Maximal Pivots on Graphs with an Application to Gene Assembly

Maxima l Pivots on Graphs with an Applicat ion to Gene Assembly Rob ert Brijder ∗ , Hendrik Jan Ho ogeb o o m L eiden Institute of A dvanc e d Computer Scienc e, L eiden University , The Netherlands Abstract W e consider principal piv ot transform ( pivot ) on graphs. W e define a natural v ariant of this op eration, called dual pivot, and show that b oth the kernel and the set of maximally applicable piv ots o f a graph are inv ariant under this op er- ation. The result is motiv ated b y a nd applicable to the theor y o f gene assembly in ciliates. Keywor ds: principal pivot transform, algebraic graph theory, ov erla p gr aph, gene assembly in ciliates 1. In tro duction The pivot ope ration, due to T uck er [18], par tially (comp onent-wise) in verts a g iven matrix. It appea rs natura lly in many ar eas including mathematical progra mming a nd numerical ana ly sis, se e [17] for a survey . Ov er F 2 (whic h is the natural setting to consider for g raphs), the piv ot op eration ha s, in addition to matrix and g raph in terpreta tions [11], a lso an interpretation in terms of delta matroids [1]. In this paper we define the dual pivot , which has a n identical effect on g raphs as the (re g ular) pivot, how ever the co ndition for it to b e applica ble differs. The main result o f the paper is that any tw o graphs in the same orbit under dual pivot hav e the same family o f maximal pivots (cf. Theorem 16), i.e., the same family o f maxima lly par tial inverses o f that matr ix. This result is obtained by combining each of the aforementioned interpretations of piv ot. This r esearch is motiv ated by the theo r y of ge ne assembly in ciliates [9], which is re c a lled in Section 7. Without the context of g ene assembly this main result (Theo rem 1 6) is surpr is ing; it is not found in the extensive literature on pivots. It fits how ever with the intuition and r esults fro m the string based mo del of gene a ssembly [4], and in this pap er we for m ulate it for the more ∗ Corresp onding author Email addr ess: rbrijder@li acs.nl (Rob ert Br ijder) Pr eprint submitted to Elsevier July 10, 2021 general graph based mo del. It is unders to o d and prov en here us ing completely different techniques, algebraica l rather than combinatorial. 2. Notation and T erminology The field with tw o elements is denoted by F 2 . Our matrix computations will be over F 2 . Hence addition is equal to t he logica l exclusiv e-o r , also denoted by ⊕ , and m ultiplication is equal to the log ic al conjunction, also denoted by ∧ . These op eratio ns carr y ov er to s ets, e.g ., for s e ts A, B ⊆ V and x ∈ V , x ∈ A ⊕ B iff ( x ∈ A ) ⊕ ( x ∈ B ). A set system is a tuple M = ( V , D ), where V is a finite set and D ⊆ 2 V is a set of subsets o f V . Let min ( D ) (max( D ), res p.) be the family of minimal (maximal, resp.) sets in D w.r.t. set inclusion, and let min( M ) = ( V , min( D )) (max( M ) = ( V , max( D )), resp.) b e the co rresp onding set sys tems. Let V be a finite set, and A be a V × V -matrix (over a n ar bitrary field), i.e., A is a matrix wher e the r ows and columns of A are identified by elements of V . Therefore, e.g., the following matrices with V = { p , q } a re equal:  p q p 1 1 q 0 1  and  q p q 1 0 p 1 1  . F o r X ⊆ V , the pr incipal submatrix o f A w.r.t. X is denoted by A [ X ], i.e., A [ X ] is the X × X -matrix obtained fro m A by restricting to rows and columns in X . Similarly , we define A \ X = A [ V \ X ]. Notions such as ma trix inv ersion A − 1 and deter minant det ( A ) are well defined for V × V -matrices. By conv ention, det( A [ ∅ ]) = 1 . A set X ⊆ V is called dep endent in A iff the columns of A co rresp onding to X are linearly dep endent. W e define P A = ( I , D ) to be the pa rtition of 2 V such tha t D ( I , r esp ectively) cont ains the dep endent (indep endent, r e s pe c tively) subsets of V in A . By co nv ent ion, ∅ ∈ I . The sets in max( I ) are called the bases of A . W e hav e that P A = ( I , D ) is uniquely determined by max( I ) (and the set V ). Sim ilar ly , P A is uniquely determined by min( D ) (and the set V ). The s e prop erties are sp ecifically us e d in matr o id theory , where a matroid may b e describ ed by its indep e ndent s ets ( V , I ), b y its family o f ba ses ( V , max( I )), or by its circuits ( V , min( D )). Mo reov er, for each bas is X ∈ max( I ), | X | is equa l to the rank r of A . W e consider undire c ted graphs without pa rallel edges, how ever we do allow lo ops. F o r a gra ph G = ( V , E ) we use V ( G ) and E ( G ) to denote its s e t of vertices V and se t of edges E , res pec tively , where for x ∈ V , { x } ∈ E iff x ha s a lo op. F or X ⊆ V , we denote the subgraph of G induced by X as G [ X ]. With a graph G one a sso ciates its adjacency ma tr ix A ( G ), which is a V × V - matrix ( a u,v ) ov er F 2 with a u,v = 1 iff { u, v } ∈ E . The matrices cor resp onding to graphs ar e precisely the symmetric F 2 -matrices; lo ops corr esp onding to diagonal 1’s. Note that for X ⊆ V , A ( G [ X ]) = ( A ( G ))[ X ]. 2 Over F 2 , vectors indexed by V ca n b e identifi ed with subs e ts of V , and a V × V -matr ix defines a linea r transfo rmation on subsets of V . The kernel (als o called null spac e) of a matrix A , de no ted by ker( A ) is determined by those linear combinations of co lumn vectors o f A that sum up to the zero vector 0. W or king in F 2 , we rega rd the e le men ts o f ker( A ) as subsets of V . Moreov er, the kernel of A is the eigenspace E 0 ( A ) on v alue 0, a nd s imilar a s ker( A ), the elements of the (only other ) eigenspa ce E 1 ( A ) = { v ∈ V | Av = v } on v alue 1 a re also considered as sets. W e will often identify a g raph with its adja c ency ma trix, so, e.g., b y the de- terminant of graph G , denoted by det G , we will mean the determinant det A ( G ) of its adjacency matr ix computed over F 2 . In the sa me vein we will o ften simply write ker( G ), E 1 ( G ), P G , etc. Let P G = ( I , D ) for some g raph G . As G is a V × V -ma trix ov er F 2 , we hav e that X ∈ D iff there is a S ⊆ X with S ∈ ker( G ) \{ ∅ } . Moreov er , min( D ) = min (k er( G ) \{ ∅ } ) and k er( G ) is the c losure of min( D ) under ⊕ (i.e., min( D ) spans ker ( G )). Consequently , min( D ) uniquely determines ker ( G ) and vice versa. As min( D ) in turn unique ly determines P G , the following holds . Corollary 1. F or gr aphs G 1 and G 2 , ker( G 1 ) = k er( G 2 ) iff the familie s of b ases of G 1 and of G 2 ar e e qual. 3. Piv ots In general the pivot op eratio n can b e studied for matrice s over a rbitrary fields, e.g ., as done in [1 7]. In this pap er we restrict ours elves to symmetric matrices ov er F 2 , which leads to a num ber of additio na l viewp oints to the same op eration, and for each of them an equiv alent definition for pivoting. Each of these definitions is known, but (to our b est knowledge) they were not b efore collected in one text. Matric es. Let A be a V × V -matrix (ov er an a rbitrary field), and let X ⊆ V b e such that A [ X ] is nonsingula r, i.e., det A [ X ] 6 = 0. The pivot of A o n X , denoted by A ∗ X , is defined as follows, see [18]. Let A =  P Q R S  with P = A [ X ]. Then A ∗ X =  P − 1 − P − 1 Q RP − 1 S − R P − 1 Q  . Matrix ( A ∗ X ) \ X = S − RP − 1 Q is called the Schur c omplement of X in A . The pivot is sometimes cons idered a partia l inv ers e, as A and A ∗ X are related by the following characteristic e q uality , where the vectors x 1 and y 1 corres p o nd to the elements of X . In fa c t, this formula defines A ∗ X g iven A and X [1 7]. A  x 1 x 2  =  y 1 y 2  iff A ∗ X  y 1 x 2  =  x 1 y 2  (1) 3 Note that if det A 6 = 0, then A ∗ V = A − 1 . By E quation (1) we se e that a pivot op eration is an in volution (i.e., op eration of o rder 2), and more gener ally , if ( A ∗ X ) ∗ Y is defined, then A ∗ ( X ⊕ Y ) is defined and they ar e equa l. The following fundamental result on pivots is due to T ucker [1 8] (see also [7, Theorem 4.1.1 ]). It is used in [3 ] to study sequences o f piv ots. Prop ositio n 2 ([1 8]). L et A b e a V × V - matrix, and let X ⊆ V b e su ch that det A [ X ] 6 = 0 . Then, for Y ⊆ V , det ( A ∗ X )[ Y ] = det A [ X ⊕ Y ] / det A [ X ] . It may b e interesting to remark here that Pr op osition 2 for the cas e Y = V \ X is called the Sc hur determinant formula and was shown alrea dy in 1917 by Issai Sch ur, see [16]. It is eas y to v er ify fro m the definition of pivot that A ∗ X is skew-symmetric whenever A is. In particular, if G is a gr aph (i.e., a symmetric matrix ov er F 2 ), then G ∗ X is als o a gra ph. F rom now on w e res tr ict our attention to g raphs. Delta Matr oids. Co nsider now a s et sy stem M = ( V , D ). W e de fine, for X ⊆ V , the twist M ∗ X = ( V , D ∗ X ), where D ∗ X = { Y ⊕ X | Y ∈ D } . Let G b e a gr aph and le t M G = ( V ( G ) , D G ) b e the set system with D G = { X ⊆ V ( G ) | det G [ X ] = 1 } . It is easy to verify that G can b e (re)constr ucted given M G : { u } is a lo o p in G iff { u } ∈ D G , and { u, v } is an edge in G iff ( { u, v } ∈ D G ) ⊕ (( { u } ∈ D G ) ∧ ( { v } ∈ D G )), see [2, Pr op erty 3.1]. In this wa y , the fa mily of graphs (with set V of vertices) ca n b e cons idered a s a s ubs e t o f the family of set sy stems (ov er set V ). Prop ositio n 2 allo ws for another (equiv alent) definition of pivot o ver F 2 . Indeed, ov er F 2 , we have by Pr op osition 2, det( A ∗ X )[ Y ] = det A [ X ⊕ Y ] for all Y ⊆ V assuming A ∗ X is defined. Ther e fore, for M G ∗ X we hav e D G ∗ X = { Z | det(( G ∗ X )[ Z ]) = 1 } = { Z | det( G [ X ⊕ Z ]) = 1 } = { X ⊕ Y | det( G [ Y ]) = 1 } = D G ∗ X , see [1]. Hence M G ∗ X = M G ∗ X is an alternative definition of the pivot op er a tion over F 2 . It tur ns out that M G has a sp ecial structure, that o f a delta matr oid , allowing a s pec ific exchange of elements b etw een any tw o sets of D G , see [1]. How ever, not every delta matro id M has a gra ph r epresentation, i.e., M may not b e of the fo rm M G for a ny g raph G (a characterization of such repr esentable delta matroids ov er F 2 is given in [2 ]). Example 3 . Let G be the gr aph depicted in the upper - left corner of Fig- ure 1 . W e have A ( G ) =     p q r s p 0 1 1 1 q 1 1 0 1 r 1 0 0 1 s 1 1 1 0     . This cor resp onds to M G = ( { p, q , r, s } , D G ), where D G = { ∅ , { q } , { p, q } , { p, r } , { p, s } , { q , s } , { r, s } , { p, q , s } , { p, q , r } , { q , r, s }} . 4 r p q s r p q s r p q s r p q s r p q s ∗{ q } ∗{ p, s } ∗{ p } ∗{ s } ∗{ r } ∗{ p, s } ∗{ q } ∗{ p, r } ∗{ r, s } ∗{ p, s } Figure 1: The or bit of G under pivo t. Only the elementary pivots are shown . F or example, { p, q } ∈ D G since det( G [ { p, q } ]) = det  0 1 1 1  = 1. Then D G ∗ { p, q } = { ∅ , { p } , { r } , { s } , { p, q } , { p, s } , { q , r } , { q , s } , { p, r, s } , { p, q , r , s }} , and the corresp onding graph is de picted on the top-right in the same Figure 1. Equiv alently , this graph is obta ined fro m G by pivot on { p, q } . Also note that we have D G ∗ { p, s } = D G , and therefor e the pivot of G on { p, s } obtains G again. The set inclusion diag r ams of M G and M G ∗{ p,q } are given in Figur e 4. Gr aphs. The pivots G ∗ X where X is a minimal element of M G \{ ∅ } w.r.t. inclusion are ca lled elementary . It is no ted in [11] that an elementary pivot X corres p o nds to either a lo op, X = { u } ∈ E ( G ), or to an edge , X = { u, v } ∈ E ( G ), w her e bo th vertices u and v are non-lo ops. Mor eov er, ea ch Y ∈ M G can be partitioned Y = X 1 ∪ · · · ∪ X n such that G ∗ Y = G ∗ ( X 1 ⊕ · · · ⊕ X n ) = ( · · · ( G ∗ X 1 ) · · · ∗ X n ) is a co mp os ition of disjoint elementary pivots. Consequently , a direct de finitio n of the elementary pivots on gra phs G is s ufficien t to define the (general) pivot o per ation. The elementary pivot G ∗{ u } on a lo op { u } is calle d lo c al c omplementation . It is the gra ph obtained from G by co mplement ing the e dges in the neighbourho o d N G ( u ) = { v ∈ V | { u, v } ∈ E ( G ) , u 6 = v } of u in G : for each v , w ∈ N G ( u ), { v , w } ∈ E ( G ) iff { v , w } 6∈ E ( G ∗ { u } ), and { v } ∈ E ( G ) iff { v } 6∈ E ( G ∗ { u } ) (the case v = w ). The other edges ar e left unchanged. The elementary pivot G ∗ { u , v } on an edg e { u, v } b etw een distinct non- lo op vertices u a nd v is called e dge c omplementation . F or a vertex x co ns ider its closed neighbour ho o d N ′ G ( x ) = N G ( x ) ∪ { x } . The e dg e { u, v } pa rtitions the v er tice s of G connected to u or v in to three sets V 1 = N ′ G ( u ) \ N ′ G ( v ), V 2 = N ′ G ( v ) \ N ′ G ( u ), V 3 = N ′ G ( u ) ∩ N ′ G ( v ). Note that u , v ∈ V 3 . The g raph G ∗ { u, v } is constr ucted by “ toggling” a ll edges b e tween differ ent V i and V j : for { x, y } with x ∈ V i and y ∈ V j ( i 6 = j ): { x, y } ∈ E ( G ) iff { x, y } / ∈ E ( G [ { u, v } ]), s e e Figure 2. The remaining edg es remain unc hanged. Note that, as a result of this o per ation, the neighbours of u and v are int er changed. Example 4 . The whole orbit of G of Exa mple 3 under pivot is given in Figure 1 . 5 V 1 V 2 V 3 u v V 1 V 2 V 3 u v Figure 2: Pi v oting { u, v } in a graph. Connection { x, y } is toggled i ff x ∈ V i and y ∈ V j with i 6 = j . Note that u and v are connect ed to all vertices i n V 3 , these edges are omitted in the diagram. The op eration does not affect edges adjacent to ve rtices outside the sets V 1 , V 2 , V 3 , nor do es it chang e any of the lo ops. It is obtained by iteratively applying elementary pivots to G . Note that G ∗ { p , q } is defined (top-rig ht) but it is not an elementary piv ot. 4. Dual Pivots In this s ection we intro duce the dual pivot and s how that it has some in ter- esting prop erties. First note that the nex t result follows dir ectly from Equation (1). Lemma 5. L et A b e a V × V -m atr ix (over some field) and let X ⊆ V with A [ X ] nonsingular. Then t he eigensp ac es of A and A ∗ X on value 1 ar e e qual, i.e., E 1 ( A ) = E 1 ( A ∗ X ) . Proof. W e hav e v ∈ E 1 ( A ) iff Av = v iff ( A ∗ X ) v = v iff v ∈ E 1 ( A ∗ X ).  F or a g raph G , we deno te G + I to b e the gr aph having adjacency matrix A ( G ) + I where I is the identit y matrix. Th us, G + I is obtained from G by replacing each lo op by a non-lo op and vice versa. Definition 6. Let G b e a graph and let X ⊆ V with det(( G + I )[ X ]) = 1 . The dual pivot of G on X , denoted b y G ¯ ∗ X , is (( G + I ) ∗ X ) + I . Note that the co nditio n det(( G + I )[ X ]) = 1 in the definition of dual pivot ensures that the express ion (( G + I ) ∗ X ) + I is defined. The dua l pivot may b e considered a s the pivot op eration c onjugate d by addition of the identit y matrix I . As I + I is the null matr ix (ov er F 2 ), w e have, similar a s for pivot, that dual pivot is an inv olution, and more generally ( G ¯ ∗ X ) ¯ ∗ Y , when defined, is equal to G ¯ ∗ ( X ⊕ Y ). By Lemma 5, we have the following result. Lemma 7. L et G b e a gr aph and let X ⊆ V su ch that G ¯ ∗ X is define d. Then ker( G ¯ ∗ X ) = ker( G ) . 6 Proof. Note that Ax = 0 iff ( A + I ) x = x . Hence, ker( G ) = E 1 ( G + I ). Since I + I is the null matrix (ov er F 2 ), we hav e also ker( G + I ) = E 1 ( G ). Therefore, w e have ker( G ¯ ∗ X ) = ker((( G + I ) ∗ X ) + I ) = E 1 (( G + I ) ∗ X ) = E 1 ( G + I ), where we used Lemma 5 is the last eq ua lity . Finally , E 1 ( G + I ) = ker( G ) and therefore we obta in k er( G ¯ ∗ X ) = ker( G ).  In particular, for the case X = V , we hav e that ker(( G + I ) − 1 + I ) = ker( G ) (the inv erse is computed ov er F 2 ) if the left-hand side is defined. Remark 8. By Lemma 7 and Corollary 1 w e ha ve that the (column) matroids asso ciated with G a nd G ¯ ∗ X are equal. Note that here the matroids ar e obtained from the co lumn vectors o f the a djacency matrice s o f G a nd G ¯ ∗ X ; this is not to be confused with gra phic ma troids which are o btained fro m the co lumn vectors of the incidence matrices o f graphs. W e call dual piv ot G ¯ ∗ X elementary if ∗ X is an elementary pivot for G + I . Equiv alently , they are the dual pivots on X for which there is no no n- empt y Y ⊂ X wher e G ¯ ∗ Y is applica ble. An element ar y dual pivot ¯ ∗{ u } is defined on a non-lo op vertex u , and an elementary dual piv ot ¯ ∗{ u, v } is defined on an edge { u, v } where b oth u and v hav e lo ops. This is the only difference b etw een pivot and its dual: both the elementary dual pivot ¯ ∗{ u } and the elementary pivot ∗{ u } have the s ame effect on the graph — b oth “ ta ke the co mplement ” of the neighbourho o d o f u . Similarly , the effect of the e le men tar y dual pivot ¯ ∗{ u, v } and the elementary pivot ∗{ u, v } is the same, only the condition when they can be applied differs. Note that the eigenspaces E 0 ( G ) = ker( G ) and E 1 ( G ) hav e a na tural inter- pretation in g raph terminolo gy . F or X ⊆ V ( G ), X ∈ E 0 ( G ) iff every vertex in V ( G ) is c onnected to an e ven num b e r o f vertices in X (lo ops do count). Also, X ∈ E 1 ( G ) iff e very vertex in V ( G ) \ X is connected to an even num ber of ver- tices in X and every vertex in X is connected to a n o dd num be r o f vertices in X (again loo ps do count). Example 9 . Let G ′ be the gra ph depicted on the upp er- left corner o f Figur e 3 . W e have A ( G ′ ) =     p q r s p 1 1 1 1 q 1 0 0 1 r 1 0 1 1 s 1 1 1 1     . Note that E 1 ( G ′ ) = { ∅ , { p, r , s }} is of dimension 1. W e can apply an ele men tar y pivot over p on G ′ . The resulting graph G ′ ∗ { p } is depicted on the upp er-r ight corne r o f Figure 3, and we hav e A ( G ′ ∗ { p } ) =     p q r s p 1 1 1 1 q 1 1 1 0 r 1 1 0 0 s 1 0 0 0     . Note that the elements of E 1 ( G ′ ) are pre- cisely the eigenv ectors (or eigens ets) o n 1 for A ( G ′ ∗ { p } ), cf. Lemma 5. The graphs G ′ + I (which is G in Example 3) and G ′ ∗ { p } + I are depicted in the 7 r p q s ∗{ p } ¯ ∗{ p } + I + I r p q s r p q s r p q s Figure 3: Dual pivot of graph G from Example 3 ( G i s shown in the low er-left corner). low er-left and lower-right co rner o f Figure 3, r e sp e c tively . By definition of the dual pivot we hav e G ¯ ∗{ p } = ( G ′ + I ) ¯ ∗{ p } = G ′ ∗ { p } + I . It is a basic fact fro m linear alg ebra that elementary row op era tions retain the kernel o f matric e s. Lemma 7 suggests that the dual pivot may p ossibly b e simulated b y elementary r ow o p e rations. W e now s how that this is indeed the case. Over F 2 the elementary r ow o per ations are 1 ) r ow sw itching and 2) adding one r ow to another (row m ultiplication over F 2 do es not change the ma trix). The elementary r ow op erations corresp onding to the dual pivot op eratio n are easily deduced by re stricting to elementary dual pivots. The dual pivot on a non-lo op vertex u cor resp onds, in the adjacency matrix , to adding the row corresp o nding to u to ea ch row cor resp onding to a vertex in the neighbourho o d o f u . Mor eov er, the dual pivot on edge { u, v } (where b oth u and v hav e lo ops) cor resp onds to 1) adding the row corres po nding to u to each row cor resp onding to a vertex in the neigh b ourho o d of v except u , 2) a dding the row cor resp onding to v to each row corr esp onding to a vertex in the neighbourho o d o f u except v , 3) switc hing the rows of u a nd v . Note that this pro cedur e allows fo r another, equiv alen t, definition o f the regular pivot: add I , apply the cor resp onding elementary row op erations, and finally add I ag ain. Note that the dual pivot ha s the pro p e rty that it transfor ms a symmetric matrix to ano ther symmetric matr ix with equal kernel. Applying elementary row op eratio ns how ever will in general not obtain symmetr ic matr ices. 5. Maximal Pivots In Sectio n 3 we recalled that the minimal elements of M G , co rresp onding to elementary pivots, form the building blo cks of (general) piv ots . In this section we show tha t the set o f maximal elements o f M G , corres po nding to “maximal pivots”, is inv a riant under dual pivot. 8 ∅ q pr pq q s ps rs pq r pq s q rs ∅ r p s q r pq ps q s prs pq r s ∅ r s pr q r rs ps pq pq r pq s q r s Figure 4: Set inclusion diagram of M G , M G ∗{ p,q } , and M G ¯ ∗{ p } for G , G ∗{ p, q } , and G ¯ ∗{ p } as given in Examples 3 and 9. F or M G = ( V , D G ), w e define F G = max( D G ). Thus, for X ⊆ V ( G ), X ∈ F G iff det G [ X ] = 1 while det G [ Y ] = 0 for every Y ⊃ X . Example 1 0. W e contin ue Example 3. Le t G be the graph on the low er-left corner of Figure 3. Then fro m the set inclus ion diagr am of M G in Figure 4 we s ee that F G = {{ p, q , s } , { p, q, r } , { q , r , s }} . Also w e see from the figure that F G ∗{ p,q } = { V } . Next we recall the Stro ng Pr incipal Mino r Theor em for (quasi- ) symmetric matrices from [12] — it is s tated here for graphs (i.e., s ymmetric matrices ov er F 2 ). 1 Prop ositio n 11. L et G b e a gr aph su ch that A ( G ) has r ank r , and let X ⊆ V ( G ) with | X | = r . Then X is indep endent for A ( G ) iff det G [ X ] = 1 . Note that the indep e nden t sets X of cardina lit y equal to the rank ar e pr e- cisely the bases of a matrix A . The following res ult is easy to see now from Prop osition 11. Lemma 12 . L et G b e a gr aph such that A ( G ) has r ank r . Each element of F G is of c ar dinality r . Proof. If there is an X ∈ F G of car dinality q > r , then the columns of A ( G [ X ]) are linea rly indep endent , a nd thus so are the columns of A ( G ) cor resp onding to X . This contradicts the ra nk of A ( G ). 1 Clearly , for a matrix A , det A [ X ] 6 = 0 implies that X is indep enden t for A . The rev erse implication is not v alid in general. 9 Finally , ass ume that there is an X ∈ F G of cardina lit y q < r . Sin ce the columns of A ( G [ X ]) are linearly independent, so are the columns of A ( G ) cor- resp onding to X . Since A ( G ) has rank r , X can b e extended to a set X ′ with car dina lit y r . Hence by Pr op osition 11 det G [ X ′ ] = 1 with X ′ ⊃ X — a contradiction of X ∈ F G .  Example 1 3. W e con tinue E xample 3. Let ag ain G b e the gr aph on the lower- left corner of Figur e 3. Then the elements F G = {{ p, q , s } , { p, q, r } , { q , r , s }} are all of cardina lit y 3 — the rank of A ( G ). Moreover, F G ∗{ p,q } = { V } and | V | = 4 is equal to the ra nk o f G ∗ { p, q } . Combining Pr op osition 11 and Lemma 12, we ha ve the following result. Corollary 14 . L et G b e a gr aph, and let X ⊆ V ( G ) . Then X is a b asis for A ( G ) iff X ∈ F G . Equiv alently , with P G = ( I , D ) from Section 2, Co rollary 14 states that max( I ) = F G . By Cor o llaries 1 and 14 w e hav e now the following. Lemma 15 . L et G and G ′ b e gr aphs. Then F G = F G ′ iff ker ( G ) = ker( G ′ ) . Recall that Lemma 7 shows that the dual pivot r etains the k er nel. W e may now conclude from Lemma 15 that also F G is retained under dual pivot. It is the main res ult of this pap er , and, as we will see in Sec tio n 7, has an imp ortant application. Theorem 16 . L et G b e a gr aph, and let X ⊆ V . Then F G = F G ¯ ∗ X if the right-hand side is define d. In particula r , the case X = V , we hav e F G + I = F G − 1 + I if G is inv ertible (ov er F 2 ). Let O G = { G ¯ ∗ X | X ⊆ V , det( G + I )[ X ] = 1 } b e the o rbit of G under dual pivot, and note that G ∈ O G . By Theore m 16, if G 1 , G 2 ∈ O G , then F G 1 = F G 2 . Note that the reverse implication do es not ho ld: e.g. O I = { I } and F I = { V } , while clearly there are many other graphs G with det G = 1 (whic h mea ns F G = { V } ). Example 1 7. W e con tinue E xample 9. Let ag ain G b e the gr aph on the lower- left cor ner of Figure 3. Then G ¯ ∗{ p } is de picted on the lower-right cor ner of Figure 3. W e have F G ¯ ∗{ p } = {{ p, q , s } , { p, q , r } , { q , r, s }} , see Figur e 4, so indeed F G = F G ¯ ∗{ p } . F or sy mmetric V × V -matrices A over F 2 , Theo r em 16 states that if A can be par tially inv erted w.r .t. Y ⊆ V , whe r e Y is max imal w.r.t. set inclusion, then this holds for every matrix obtained from A by dua l pivot. 10 6. Maximal Contra ctions F or a g raph G , w e define the c ontr actio n of G on X ⊆ V with det G [ X ] = 1, denoted by G ∗ \ X , to b e the graph ( G ∗ X ) \ X — the pivot on X follow ed by the remov al o f the vertices of X . Equiv alent ly , co n tra ction is the Sch ur complement applied to gra phs. A contraction of G on X is maximal if there is no Y ⊃ X such that det G [ Y ] = 1 , hence if X ∈ F G . The graph obtained by a maximal contraction on X is a discre te graph G ′ (without lo ops). Indeed, if G ′ were to hav e a lo o p e = { u } o r an edg e e = { u, v } b etw een tw o non-lo op vertices, then, since det G [ X ⊕ e ] = det(( G ∗ X )[ e ]) = det(( G ∗ \ X )[ e ]) = 1, X ⊕ e ⊃ X would b e a cont ra diction of the maximality of X . Mo reov er, by L e mma 12, the nu mber of vertices of G ′ is equal to the nullit y (dimension of the kernel, which equals the dimension of the matrix minus its ra nk) of G . Remark 18. In fact, it is known that any Schur complement in a matrix A has the same nullit y as A itself — it is a consequence of the Guttman rank a dditivit y formula, see, e.g., [19, Sec tion 6 .0.1]. The r efore the the nullit y is inv aria nt under contraction in g eneral (not only maximal contraction). By Theor e m 16 we hav e the following. Corollary 19 . The set of discr ete gr ap hs obtainable thr ough c ontr actions is e qual for G and G ¯ ∗ X for al l X ⊆ V with det( G + I )[ X ] = 1 . In this s ense, a ll the e lemen ts of the o rbit O G hav e equal “b ehaviour” w.r.t. maximal contractions. Example 2 0. W e contin ue the example. Recall that, fro m Example 17, F G = F G ¯ ∗{ p } = {{ p, q , s } , { p, q , r } , { q , r, s }} . The elementary co nt ra ctions starting from G and G ¯ ∗{ p } are given in Figure 5. Notice that the ma ximal contractions of G and G ¯ ∗{ p } obtain the same set o f (discrete) graphs. It is impo r tant to rea lize that while the maximal c ontractions (corresp ond- ing to F G ) are the sa me for gr aphs G a nd G ¯ ∗ X , the whole set of contractions (corresp onding to M G ) may be sp ectacula rly different. Indeed, e.g., in Exa m- ple 20, the elementary pivots for G are ∗ { q } , ∗{ p, s } , ∗{ p, r } , and ∗{ r , s } , while the elementary pivots for G ¯ ∗{ p } ar e ∗ { r } , ∗{ s } , and ∗{ p, q } (see Figure 4). 7. Application: Gene Assembly Gene a s sembly is a highly inv olved and par allel pro ce s s o ccurring in o ne - cellular organisms c a lled ciliates. During gene assembly a nucleus, called mi- cronucleus (MIC), is transformed into another nucleus ca lled macronucleus (MAC). Segments of the genes in the MAC o cc ur in scra mbled o rder in the MIC [9]. Dur- ing g ene assembly , re combination takes place to “so rt” these g ene segments in the MIC in the rig h t o rientation and o rder to o btain the MAC g ene. The trans- formation of single genes from their MIC form to their MAC form is formally 11 r p q s r p s p q s r p q q s r s p r q r p q p s s r p r p q s ¯ ∗{ p } Figure 5: Elementa ry contract ions starting f rom G and G ¯ ∗{ p } . mo delled, se e [8, 10, 9], as both a string based mo del a nd a (almost equiv alent) graph ba sed mo del. It is observed in [3] that t wo of the three ope r ations in the graph ba sed mo del are exactly the tw o elementary principal pivot transfor m (PPT, or simply pivot ) ope rations on the co rresp onding adja c ency matrices considered ov er F 2 . The third o per ation simply remov es isolated vertices. Maximal contractions ar e esp ecially imp ortant within the theor y o f g ene assembly in ciliates — such a maximal se q uence determines a complete trans- formation of the gene to its MAC form. W e first recall the string rewriting system, and then rec a ll the generaliza tio n to the g raph rewriting system. Let A b e an ar bitr ary finite alphab et. The se t of letters in a string u o ver A is denoted by L ( u ). String u is called a double o c curr enc e string if each x ∈ L ( u ) o ccurs exactly twice in u . F or e x ample, u = 41215 425 is a double o ccur rence string ov er L ( u ) = { 1 , . . . , 5 } . Let ¯ A = { ¯ x | x ∈ A } with A ∩ ¯ A = ∅ , and let ˜ A = A ∪ ¯ A . W e use the “bar op era tor” to mov e from A to ¯ A and back from ¯ A to A . Hence, for x ∈ ˜ A , ¯ ¯ x = x . F or a str ing u = x 1 x 2 · · · x n with x i ∈ A , the inverse of u is the string ¯ u = ¯ x n ¯ x n − 1 · · · ¯ x 1 . W e define the morphism k · k : ( ˜ A ) ∗ → A ∗ as follows: for x ∈ ˜ A , k x k = x if x ∈ A , and k x k = ¯ x if x ∈ ¯ A , i.e., k x k is the “unbarred” v ariant of x . Hence, e.g., k 2 ¯ 5 ¯ 3 k = 2 53. A le gal st ring is a string u ∈ ( ˜ A ) ∗ where k u k is a double o ccurrence string. W e denote the empty string by λ . Example 2 1. The str ing u = q ps ¯ q r psr ov er ˜ A with A = { p, q , r , s } is a lega l string. As ano ther example, the leg al string 34 45675 6789 ¯ 3 ¯ 2289 over ˜ B with 12 B = { 2 , 3 , . . . , 9 } repr e s ent s the micr onuclear for m of the gene corr esp onding to the actin protein in the stichotrichous ciliate S terkiel la nova , see [14, 6]. It is p os tulated that g ene assembly is p erfor med by three types of e le men tar y recombination op erations, called lo op, hair pin, and double-lo o p reco mbin atio n on DNA, see [15]. These three recombination op er ations ha ve b een mo deled as three type s of string rewr iting rules oper ating on legal s trings [8, 9] — together they form the string p ointer r eduction system. F or all x, y ∈ ˜ A with k x k 6 = k y k we define: • the st ring ne gative rule for x by snr x ( u 1 xxu 2 ) = u 1 u 2 , • the st ring p ositive ru le for x by spr x ( u 1 xu 2 ¯ xu 3 ) = u 1 ¯ u 2 u 3 , • the string double ru le for x, y by sdr x,y ( u 1 xu 2 y u 3 xu 4 y u 5 ) = u 1 u 4 u 3 u 2 u 5 , where u 1 , u 2 , . . . , u 5 are arbitra ry (p ossibly empty) str ings o ver ˜ A . Example 2 2. Let aga in u = q ps ¯ q r psr be a le g al string. W e have spr q ( u ) = ¯ s ¯ pr psr . And more over, spr ¯ r spr ¯ p spr q ( u ) = ¯ s ¯ s . Finally , s nr ¯ s spr ¯ r spr ¯ p spr q ( u ) = λ . W e now define a g raph for a leg al string repr esenting whether or not interv a ls within the leg a l string “ov erla p” . Let u = x 1 x 2 · · · x n be a legal s tring with x i ∈ ˜ A for 1 ≤ i ≤ n . F or letter y ∈ L ( k u k ) let 1 ≤ i < j ≤ n b e the po sitions of y in u , i.e., k x i k = k x j k = y . The y -interval of u , denoted by int v y , is the subs tring x k x k +1 · · · x l where k = i if x i = y a nd k = i + 1 if x i = ¯ y , and similarly , l = j if x j = y and l = j − 1 if x j = ¯ y (i.e., a b or der of the interv al is included in ca se o f y and e x cluded in ca se of ¯ y ). Now the overlap gr aph of u , denoted by G u , is the graph ( V , E ) with V = L ( k u k ) a nd E = {{ x, y } | x o ccurs exactly once in k in tv y k} . Note that E is well defined as x o ccurr ing exactly once in the y -interv a l of u is equiv alent to y o ccurr ing exactly onc e in the x -interv al of u . Note that we hav e a lo op { x } ∈ E iff b oth x and ¯ x o ccur in u . The ov erla p gr aph as defined her e is an extension of the usual definition o f overlap gr aph (also calle d circle gra ph) from simple graphs (without lo ops) to gr aphs (where loo ps are allow ed). See [13, Section 7 .4 ] for a brief ov erview of (simple) ov erla p graphs . Example 2 3. The ov er lap gra ph G u of u = q ps ¯ q r psr is exa ctly the graph G of Example 3. It is shown in [8 , 1 0], see also [9 ], that the string rules snr x , spr x , and sdr x,y on leg al strings u can b e simulated as g raph rules gnr x , gpr x , and gdr x,y on ov erlap g r aphs G u in the sense that G spr x ( u ) = gpr x ( G u ), wher e the le ft-hand side is defined iff the right-hand s ide is defined, a nd similarly for gdr x,y and 13 gnr x 2 . It was s hown in [3] that gpr x and g dr x,y are exa ctly the t wo t yp es of contractions o f elementary pivots ∗\{ x } and ∗\{ x, y } on a lo op { x } and a n edge { x, y } without lo ops, resp ectively . The gnr x rule is the re mov al of isolated vertex x . Example 2 4. The s equence spr ¯ r spr ¯ p spr q applicable to u given in Exam- ple 22 cor resp onds to a max ima l contraction of gra ph G = G u of Example 3 as can be seen in Figure 5. Within the theor y of ge ne a s sembly one is interested in maximal r ecombina- tion str ategies o f a g ene. These stra tegies corres po nd to maximal contractions of a graph G (hence de c o mpo sable into a sequence ϕ 1 of contractions of elemen- tary pivots gpr and g dr applicable to (defined on) G ) follow ed by a sequence ϕ 2 of gnr r ules, r emoving isolated vertices, until the empty g raph is obtained. Here we ca ll these sequences ϕ = ϕ 2 ϕ 1 of gra ph rules c omplete c ontr actions . If we define the set of vertices v of ϕ used in g nr v rules by gnrdom( ϕ ), then the following result ho lds b y Cor o llary 1 9. Theorem 25 . L et G 1 , G 2 ∈ O G for some gr aph G , and let ϕ b e a c omple te c ontr action of gr aph G 1 . Then ther e is a c omplete c ontr action ϕ ′ of G 2 such that gnr dom( ϕ ) = g nr dom( ϕ ′ ) . Hence, Theo rem 25 shows that all the elements o f O G , for any gra ph G , hav e the same b ehaviour w.r .t. the applicability o f the rule gnr x . A simila r result a s Theor em 25 was shown for the string rewriting mo de l, see [4, Theorem 34] 3 . It should be stressed how ever that Theorem 25 is r eal generaliza tion o f the result in [4] as no t every gr aph has a string repre s ent atio n (i.e., not every graph is an ov er la p g raph), and mor eov er it is obtained in a very different wa y: here the result is obtained using techniques from linear algebra. 8. Discussion W e intro duced the concept of dual pivot and hav e shown that it has in- teresting pr op erties: it has the same e ffect as the (regular) pivot and can b e simulated b y elementary r ow op erations — consequently it keeps the kernel in- v ariant. The dual pivot in this way allows for an alter native definition of the (regular) pivot op eration. F urthermo re, we have shown that tw o graphs hav e 2 There is an exception for gnr x : although G snr x ( u ) = gnr x ( G u ) holds if the l eft-hand side is defined, there are cases where the right-hand side is defined ( x i s an isolated vertex in G u ) while the l eft-hand side i s not defined ( u do es not hav e substring xx ). This is why the string and gr aph models are “almost” equiv alen t. This difference in mo dels is not relev an t for our purposes. 3 This result states that tw o legal strings equiv alent modulo “dual” string rules hav e the same reduction graph (up to isomorphism). It then foll o ws f rom [5, Theorem 44] that these strings hav e complete contract ions with equal snr dom, the stri ng equiv alent of gnrdom . 14 equal kernel precisely when they have the same s et o f maximal pivots. F rom this it follows that the set of maximal pivots is inv ariant under dual pivot. This main result is motiv ated by the theory o f gene a ssembly in ciliates in which maximal contractions cor resp ond to c o mplete tra nsformations of a gene to its macro nuclear fo rm. How ever, as a pplying a maximal pivot cor resp onds to calculating a maximal par tial inv erse o f the ma tr ix, the res ult is also interesting from a purely theoretical p oint of view. A cknow le dgements W e thank Lorenzo T raldi and the tw o a nonymous referee s for their v aluable comments on the pap er. R.B. is supp or ted by the Netherlands Or ganization for Scient ific Research (NWO), pr o ject “Annotated graph mining”. References [1] A. Bouchet. Repr esentabilit y of ∆-matroids . In Pr o c. 6th Hungarian Col- lo quium of Combinatorics, Col lo quia Mathematic a So cietatis J ´ anos Bolya i , volume 52 , pag es 167– 1 82. North-Holla nd, 1987. [2] A. Bo uchet a nd A. Duchamp. Representabilit y of ∆-matroids o ver GF (2). Line ar Alge br a and its Applic ations , 146:6 7–78, 19 91. [3] R. Br ijder, T. Har ju, and H.J. Hoo geb o om. P ivots, determinants, and per fect matchings o f graphs. Submitted, [ar Xiv:0811.3 500], 2008. [4] R. Brijder and H.J. Ho o geb o om. The fibers and range of reduction g r aphs in ciliates. Ac ta Informatic a , 45:3 8 3–40 2 , 200 8. [5] R. Brijder, H.J. Ho ogeb o om, a nd M. Muskulus. Strategies of lo op r ecom- bination in cilia tes. Discr ete Applie d Mathematics , 156 :1736– 1753, 2 008. [6] A.R.O. Cav alcanti, T.H. Cla rke, and L.F. Landweber. MDS IES DB: a database of macronuclear and micronuclear genes in spiro trichous ciliates. Nucleic A cids R ese ar ch , 3 3:D396–D3 98, 2005 . [7] R.W. Cottle, J.-S. Pang, and R.E. Stone. The Line ar Complementarity Pr oblem . Academic Pre s s, Sa n Diego, 1992. [8] A. Ehrenfeuch t, T. Harju, I. Petre, D.M. P rescott, and G. Rozenber g . F or - mal systems for g ene assembly in cilia tes. The or etic al Computer S cienc e , 292:19 9–219 , 200 3. [9] A. Ehr enfeuc ht, T. Harju, I. Petre, D.M. Pres cott, and G. Rozenber g. Com- putation in Living Ce l ls – Gene A ssembly in Ciliates . Springer V erlag, 2004. [10] A. Ehrenfeuch t, I. Petre, D.M. Pr e scott, and G. Rozenber g . String and graph reduction sys tems for gene ass e m bly in ciliates. Mathematic al Struc- tur es in Computer Scienc e , 12:113– 134, 20 0 2. 15 [11] J .F. Geelen. A g eneralizatio n of Tutte’s characteriza tion of totally uni- mo dular matr ices. J ournal of Combinatori al The ory, Series B , 7 0:101 – 117, 1997. [12] V. Ko diyalam, T. Y. Lam, and R. G. Swan. Determinantal idea ls, Pfaffian ideals, and the princ ipa l mino r theorem. In Nonc ommutative Rings, Gr oup Rings, D iagr am Algebr as and Their A pplic ations , pages 3 5–60. American Mathematical So ciety , 2008. [13] T.A. McKee and F.R. McMorr is. T op ics in Int erse ction Gr aph The ory . So ciety fo r Industrial Mathematics, 1999 . [14] D.M. Prescott and M. DuBois. Int erna l eliminated s e gments (IE Ss ) of oxytric hidae. J ournal of Eu karyotic Micr obio lo gy , 43 :432–4 41, 1 996. [15] D.M. Pr escott, A. Ehr enfeuch t, and G. Rozenberg . Molecular op erations for DNA pro cess ing in hypotrichous ciliates. Eur op e an Journal of Pr otistolo gy , 37:241 –260, 2 0 01. [16] J . Sch ur. ¨ Uber Potenzr eihen, die im Inner n des Einheitskreis e s b eschr¨ ankt sind. Journal f ¨ ur die r eine und angewandte Mathematik , 14 7:205 –232, 191 7. [17] M.J. Tsatsomer os. Principal pivot transforms: prop erties a nd a pplica tions. Line ar Alge br a and its Applic ations , 307(1 - 3):151– 165, 2000. [18] A.W. T uc ker. A combinatorial equiv alence of matr ic es. In Combinato- rial Analysis, Pr o c e e dings of Symp osia in Applie d Mathematics , volume X, pages 129 –140. America n Mathematical So ciety , 1960. [19] F. Zhang. The Schur Complement and Its Applic ations . Springer, 19 92. 16