is a greedoid for every forest G . The cases of bipartite graphs and triangle-free graphs w ere analyzed in [5] and [6], respectively . In this pap er w e give necessary and sufficient conditions for Ψ( G ) to form a greedoid, where G is: (a) the disjoin t u nion of a family of graphs; (b) the Zyko v sum of a famil y of graphs; (c) the corona X ◦ { H 1 , H 2 , ..., H n } obtained by joining each vertex x of a graph X to all the vertices of a graph H x . Keywords: Corona, Zyko v sum, greedoid, local maximum stable set 1 In tro duction Throughout this pap er G = ( V , E ) is a simple (i.e., a finite, undirected, loo pless and without m ultiple edges) graph with vertex set V = V ( G ) and edge set E = E ( G ) . If X ⊂ V , then G [ X ] is the subgra ph of G spanned by X . By G − W w e mean the subgr aph G [ V − W ], if W ⊂ V ( G ). W e also denote by G − F the par tial subgraph of G obtained b y deleting the edges of F , for F ⊂ E ( G ), and w e write shortly G − e , whenever F = { e } . The neighb orho o d of a vertex v ∈ V is the set N G ( v ) = { w : w ∈ V and vw ∈ E } . W e denote the n eig hb orho o d of A ⊂ V b y N G ( A ) = { v ∈ V − A : N ( v ) ∩ A 6 = ∅ } and its close d neighb orho o d b y N G [ A ] = A ∪ N ( A ), or shortly , N ( A ) and N [ A ], if no ambiguity . K n , P n , C n denote respe ctiv ely , the complete g raph on n ≥ 1 vertices, the c hordless path on n ≥ 2 v ertices, and the chordless cycle on n ≥ 3 vertices, respec tiv ely . A stable set in G is a set of pair wise non-a djacen t vertices. A stable set of maximum s ize will b e referred to a s a maximum stable set of G , and the stability nu mb er of G , denoted b y 1 α ( G ), is the cardinality of a maximum stable set in G . In the sequel, b y Ω( G ) we denote the set of all maximum stable sets of the graph G . An y stable set S is max ima l (with resp ect to set inclusio n) in G [ N [ S ]], but is no t necessa r- ily , a maximum one. A set A ⊆ V ( G ) is a lo c al maximum stable set of G if A is a maximum stable set in the subgr aph induced b y N [ A ], i.e., A ∈ Ω( G [ N [ A ]]), [4]. Let Ψ( G ) stand for the set o f all lo cal maximum stable sets o f G . Clearly , every stable set containing only p endan t vertices belongs to Ψ( G ). Nevertheless, there exis t lo cal maximum stable sets that do not contain penda n t vertices. F or instance, { e, g } ∈ Ψ( G ), where G is the graph from Figur e 1. ✇ ✇ ✇ ✇ ✇ ✇ ✇ ❅ ❅ ❅ a b c d g f e W Figure 1: A graph having v arious lo cal maximum stable sets . The following theor em c o ncerning ma xim um sta ble sets in gener al gr aphs, due to Nemhauser and T ro tter J r. [10], shows that some stable sets ca n b e enlar g ed to ma xim um stable sets. Theorem 1. 1 [10] Every lo c al maximum stable set of a gr aph is a subset of a maximu m stable set. Nemhauser and T rotter Jr. int erpret this as sertion as a sufficient lo cal optimality con- dition for a binar y integer progra mming formulation of the weigh ted max im um stable s et problem, a nd use it to prove an outstanding re sult claiming that integer parts of solutions of the corresp onding linea r programming r elaxation main tain the same v alues in the optimal solutions of its binary int eger prog ramming counterpart. In other words, it mea ns tha t a well-kno wn branch-and-bound heuristic for general integer prog r amming problems turns o ut to be an exa c t a lgorithm so lving the weigh ted max imum stable set problem. The graph W fro m Figure 1 has the pro p erty that every S ∈ Ω( W ) contains so me lo- cal maximum sta ble set, but these lo cal maximum s ta ble sets a re of different cardina lities: { a, d, f } ∈ Ω( W ) and { a } , { d, f } ∈ Ψ( W ), while for { b, e, g } ∈ Ω( W ) only { e, g } ∈ Ψ( W ). How ev er, there e x ists a graph G satisfying Ψ( G ) = Ω( G ), e.g., G = C n , for n ≥ 4. A greedoid is a set system generalizing the no tion of a matroid. Definition 1.2 [1], [2] A gr e e doid is a p air ( E , F ) , wher e F ⊆ 2 E is a non-empty set system satisfying the fol low ing c onditions: A c c essibility: for every n on-empty X ∈ F ther e is an x ∈ X such that X − { x } ∈ F ; Exchange: for X , Y ∈ F , | X | = | Y | + 1 , ther e is an x ∈ X − Y such that Y ∪ { x } ∈ F . Let us observe that { d, g } ∈ Ψ( W ), while { d } , { g } / ∈ Ψ( W ), where W is the gra ph depicted in Figure 1. How ev er, it is worth mentioning that if Ψ( G ) is a gr eedoid and S ∈ Ψ( G ), | S | = k ≥ 2, then acco rding to the accessibility pr operty , one can build a chain { x 1 } ⊂ { x 1 , x 2 } ⊂ ... ⊂ { x 1 , ..., x k − 1 } ⊂ { x 1 , ..., x k − 1 , x k } = S 2 such that { x 1 , x 2 , ..., x j } ∈ Ψ( G ), for all j ∈ { 1 , ..., k − 1 } . F or example, { a } ⊂ { a, b } ⊂ S is an a ccessibilit y chain of the set S = { a, b, c } ∈ Ψ( G 2 ), where G 2 is prese n ted in Figure 2. In [4] it is pr o v ed the following re sult. Theorem 1. 3 F or every tr e e T , Ψ( T ) is a gr e e doid on its vertex set. The case of bipar tite g r aphs owning a unique cycle , w ho se family of lo cal maxim um sta ble sets forms a greedo id is a nalyzed in [3] (for an example, see the gr a ph G 1 from Figure 2). I n general, lo cal maximum stable sets of bipartite gr aphs were treated in [5], while for triangle- free graphs w e r efer the reader to [6] for details. Nev ertheless, there exist non-bipartite and also non-triang le-free graphs whose families of lo cal maxim um s ta ble s ets form gr eedoids. F o r instance, the families Ψ( G 2 ) , Ψ( G 3 ) , Ψ( G 4 ) of the graphs in Figure 2 a re greedoids. ✇ ✇ ✇ ✇ ✇ ✇ ✇    G 1 ✇ ✇ ✇ ✇ ✇    a b c G 2 ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅ G 3 ✇ ✇ ✇ ✇ ✇ ✇    ❅ ❅ ❅   ❅ ❅ ❅ G 4 Figure 2: Graphs whos e family o f loca l max imum stable sets for m g reedoids. In this note we present ” if and only if ” conditions for Ψ( G ) to be a greedoid, where G is the disjoint union, or the Zykov sum, o r the co rona of a family of g r aphs. 2 Disjoin t union and Zyk o v sum of graphs Let G b e the disjoint union of the family of graphs { G i : 1 ≤ i ≤ p } , p ≥ 2, i.e., V ( G ) = V ( G 1 ) ∪ V ( G 2 ) ∪ ... ∪ V ( G p ) and E ( G ) = E ( G 1 ) ∪ E ( G 2 ) ∪ ... ∪ E ( G p ) , under the as s umption that V ( G i ) ∩ V ( G j ) = ∅ , 1 ≤ i < j ≤ p . Clea rly , α ( G ) = α ( G 1 ) + α ( G 2 ) + ... + α ( G p ) and S ⊆ V ( G ) is stable if and only if every S ∩ V ( G i ) , 1 ≤ i ≤ p , is sta ble. Moreov er, o ne c a n ea sily prov e the following result. Prop osition 2.1 If G is the disjoint u nion of the family of gr aphs { G i : 1 ≤ i ≤ p } , p ≥ 2 , then: (i) S ∈ Ψ( G ) if and only if S ∩ V ( G i ) ∈ Ψ ( G ) , 1 ≤ i ≤ p ; (ii) Ψ( G ) is a gr e e doid if and only if every Ψ( G i ) , 1 ≤ i ≤ p , is a gr e e doid. Recall that the Zyko v su m of the graphs G i , 1 ≤ i ≤ p, p ≥ 2 , is the graph Z = Z [ G 1 , ..., G p ] = G 1 + G 2 + ... + G p having V ( Z ) = V ( G 1 ) ∪ ... ∪ V ( G p ) , E ( Z ) = E ( G 1 ) ∪ ... ∪ E ( G p ) ∪ { v i v j : v i ∈ V i , v j ∈ V j , 1 ≤ i < j ≤ p } . 3 ✇ ✇ ✇ ✇ ✇ ✟ ✟ ✟ ✟ ✟ ✟       ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ G 1 ✇ ✇ ✇ ✇ ✇ ✇ ✟ ✟ ✟ ✟ ✟ ✟       ❍ ❍ ❍ ❍ ❍ ❍ ✟ ✟ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ❍ ❍ ❅ ❅ ❅ ❅ ❅ ❅ G 2 Figure 3: G 1 = Z [ K 2 , P 3 ] and G 2 = Z [ P 3 , P 3 ]. Clearly , α ( Z ) = max { α ( G i ) : 1 ≤ i ≤ p } . If a ll G 1 , G 2 , ..., G p , p ≥ 2 , ar e complete graphs, then Z is complete, as well. In this ca s e, we hav e Ψ( Z ) = Ω( Z ) = {{ v } : v ∈ V ( Z ) } and Ψ( Z ) is, evidently , a greedo id. Lemma 2.2 If Z = Z [ G 1 , ..., G p ] , then min {| S | : S ∈ Ψ( Z ) } ≥ max 2 { α ( G i ) : 1 ≤ i ≤ p } , wher e max 2 { α i : 1 ≤ i ≤ p } is a se c ond lar gest numb er of the se quen c e. Pro of. Notice that if S ⊆ V ( Z ) is stable, then ther e is s o me i ∈ { 1 , 2 , ..., p } such that S ⊆ V ( G i ). Hence, if S ∈ Ψ( Z ), then S ∈ Ψ( G k ) for some k ∈ { 1 , 2 , ..., p } , a nd, in addition, | S | ≥ max { α ( G i ) : 1 ≤ i ≤ p, i 6 = k } . Since max 2 { α ( G i ) : 1 ≤ i ≤ p } = min 1 ≤ k ≤ p (max { α ( G i ) : 1 ≤ i ≤ p, i 6 = k } ) , we get that min {| S | : S ∈ Ψ( Z ) } ≥ max 2 { α ( G i ) : 1 ≤ i ≤ p } , which c ompletes the pro of. Let us observe that for the g raphs G 1 = Z [ K 2 , P 3 ] and G 2 = Z [ P 3 , P 3 ] (depicted in Figure 3), Ψ( G 1 ) is a greedoid, while Ψ( G 2 ) is not a gr eedoid, b ecause { v } / ∈ Ψ( G 2 ), for every v ∈ V ( G 2 ). Prop osition 2.3 L et Z = Z [ G 1 , ..., G p ] b e such that α ( Z ) > 1 . Then Ψ( Z ) is a gr e e doid if and only if the fol lowing assertions ar e true: (i) al l Ψ( G i ) , 1 ≤ i ≤ p , ar e gr e e do ids; (ii) ther e is a unique k ∈ { 1 , 2 , .., p } su ch that G k is not c omplete; (iii) Ψ( Z ) = Ψ ( G k ) . Pro of. T aking in to account the definition of Z , it follows that at least one of the gr a phs G i is not complete, beca us e a nd α ( Z ) > 1. Assume that Ψ( Z ) is a gr e e doid and le t { a } ∈ Ψ( Z ). Hence we infer that min {| S | : S ∈ Ψ( Z ) } = 1 . 4 Consequently , by Lemma 2.2, w e get 1 ≥ ma x 2 { α ( G i ) : 1 ≤ i ≤ p } . Thu s a ll G i , 1 ≤ i ≤ p but o ne must b e co mplete graphs. Suppo se G k is the unique non-complete gra ph. Then a ∈ V ( G k ) and α ( Z ) = α ( G k ). Clearly , all Ψ( G i ) , 1 ≤ i ≤ p, i 6 = k , ar e g reedoids. In addition, { v } / ∈ Ψ ( Z ), for every v ∈ V ( Z ) − V ( G k ), b ecause V ( G k ) ⊆ N Z ( v ) and α ( G k ) > 1. It follows that S ⊆ V ( G k ), for every S ∈ Ψ( Z ). Mor eo v er, one c a n say that S ∈ Ψ( G k ), i.e., Ψ ( Z ) ⊆ Ψ( G k ). Otherwise, if so me A ∈ Ψ ( Z ) do es not belo ng to Ψ( G k ), it follows tha t there is a stable set B in N G k [ A ], lar ger than A . Since B is stable in Z , a s well, and B ⊆ N G k [ A ] ⊆ N Z [ A ], it implies A / ∈ Ψ ( Z ), in contradiction with the choice o f A . O n the other hand, taking int o account that no stable se t in Z can meet bo th V ( G k ) and V ( Z ) − V ( G k ), it follows that Ψ( G k ) ⊆ Ψ( Z ) is true, as w ell. In other words, we infer that Ψ( Z ) = Ψ( G k ), whic h ensures that Ψ( G k ) is a gr eedoid. The conv erse is clear. 3 Corona of graphs Let X be a gr aph with V ( X ) = { v i : 1 ≤ i ≤ n } , a nd { H i : 1 ≤ i ≤ n } be a family of graphs. Jo ining e ac h v i ∈ V ( X ) to all the v ertices of H i , we obtain a new gr aph, which w e denote by G = X ◦ { H 1 , H 2 , ..., H n } (see Figur e 4 for an example, where X = K 3 + v 3 v 4 ). If H 1 = H 2 = ... = H n = H , we write G = X ◦ H , and in this case, G is ca lled the c or ona of X and H . ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁                ❅ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ x y z u t v 1 v 2 v 3 v 4 K 3 K 2 P 3 K 1 G Figure 4: G = ( K 3 + v 3 v 4 ) ◦ { K 3 , K 2 , P 3 , K 1 } is a well-cov ered gr a ph. Let us notice that G = X ◦ { H 1 , H 2 , ..., H n } has α ( G ) = α ( H 1 ) + α ( H 2 ) + ... + α ( H n ). Let us consider the g raph G depicted in Figur e ?? . Notice that: • S 1 = { x, z , v 4 } ∈ Ψ( G ) a nd a lso S 1 ∩ V ( H ) ∈ Ψ ( H ), for every H ∈ { K 3 , K 2 , P 3 , K 1 } ; • the set S 2 = { y , v 2 } is stable, but S 2 / ∈ Ψ( G ), b ecause { y , u, v 3 } ⊆ N G [ S 2 ] a nd it is stable and larg er than S 2 ; • { v 4 } , { v 2 , v 4 } ∈ Ψ( K 3 + v 3 v 4 ), but { v 4 } , { v 2 , v 4 } / ∈ Ψ( G ); • { y , v 4 } / ∈ Ψ( G ), sinc e { y , t, v 3 } ⊆ N G [ { y , v 4 } ] and it is stable and larger than { y, v 4 } ; • { y } ∈ Ψ ( K 3 ) , { x, z } ∈ Ψ( P 3 ) and also { x, y , z } ∈ Ψ ( G ); • the set S 3 = { y , v 3 } is stable a nd S 2 ∩ V ( H ) ∈ Ψ( H ), for each H ∈ { K 3 , K 2 , P 3 , K 1 } , but S 3 / ∈ Ψ( G ). 5 Lemma 3.1 L et G = X ◦ { H 1 , H 2 , ..., H n } , wher e V ( X ) = { v i : 1 ≤ i ≤ n } , n ≥ 2 . Then the fol lowing assertions ar e tr ue : (i) Ψ( H i ) ⊆ Ψ( G ) , 1 ≤ i ≤ n ; (ii) if v i ∈ S ∈ Ψ( G ) , then H i is c omplete, and S ∩ V ( H k ) 6 = ∅ , for e ach v k ∈ N X ( v i ) ; (iii) if S ∈ Ψ( G ) , then S ∩ V ( H i ) ∈ Ψ( H i ) , 1 ≤ i ≤ n ; (iv) if S is a stable set in G su ch that: S ∩ V ( H i ) ∈ Ψ( H i ) , 1 ≤ i ≤ n , and for every v i ∈ S ∩ V ( X ) , H i is a c omplete gr aph , while S ∩ V ( H k ) 6 = ∅ , for al l v k ∈ N X ( v i ) , then S ∈ Ψ( G ) . Pro of. (i) Let A ∈ Ψ( H i ). Then, N G [ A ] = N H i [ A ] ∪ { v i } and, thus, A is a maximum sta ble set in N G [ A ], as well, i.e., A ∈ Ψ ( G ). C o nsequen tly , Ψ( H i ) ⊆ Ψ( G ) for each i ∈ { 1 , 2 , ..., n } . (ii) If v i ∈ S and there ar e non-a djacen t vertices x, y ∈ V ( H i ), then the s et S ∪ { x, y } − { v i } is stable in N G [ S ], lar ger than S , in contradiction with S ∈ Ψ( G ). Therefor e, H i m ust be a complete gr aph. Assume that S ∩ V ( H k ) = ∅ for some v k ∈ N X ( v i ). If N ( v k ) ∩ S = { v i } , then for every x ∈ V ( H i ), the se t S ∪ { v k , x } − { v i } is stable in N G [ S ] and larger than S , in contradiction with S ∈ Ψ( G ). If N ( v k ) ∩ S = { v i , v j 1 , v j 2 , ..., v j q } , then S ∪ { x, v k } ∪ { x j 1 , x j 2 , ..., x j q } − { v i , v j 1 , v j 2 , ..., v j q } is a stable set in N G [ S ] for every x ∈ V ( H i ) and each x j t ∈ V ( H j t ) , 1 ≤ t ≤ q , larg er than S , in contradiction with S ∈ Ψ( G ). Consequently , S ∩ V ( H k ) 6 = ∅ , for every v k ∈ N X ( v i ). (iii) Assume that S ∈ Ψ( G ). If S j = S ∩ V ( H j ) / ∈ Ψ( H j ), then v j / ∈ S (beca use S j 6 = ∅ ) and there is some stable se t A j ⊆ N H j [ S j ] larg er than S j . Since N H j [ S j ] ∪ { v j } = N G [ S j ] and v j / ∈ A , we get that ( S − S j ) ∪ A is a s table set included in N G [ S ] and | S | < | ( S − S j ) ∪ A | , in contradiction with S ∈ Ψ( G ). The r efore, S ∩ V ( H i ) ∈ Ψ ( H i ) for every i ∈ { 1 , 2 , ..., n } . (iv) W e ha ve to prove that | A | ≤ | S | for every stable set A ⊆ N G [ S ]. Let us define the following partitions o f the sets A and S : A = A 1 ∪ A 2 ∪ A 3 , S = S 1 ∪ S 2 ∪ S 3 where A 1 = S v j ∈ ( A − S ) [ A ∩ ( V ( H j ) ∪ { v j } )] , S 1 = S v j ∈ ( A − S ) [ S ∩ ( V ( H j ) ∪ { v j } )] A 2 = S v j ∈ V ( X ) − ( A ∪ S ) [ A ∩ ( V ( H j ) ∪ { v j } )] , S 2 = S v j ∈ V ( X ) − ( A ∪ S ) [ S ∩ ( V ( H j ) ∪ { v j } )] A 3 = S v j ∈ S [ A ∩ ( V ( H j ) ∪ { v j } )] , S 3 = S v j ∈ S [ S ∩ ( V ( H j ) ∪ { v j } )] . Our in ten t is to show that | A k | ≤ | S k | , 1 ≤ k ≤ 3 , 6 which w ill lea d us to the co nc lus ion that | A | ≤ | S | . Case 1. v j ∈ A − S . Since S is maximal in N G [ S ], we infer that v j y ∈ E ( G ), for some y ∈ S . If y ∈ V ( H j ), then S ∩ V ( H j ) 6 = ∅ . Otherwise , y ∈ V ( X ) and according with the hypo thesis on S , ag ain S ∩ V ( H j ) 6 = ∅ . Therefo r e, we get that | A ∩ ( V ( H j ) ∪ { v j } ) | = 1 ≤ | S ∩ ( V ( H j ) ∪ { v j } )) | , which implies | A 1 | ≤ | S 1 | . Case 2. v j ∈ V ( X ) − ( A ∪ S ). Since S ∩ V ( H j ) ∈ Ψ( H j ), w e hav e that | A ∩ V ( H j ) | ≤ | S ∩ V ( H j ) | . T ogether with the condition v j ∈ V ( X ) − ( A ∪ S ) it gives | A ∩ ( V ( H j ) ∪ { v j } ) | ≤ | S ∩ ( V ( H j ) ∪ { v j } )) | . Therefore, it follows that | A 2 | ≤ | S 2 | . Case 3. v j ∈ S . According with the hypo thes is o n S , H j is a clique. Consequently , we obta in | A ∩ ( V ( H j ) ∪ { v j } ) | ≤ 1 ≤ | S ∩ ( V ( H j ) ∪ { v j } )) | , which e nsures that | A 3 | ≤ | S 3 | . The following theorem generalizes some par tial findings from [7], [8], [9]. Theorem 3. 2 If G = X ◦ { H 1 , H 2 , ..., H n } and H 1 , H 2 , ..., H n ar e non-empty gr ap hs, t hen Ψ( G ) is a gr e e doi d if and only if every Ψ ( H i ) , i = 1 , 2 , ..., n , is a gr e e doid. Pro of. Assume that Ψ( G ) is a gr eedoid. According to Lemma 3 .1 (i) , (iii) , w e get that Ψ( H i ) = { S ∩ V ( H i ) : S ∈ Ψ ( G ) } , 1 ≤ i ≤ n. Hence, ev ery Ψ( H i ) s a tisfies b oth access ibilit y prop ert y and exc hange pr operty , i.e ., Ψ( H i ) is a greedoid. Conv ersely , suppose tha t ev ery Ψ( H i ) , 1 ≤ i ≤ n , is a greedoid. Firstly , we show that Ψ( G ) satisfies the a ccessibilit y pro perty . Let S ∈ Ψ( G ) and S 6 = ∅ . If v i ∈ S ∩ V ( X ), then N X ( v i ) ∩ S = ∅ , V ( H i ) ∩ S = ∅ , while, by Lemma 3.1 (ii) , S ∩ V ( H k ) 6 = ∅ ho lds for every v k ∈ N ( v i ). Hence, we ma y infer that S − { v i } ∈ Ψ( G ). If S ∩ V ( X ) = ∅ , then there is so me i ∈ { 1 , 2 , ..., n } , s uc h that S i = S ∩ V ( H i ) 6 = ∅ and S i ∈ Ψ( H i ), a ccording to Lemma 3.1 (iii) . Since Ψ( H i ) is a greedoid, there is some x ∈ S i such that S i − { x } ∈ Ψ( H i ). Since N G [ S − { x } ] ∩ V ( H i ) = N H i [ S i − { x } ] , while N G [ S − { x } ] ∩ V ( H j ) = N G [ S ] ∩ V ( H j ) for every j 6 = i , w e ma y conclude that S − { x } ∈ Ψ( G ). W e check now the exchange prop erty . Let S 1 , S 2 ∈ Ψ( G ) b e with | S 1 | = | S 2 | + 1. 7 Case 1 . S 1 ∩ V ( H j ) = S 2 ∩ V ( H j ) for all j ∈ { 1 , 2 , ..., n } . Then there is some v i ∈ S 1 − S 2 , bec ause | S 1 | > | S 2 | . Hence, it follo ws S 1 ∩ V ( H i ) = ∅ , whic h ensures that also S 2 ∩ V ( H i ) = ∅ . By Lemma 3.1 (ii) , we hav e that, for every v k ∈ N G ( v i ), S 1 ∩ V ( H k ) 6 = ∅ which implies that also S 2 ∩ V ( H k ) 6 = ∅ . Co nsequen tly , using Lemma 3.1 (iv) , we may infer that S 2 ∪ { v k } ∈ Ψ( G ). Case 2. There is some i ∈ { 1 , 2 , ..., n } , such that A 1 = S 1 ∩ V ( H i ) is larg er than A 2 = S 2 ∩ V ( H i ). Since A 1 , A 2 ∈ Ψ ( H i ) and Ψ( H i ) is a greedoid, there must ex ist s ome x ∈ A 1 − A 2 , such that A 2 ∪ { x } ∈ Ψ( H i ). Hence, we get also that S 2 ∪ { x } ∈ Ψ( G ). Consequently , Ψ( G ) satisfies the exchange prop ert y . In conclusion, Ψ( G ) for ms a greedoid on the vertex set of G . Corollary 3.3 Ψ( X ◦ H ) is a gr e e doid if and only if Ψ ( H ) is a gr e e doid. 4 Conclusions and future w ork Let { H 1 , ..., H n } b e a family o f graphs indexed b y the vertex set { 1 , 2 , .., n } o f a gra ph H 0 . The gra ph denoted by H 0 [ H 1 , H 2 , ..., H n ] is defined a s: V ( H 0 [ H 1 , H 2 , ..., H n ]) = { 1 } × V ( H 1 ) ∪ ... ∪ { n } × V ( H n ) , and ( i, x ) , ( j, y ) ∈ V ( H 0 [ H 1 , H 2 , ..., H n ]) are adjac en t if a nd only if either (i) ij ∈ E ( H 0 ) or (ii) i = j and xy ∈ E ( H i ). F or instance, K n [ H 1 , H 2 , ..., H n ] is the disjoin t union of the graphs H 1 , ..., H n ; K n [ H 1 , H 2 , ..., H n ] is the Zyko v sum of H 1 , ..., H n ; while if H 1 = H 2 = ... = H n , then H 0 [ H 1 , H 2 , ..., H n ] is known as lexicogra phic pro duct H 0 • H 1 . 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