On matroid generation R. J. Kingan*, S. R. Kingan**, Wendy Myrvold† Abstract A matroid is said to be representable over a finite field GF (q), for some prime power q, if we can find a matrix A with entries from GF (q), such that the sets of independent columns of A correspond to the independent sets in M . A simple matroid M with ground set E, also called a combinatorial geometry, is one in which every singleton subset and every 2-element subset of E is independent. A GF (q)-representable combinatorial geometry is a subset of the projective geometry P G(r − 1, q). We present an algorithm for generating isomorph-free GF (q)-representable combinatorial geometries. In the scheme outlined by McKay [14], this algorithm is in the class of algorithms for generation via “canonical construction path.” This algorithm generates matroids exhaustively when q = 2 or 3. For fields of higher order the presence of inequivalent representations renders the canonical contruction path inexhaustive. Finally, we extend Acketa’s work on binary matroid generation [1] and also present some new enumerative results on ternary matroids. 1. Introduction This is a paper on generation of isomorph-free simple matroids with an emphasis on the structure of the matroid in terms of representability. We follow the matroid terminology in Oxley [15]. Let E be a finite set and r be a function that maps 2E into the set of non-negative integers and satisfies the following properties: i. If X ⊂ E, then 0 ≤ r(X) ≤ |X|; ii. If X ⊆ Y ⊆ E, then r(X) ≤ r(Y ); and iii. If X and Y are subsets fof E, then r(X ∪ Y ) + r(X ∩ Y ) ≤ r(X) + r(Y ). Then r is the rank function of a matroid M on ground set E. We define the rank of M , denoted by r(M ), to be r(E). We say X ⊆ E is independent if * Kingan Analytics, Inc., Hershey, PA 17033 ** Pennsylvania State University, Middletown, PA 17057 † University of Victoria, Victoria, B.C. V8W3P6, Canada. 1

|X| = r(X), X is a basis if |X| = r(X) = r(M ), X is a circuit if X 6= φ and for all x ∈ X, r(X − x) = |X| − 1 = r(X). We define the closure of X, denoted by cl(X), as the set of all x ∈ E such that r(X ∪ x) = r(X). We say X ⊆ E is a flat if cl(X) = X, X is spanning if cl(X) = E, and X is a hyperplane if it is a closed set with rank r(M ) − 1. A single-element circuit is called a loop and a rank 1 circuit with two or more elements is called a parallel class. A simple matroid is one without loops or parallel classes. Simple matroids are also called combinatorial geometries. A rank r matroid M with ground set E is representable over a field F if there is a rank preserving map φ : E −→ V (r, F ). If F is finite, then F is the unique Galois field, denoted by GF (q), where q is the power of a prime. If M is representable, then we can find a matrix A with entries from F that represents M . We can write the matrix in standard form A = [Ir |D], where the columns of A correspond to a non-zero representative vector from one-dimensional subspaces of V (r, F ). We denote the matroid M as M [A] to highlight the fact that A represents M . Observe that if M is simple then A has no zero column and no columns are repeated. In this case the columns of A can be viewed as a subset of the projective geometry of rank r, P G(r − 1, F ). Let M be a matroid on ground set E. The dual of M , denoted by M ∗ is the matroid on E with rank function r∗ defined as r∗ (X) = |X| − r(M ) + r(E − X). In terms of matrices the dual of an n-element, rank r matroid represented by the matrix [Ir |D] is [−DT |In−r ]. The circuits of M ∗ are usually referred to as cocircuits. Likewise for other matroid definitions. If M and N are matroids on the sets E ∪ e and E, respectively, where e 6∈ E, then M is a single-element extension of N if M \e ∼ = N . In this case we call N a single-element deletion of M . We say M is a singleelement coextension of N if M ∗ is a single-element extension of N ∗ , that is, M/e ∼ = N . In this case we call N a single-element contraction of M . We say two matroids, M1 and M2 on ground sets E1 and E2 , respectively, are isomorphic is there is a bijection f from E1 to E2 such that, for all X ⊆ E1 , f (X) is independent in M2 if and only if X is independent in M1 . Central to this paper is the issue of determining when two GF (q)-representable matroids are isomorphic. Since the matroids are representable over GF (q), the matrices representing them are subsets of the projective geometry P G(r − 1, q). Roughly speaking, two such matrices are isomorphic if one is the image of the other under an automorphism of P G(r − 1, q), where an automorphism of a projective geoemtry is a permutation of its set of points that maps lines into lines [15, pg 185]. Let M be a GF (q)-representable matroid with rank at least 3 and let 2

A be a matrix over F representing M . We say another matrix B is equivalent to A if B can be obtained from A by a sequence of the following operations: interchange two rows, multiply a row by a non-zero member of GF (q), replace a row by the sum of that row and another, delete a zero row (unless it is the only row), interchange two columns (the labels moving with the columns), multiply a non-zero column by a non-zero member of GF (q), and replace each entry of the matrix by its image under some automorphism of GF (q). For a fixed finite field GF (q) and fixed rank r, subsets of columns of P G(r − 1, q) that are equivalent give rise to isomorphic matroids. An n-element rank-r matroid M is called uniquely F -representable if all the r by n matrices representing M over F are equivalent. Binary and ternary matroids are uniquely representable. Moreover, the last operation is not needed [15, p. 188]. However, higher order fields lack unique representablity, that is two matroids may be isomorphic, but there may be no way to get from one to the other by the above operations. In this case, building up matroids via single-element extensions is not exhaustive. It is possible that a rank-r GF (q)-representable matroid M [A] may be a singleelement extension of another rank-r GF (q) representable matroid M [B], yet there will be no column in P G(r − 1, q) that can be added to B to obtain A. In this paper we focus on generating isomorph-free matroids representable over GF (q). We will focus only on generating combinatorial geometries (simple matroids) because a matroid is representable over GF (q) if and only if the corresponding simple matroid is representable over GF (q). Such generation, besides establishing a new mathematical sequence of numbers, is usually undertaken to study the structure of matroids. In Section 2 we give a brief history of enumeration in matroid theory. In Section 3 we present the algorithm for generating isomorph-free combinatorial geometries representable over GF (q). The generation is exhaustive when q = 2 and 3. We will also outline the procedures used to test and verify the accuracy of the code. In Section 4, we put the generation algorithm for matroids in context with generation algorithms for other combinatorial objects. In the scheme outlined by McKay [14], this algorithm is in the class of algorithms for generation via “canonical construction path.” We will describe the similarities to MacKay’s algorithm. In Section 5, we present the output of the algorithm. We obtain some previously unknown sequences of numbers associated with binary and ternary matroids. Note that the generation program gives, not just sequences, but actual matroids which can be used for further analyses. Moreover, the program also generates vast numbers of geometries over higher fields. 3

2. A brief history of enumeration in matroid theory Matroids were defined by Whitney in 1935 [17] in an attempt to capture the properties of dependence common to graphs and matrices. Through out this paper n will denote the number of elements in the matroid. In 1967 Doyen constructed all the isomorph-free rank 3 combinatorial geometries on n ≤ 9 elements [1]. In 1971, Fournier verified that all rank 3 combinatorial geometries with n ≤ 8 are representable [8]. Vamos did similar work in an unpublished manuscript. In 1973 Blackburn, Crapo, and Higgs [4] generated exhaustively the isomorph-free combinatorial geometries on n ≤ 8 elements. Not much was done on matroid generation for the next decade. In 1984, Acketa [1] used their list to obtain all the isomorph-free, binary combinatorial geometries on n ≤ 8 elements. He also determined all the matroids on n ≤ 8 elements. In 1990, Betten and Glynn generated exhaustively the isomorph-free rank 3 combinatorial geometries on n ≤ 10 elements. Pietsch extended the generation of rank 3 geometries to 11 elements and Betten and Betten went to 12 elements [3]. Similar work was independently undertaken by Grieg [9, 10]. In 1994 binary combinatorial geometries were generated independently by Dharmatilake [5] and Kingan [11] in their dissertations for the purpose of obtaining structural results. In 2000, Dukes presented several interesting enumerative results in his dissertation [7]. In the table below we summarize the known information on the number of isomorph-free combinatorial geometries on n ≤ 8 elements. The columns give the rank and the rows give the number of elements. A similar table by Acketa is available for binary combinatorial geometries on n ≤ 8 elements [1]. In Table 2 of Section 5 we extend his results to 11 elements. Table 3 gives similar results for ternary combinatorial geometries. 2 3 4 5 6 2 1 1 1 1 1 3 1 2 4 9 4 1 3 11 5 1 4 6 1 7 8

7 1 23 49 22 5 1

8 1 68 317 217 40 6 1

Table 1: The number of known isomorph-free combinatorial goemetries 4

Additionally, the number of rank 3 combinatorial geometries is known for n ≤ 12 elements. There are 383 rank 3, 9-element geometries, 5250 10-element geometries, 232,929 11-element geometries, and 28,872,973 12element geometries. 3. The algorithm In this section we will describe the matroid generation algorithm briefly before giving the formal algorithm. For a fixed finite field GF (q), we generate non-isomorphic matroids representable over GF (q) using single-element extensions and a matroid isomorphism checker. The isomorphism checker computes, for two matroids, classes of elements based on membership in circuits, independent sets and spanning sets. If the two matroids agree both in the sizes of their classes and their numbers of circuits, independent sets and spanning sets of various sizes, the checker tests each map from one matroid to the other that preserves classes, to see if a map exists that preserves all bases. Although it does not run in polynomial time, it is fast enough for isomorphism testing for this application. The input to the generation algorithm is the order q of the field and a set {M1 , . . . , Mk } of n-element rank r combinatorial geometries in matrix form, called seed matrices. Each Mi is representable over GF (q) and is in standard form [Ir |D], where D has n − r columns. The program begins by generating the matrix corresponding to P G(r−1, q). Note that P G(r−1, q) has (q r − 1)/(q − 1) columns and this is the maximum size for a rank r, GF (q)-representable geometry. For each Mi the program first compares the columns of Mi with the columns of P G(r − 1, q) and taking into account scalar multiples, expresses Mi as a minor of P G(r − 1, q). Starting with this representation, the program determines the list of columns from P G(r −1, q) absent in Mi . It adds each missing column in turn to Mi to generate single-element extensions, and discards duplicates using the isomorphism checker. This gives all of the GF (q)-representable single-element extensions of Mi . We also need to be able to eliminate extensions that have occurred as extensions of other earlier seed matroids. Once the non-isomorphic singleelement extensions of Mi are obtained, the single-element deletions (possible parents) of each extension are computed. Each extension’s singleelement deletions are then compared, using the isomorphism checker, to the previous seed matroids M1 , . . . , Mi−1 . If any single-element deletion of the new extension is isomorphic to a previously generated seed matroid, the extension is discarded. By comparing single-element deletions of newly 5

generated extensions to previously considered seed matroids, the number of calls to the isomorphism checker is reduced. This method is exhaustive for fields like GF (2) and GF (3) where unique representability holds. Finally, once all the non-isomorphic (n + 1)-element matroids are obtained they are used in turn as seed matroids for the next round. If we start with the identity matrix [Ir ] and complete [(q r − 1)/(q − 1)] − r rounds, we will get all of the rank-r, GF (q)-representable geometries culminating with P G(r − 1, q) itself, subject to the restrictions above. Algorithm 2.1 Require: r = rank of matroids n0 = size of seed matroids q = size of base field seed1 , . . . , seedk0 = seed matroids k0 = number of seed matroids P G ← generateP G(r − 1, q) nmax ← |E(P G)| n ← n0 for i ← 0 to k0 − 1 do extMi ← seedi end for knew ← k0 while n < nmax for i ← 0 to knew − 1 currentMi ← extMi end for k ← knew knew ← 0 output currentM1 , . . . , currentMk for i ← 0 to k − 1 do {currentExt0 , . . . , currentExtkext −1 } ← generateSingleElementExtensions(currentMi , P G) for j ← 0 to kext − 1 do f lag ← f alse for m ← 1 to |currentExtj | do del ← deleteElement(currentExtj , m) 6

if rank(del) = rank(currentExtj ) then for l ← 0 to i − 1 do if checkIsomorphism(del, currentMl ) then f lag ← true break end if end for end if if f lag then break end if end for if not(f lag) then extMknew ← currentExtj knew ← knew + 1 end if end for end for n←n+1 end while The matroid isomorphism checker relies on structural properties of matroids, specifically circuits, independent sets and bases, to reject candidates that are clearly different, and also to narrow down the list of mappings to test in order to verify an isomorphism. It does not rely on any matroid properties specific to GF (q)-representable matroids, and thus can be used even when inequivalent representations are present. However, the cannonical path contruction fails in that case. In other words, the isomorphism checker can be used in a broader setting and has applications beyond matroid generation. Let E = {e1 , e2 , . . . , en } be a finite nonempty set, and let F be a family of subsets of E. We denote by chain(E, F) and degrees(E, F) the vectors (c0 , c1 , . . . , cn ) and (d1 , . . . , dn ), respectively, where ci = |{s ∈ F : |s| = i}| and di = |{s ∈ F : ei ∈ s}|. 7

Now let F1 , F2 , . . . , Fm be families of subsets of E, and let dk = degrees(E, Fk ). Define a n × k matrix D by: D = [dT1 | · · · |dTk ] Given two matroids M1 and M2 , the isomorphism checker determines chain and degree vectors for the family C(M ) of circuits, the family I(M ) of independent sets, and the family B(M ) of bases. Circuits are determined by examining all subsets of the matroid of sizes 1 through r(M ) + 1, organized by size, and rejecting independent sets and sets that contain a smaller circuit. Independent sets and bases are determined (for GF (q)-representable matroids) by Gaussian reduction on subsets of columns of the matroid’s base matrix. If all of these vectors agree (the degree vectors must agree up to order only), then matrices D1 and D2 are determined as defined above from the degree vectors of M1 and M2 . Since all of the degree vectors agree, D1 and D2 will contain exactly the same rows, although possibly in different order. Furthermore, it is clear that any isomorphism φ : M1 → M2 must preserve the circuit, independent set and basis degrees of each element. Let np denote the number of different rows present in D1 (and D2 ), and define partitions {p1 , . . . , pnp } and {q1 , . . . , qnp } of {1, . . . , n} so that all of the rows of D1 with indices in pi and all of the rows of D2 with indices in qi are identical. Then |pi | = |qi | for all i, so let Φi be the set of oneto-one and onto functions from pi to qi for each i. By choosing one map φi from each Φi , we may define a mapping φ : E(M1 ) → E(M2 ). There are |p1 |!|p2 |! · · · |pnp |! such candidate mappings. The isomorphism checker enumerates all such candidate mappings, using Dijkstra’s [6] algorithm for enumerating permutations, and for each mapping φ, checks whether φ maps each basis of M1 to a basis of M2 . If a map is found that preserves bases, the algorithm concludes that the matroids are isomorphic. Algorithm 2.2 Require: Matroids M1 and M2 n ← |v(M1 )| if not(n = |v(M2 )|) then answer ← f alse STOP end if r ← r(M1 ) 8

if not(r = r(M2 )) then answer ← f alse STOP end if I1 ← I(M1 ) chain1 ← chain(E(M1 ), I1 ) I2 ← I(M2 ) chain2 ← chain(E(M2 ), I2 ) if not(chain1 = chain2 ) then answer ← f alse STOP end if degrees11 ← degrees(E(M1 ), I1 ) degrees21 ← degrees(E(M2 ), I2 ) if not(sort(degrees11 ) = sort(degrees21 )) then answer ← f alse STOP end if C1 ← C(M1 ) chain1 ← chain(E(M1 ), C1 ) C2 ← C(M2 ) chain2 ← chain(E(M2 ), C2 ) if not(chain1 = chain2 ) then answer ← f alse STOP end if degrees12 ← degrees(E(M1 ), C1 ) degrees22 ← degrees(E(M2 ), C2 ) if not(sort(degrees12 ) = sort(degrees22 )) then answer ← f alse STOP end if B1 ← B(M1 ) B2 ← B(M2 ) if not(|B1 | = |B2 |) then answer ← f alse STOP end if 9

degrees13 ← degrees(E(M1 ), B1 ) degrees23 ← degrees(E(M2 ), B2 ) if not(sort(degrees13 ) = sort(degrees23 )) then answer ← f alse STOP end if P1 ← classes(E(M1 ), degrees11 , degrees12 , degrees13 ) P2 ← classes(E(M2 ), degrees21 , degrees22 , degrees23 ) np ← |P1 | Denote P1 by {p1 , . . . , pnp }, P2 by {q1 , . . . , qnp } for i ← 1 to np do πi ← f irstM ap(pi , qi ) end for hasM ore ← true while hasM ore f ← combineF unctions(π1 , . . . , πnp ) f ound ← true for all B ∈ B1 do if not(f (B) ∈ B2 ) then f ound ← f alse break end if end for if f ound then output f answer ← true STOP end if j ← −1 for i ← 1 to np do if not(πi = lastM ap(pi , qi )) then j←i break end if end for if j > 0 then πj ← nextM ap(pj , qj , πj ) for i ← 1 to j − 1 do πi ← f irstM ap(pi , qi ) 10

end for else hasM ore ← f alse end if end while answer ← f alse STOP We end this section with a brief description of software testing conducted on this program. This matroid generation program and the isomorphism checker are part of the Oid software system, a framework for combinatorial objects. The system has a large library of small reusable pieces of code. The lower-level routines, including the routines for independence testing, subset enumeration, permutation enumeration, and generation of circuits, bases and independent sets were unit-tested with a variety of matroids of various ranks, represented over various finite fields. Some of these, in particular the subset enumeration and permutation enumeration algorithms, rely on published algorithms. The isomorphism checker was also unit tested with a variety of examples. Most of the routines have been used in other applications. The results of the matroid generation program were verified by hand for several small values of r and q, and the total numbers of matroids generated were compared to all available previously published values. 4. The canonical construction path In [14] McKay provides a general technique for generating exhaustively families of combinatorial objects without isomorphs. Like the algorithms described above, this technique proceeds by beginning with smaller objects and extending along certain paths to obtain larger objects. For each object X, the technique posits a set L(X), essentially of paths to possible parents of X, and a set U (X) of paths to possible children of X. The technique’s algorithms proceed in a depth-first fashion. In one version, the technique’s algorithm iterates through U (X), noting not only each possible child, but the method in which the child was obtained. Each possible child, along with the extension method, is evaluated against a canonical form that specifies the form an object must take and highlights a feature of the object corresponding to the extension method. If the child, when put into the canonical form, exhibits its extension method as the canonical feature, it is kept. 11

In the example of graphs, McKay indicates that the canonical form could be a specific choice of labelling for a graph; a child graph noting its newly added vertex would be tested by applying the canonical labelling and checking whether the newly added vertex received the first label. In this version of the algorithm, all of the “true” children of a given object are generated and then duplicates are removed by isomorphism checking. In another version, instead of checking every element of U (X), the action of the automorphism group Aut(X) on X is extended to U (X), and only one element of each orbit is considered, in order to prevent duplication. In considering this technique for the generation of representable matroids, one notes that the problem is somewhat harder than graph generation, in that the set of objects to be generated, namely GF (q)-representable matroids, does not have a canonical form. Two completely different r × n matrices over GF (q) may represent the same matroid. Even if one were to consider the action of the operations listed in the definition of equivalent representations (row operations, column scalings, column permutations, and field automorphisms) on the set of representable matroids, the problem of unique representability mentioned above would prevent the identification of a canonical matrix form for a matroid. Furthermore, even in cases where unique representability does not pose a problem, conversion to a canonical form would be computationally intensive. Our answer to this problem has been to substitute the generation of all single-element extensions of a matroid, and comparison by isomorphism checking, for iteration through orbits of the action of Aut(X) on U (X). (This is similar to the first version of McKay’s algorithm described above). For the identification of “true” children using a canonical form, we substitute comparison of single-element deletions of candidate children to seed matroids considered earlier, again using isomorphism checking. Some calls to the isomorphism checker have been eliminated by regarding each matroid as a minor of P G(r − 1, q) (in particular two matroids are initially compared by determining if they contain exactly the same columns). 5. Applications of the generation program In this section we extend Acketa’s work on binary matroid generation. Table 2 gives some numbers of n-element, rank-r, isomorph-free binary combinatorial geometries. Acketa presented such numbers for n ≤ 8. We extend it to n ≤ 11 and in some cases up to n ≤ 12 elements. These numbers are exhaustive. Note that the number of elements is along the columns of the table and the rank is along the rows. Clearly there will 12

always be exactly one combinatorial geometry with rank r and r elements, viz, the identity matrix, and r − 1 combinatorial geometries with rank r and r + 1 elements.

2 3 4 5 6 7 8 9 10 11 12

2 3 4 1 1 0 1 2 1

5 0 1 3 1

6 0 1 4 4 1

7 0 1 5 8 5 1

8 0 0 6 15 14 6 1

9 0 0 5 29 38 22 7 1

10 0 0 4 46 105 80 32 8 1

11 0 0 3 64 273 312 151 44 9 1

12 0 0 2 89

266 59 10 1

Table 2: The number of isomorph-free binary combinatorial geometries. Table 3 gives the number n-element, rank-r, isomorph-free ternary combinatorial geometries for n ≤ 10 and r ≤ 7. In some cases we can go up to 12 elements. 2 3 4 5 6 7 2 1 1 0 0 0 0 3 1 2 3 4 4 4 1 3 8 19 5 1 4 15 6 1 5 7 1

8 0 3 44 61 26 6

9 0 3 91 277 162 40

10 0 2 199 1439 1381 375

11 0 1 401

12 0 1 806

Table 3: The number of isomorph-free ternary combinatorial geometries. We will end this paper with a short discussion of matroid generation when the seed matroids are 3-connected. A matroid is 3-connected if it is connected and E cannot be partitioned into subsets X and Y , each having at least two elements, such that r(X) + r(Y ) − r(M ) = 1. If N is a 3connected matroid, then an extension M of N is 3-connected provided e is not in a 1- or 2- element circuit of M and e is not a coloop of M . Likewise, M is a 3-connected coextension of N if M ∗ is an extension of N ∗ . Thus if the seed matroids are 3-connected then the extensions computed are all 13

3-connected. Let µ be a class of matroids that is closed under minors and isomorphism. A member M of µ is called a splitter for µ if µ has no 3-connected member having a proper M -minor. For a fixed finite field GF (q), a generation program like the one described above is most useful for finding splitters [11, 12]. Given a 3-connected matroid M and a 3connected minor N , the well-known Splitter Theorem by Seymour [15, p. 356] guarantees that there is a path from N to M via a sequence of singleelement extensions and coextensions. Therefore, in order to obtain all the 3-connected matroids with a specified 3-connected minor N one needs to begin with N and compute single-element extensions and coextensions. Similarly, we can also use the generation program to obtain 3-connected matroids without a specified set of minors. References [1] Acketa, D. M. (1984). A construction of non-simple matroids on at most 8 elements. J. Combin. Inform. System Sci. 9, 121-132. [2] Batten L. M. and Beutelspacher A. (1993), The theory of finite linear spaces, Cambridge University Press. [3] Betten, A. and Betten D. (1999), Linear spaces with at most 12 points, J. Combin. Des. 7 no. 2, 119–145. [4] Blackburn, J. E., Crapo, H. H., and Higgs, D. A. (1973), A catalogue of combinatorial geometries, Math. Comp. 27, 155-166. [5] Dharmatilake, J. S. (1994), Binary Matroids of Branch-width 3, Ph.D. dissertation, Ohio State University. [6] Dijkstra, E.W. (1976), A Discipline of Programming, Prentice-Hall, Upper Saddle River, New Jersey. [7] Dukes (2000), Counting and Probability in Matroid Theory, Ph.D. Dissertation, Trinity College Dublin. [8] Fournier, J. C. (1971), Representation sur un corps des matroides d’ordre ¡= 8. In Theorie des matroides (ed. Bruter, C. P.), Lecture Notes in Math. Vol. 211, pp. 50-61. Springer-Verlag, Berlin. [9] Greig, Malcolm (1999), Finite linear spaces. I. Proceedings of the Thirtieth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1999). Congr. Numer., 139, 167–200. [10] Greig, Malcolm (2002), Finite linear spaces. II. Special issue in honour of Ronald C. Mullin, Part II. Des. Codes Cryptogr., 27, no. 1-2, 25–47. [11] Kingan, S. R. (1994), Matroid Stucture, Ph.D. dissertation, Louisiana State University. 14

[12] Kingan, S. R. (1996), Binary matroids without prisms, prism duals and cubes, Discrete Math. 152, 211-224. [13] Kingan, S. R. (1997), A generalization of a graph result of D. W. Hall, Discrete Math. 173, 129-135. [14] McKay, B. D. (1998), Isomorph-free exhaustive generation J. Algorithms, 26, 306–324. [15] Oxley, J. G. (1992), Matroid Theory. Oxford University Press, New York. [16] P. D. Seymour (1980), Decomposition of regular matroids. J. Combin. Theory Ser. B 28,305-359. [17] Whitney, H. (1935), On the abstract properties of linear dependence. Amer. J. Math. 57, 509-533.

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