P.D. Seymour. Decomposition of regular matroids. Journal of Combinatorial Theory (B), 28:305--359, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ABSTRACT: The key to Seymour s Regular Matroid Decomposition Theorem is his result that each 3 connected regular matroid with no R 10 or R 12 minor is graphic or cographic. We present a proof of this in terms of signed graphs. 1 Introduction Seymour s Regular Matroid Decomposition Theorem [3] says that each regular matroid can be obtained from graphic matroids, their duals, and copies of R 10 by taking 1 , 2 , and 3 sums. The key part of his proof is the following result. 1) Theorem: Each 3 connected regular matroid that is neither graphic nor cographic contains an R 10 or R 12 ....

....3G1. CWI, Postbus 94079, 1090 GB Amsterdam, The Netherlands and Department of Mathematics and Computer Science, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands. quite easy to nd. The only two proofs of Theorem (1) known to us, Seymour s original proof in [3] (see Truemper [5] for a shorter version) and the proof in this paper, are both more complicated than existing proofs of Tutte s characterization of graphic matroids [6] even though both proofs use that characterization. This may appear unexpected. However, the union of two matroid classes may ....

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P.D. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory, Series B 28 (1980) 305-359.

.... by i c or dy i e that we have jy i z i j = j2W i j 1 3 Extension to Larger Hereditary Discrepancies Seymour proved that any totally unimodular matrix (and therefore any hypergraph of hereditary discrepancy 1) can be built out of matrices coming from a relatively simple collection [12]. Unfortunately, we have no such decomposition theorem for hypergraphs of 3 hereditary discrepancy d where d is xed and larger than 1. Furthermore, there appears to be little discussion of constructions for rich classes of such hypergraphs in the literature. One natural way to construct ....

P. D. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory (B) 28 (1980), 305-359.

.... The following claims and propositions provide the sufficient conditions for a linear programming problem in order to yield the optimal integer solution (ILP) Totally Unimodular Matrix (TU) A matrix A is totally unimodular (TU) if every square sub matrix of A has determinant 1, Gamma1, or 0 [4, 5]. Lemma 1 If matrix A is TU , the linear programming relaxation can solve the ILP [2, 4] A sufficient condition for matrix A = a i j ) m Thetan to be totally unimodular, proposed by Heller and Tompkins [4] is Lemma 2 A matrix Am Thetan = a i j ) m Thetan is TU if ffl a i j 2 f Gamma1; ....

P. D. Seymour. "Decomposition of Regular Matroids". In Journal of Combinatorial Theory, B28, pp. 305-359, 1980.

....and relatively transparent structure or one of a number of prescribed structural faults, along which it can be decomposed. An early example is the Kronecker Decomposition Theorem for Abelian groups; a celebrated example in combinatorics is Paul Seymour s decomposition theorem for regular matroids [38]. Berge s notions of balanced matrices and perfect graphs have been treated this way. Conforti, Cornuejols, and Rao [22] proved that every balanced matrix is either totally unimodular (and therefore decomposable in its own right by virtue of Seymour s theorem) or has a structural fault called a ....

P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305--359.

.... Yannakakis [31] for restricted unimodular matrices, the results of Anstee and Farber [1] Hoffman, Kolen and Sakarovitch [27] Golumbic and Goss [24] for totally balanced matrices, the results of Conforti and Rao for strongly balanced matrices [16] and linear balanced matrices [17] and Seymour s [29]characterization of totally unimodular matrices are all in this spirit. Restricted balanced graphs A graph is restrictedbalanced if every cycle is balanced. Restricted balanced graphs were introduced by Commoner [10] A graph is basic if it is bipartite and all the nodes in one side of the ....

P. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory B 28 (1980) 305-359.

....this common extremal matroid is obtained by adding a point freely on a 3 point line of M(K r 2 ) contracting that point, and simplifying the resulting matroid. The deeper motivation for this paper is the desire to obtain analogues of Seymour s decomposition theory for regular matroids [7]. The results of [3, 4] and this paper present the tantalising hope that such analogues are feasible. In such a development, minors of the extremal matroids for dyadic, near regular, and 6 p 1matroids would be expected to play a role similar to that played by graphic matroids in Seymour s ....

Seymour, P. D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305-359.

....(U 1 ; U 2 ) of U such that jU 1 j k, jU 2 j k and r(U 1 ) r(U 2 ) r(U) k 1. The matroid M is k connected if it has no (k 1) separation. The k separation is strict if jU 1 j k, jU 2 j k. A matroid is internally k connected if it has no strict (k 1) separation. Theorem 6. 11 ( Seymour [59]) Every 3 connnected, internally 4 connected regular matroid is graphic, cographic or a 10 element matroid R 10 . Theorem 6.12 ( Seymour [60] Let M be a 3 connected binary matroid with no F 7 minor. Then M is regular or M = F 7 . 6.3 Signed Matroid Let M be a binary matroid and S V (M) a ....

P. Seymour, Decomposition of Regular Matroids, Journal of Combinatorial Theory B 28 (1980) 305-359.

....P , that is, by declaring P to be independent and leaving the remaining independent sets the same. Note that W 2 is the matroid U 2;4 . The terms rim and spoke will be used in the obvious way in W n . The following result is a straightforward consequence of Seymour s Split4 ter Theorem [6]. For a discussion of this theorem and its consequences see [5, Chapter11] 2.8) Let M(E) is be non binary, 3 connected matroid. If M is not a whirl, there exists x 2 E such that either Mnx or M=x is non binary and 3connected. 3 The 3 connected Case of Theorem1.1 In this section we consider the ....

P. D. Seymour. Decomposition of regular matroids. J. Combin. Theory Ser. B, 28:305-359, 1980.

....except possibly GF (2) while regular matroids are the ones representable over all elds. It would be of interest to know what results for regular matroids extend to near regular matroids. Oxley [12, Problem 14.1. 10] asks if there is an analogue of Seymour s regular matroid decomposition theorem [14] for the class of matroids representable over all odd primes. This is just the class of dyadic matroids. I believe that this is a very interesting question. As the class of dyadic matroids contains the class of near regular matroids it is natural to begin by seeking a Seymour type decomposition ....

....with the property that si(M=a) si(M=b) si(M=c) si(M=a; b) and si(M=a; c) are 5 all non binary and 3 connected. 2 The above result is also essential in this paper. It is used in the proofs of Theorem 5.1 and Lemma 5.2. It is assumed that the reader is familiar with Seymour s Splitter Theorem [14]. For a good discussion of this theorem and its consequences see [12, Chapter 11] One consequence that is used several times in this paper is (2.2) Let M be a non binary, 3 connected matroid. If M is not a whirl, then there exists x 2 E(M) such that either Mnx or M=x is non binary and ....

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Seymour, P. D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305-359.

....degrees, they are quite well understood. For example, excluded minor characterizations are known for binary matroids [20] regular matroids [20] ternary matroids [1, 18] 6 p 1 matroids [8] and near regular matroids [7] There is a very satisfactory decomposition theory for regular matroids [19], and a decomposition theory for 6 p 1 matroids in terms of near regular matroids [9] In the light of the above and the recent excluded minor characterization of quaternary matroids [8] it is natural to turn attention to sets of elds containing GF (4) and to attempt to describe the classes ....

Seymour, P. D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305-359.

....: p r g be the vertices of an r simplex. Then W r is obtained by placing a point freely on each of the lines fp 1 ; p 2 g; fp 2 ; p 3 g; fp r 1 ; p r g and fp r ; p 1 g. If r = 2, then W 2 = U 2;4 . The following results are straightforward consequences of Seymour s Splitter Theorem [20]. For a discussion of this theorem and its consequences see [17, Chapter 11] 2.4) Let M and N be 3 connected matroids with the property that N is a non binary minor of M , jE(N)j 4, and if N is a whirl, then M has no larger whirl as a minor. Then there is a sequence M 0 ; M 1 ; M n of ....

Seymour, P. D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305-359.

....1; 1) T ; x b = 1; 1; 1; 0) T and x c = 1; 1; 1; 1) T . 3) R 10 : x a = 1; 0; 1; 0; 0) T ; x b = 1; 0; 1; 0; 1) T ; x c = 1; 0; 1; 1; 0) T ; x d = 1; 0; 1; 1; 1) T ; x e = 1; 1; 0; 0; 0) T ; x f = 1; 1; 1; 1; 1) T . Note that (1) 2) are easy and (3) can by found in [24] (p. 357) The rows of the matrix [BF7 jx b ] resp. BF7 jx c ] span the circuit space of a matroid known as AG(3; 2) resp. S 8 ) If [BN jx] is a matrix whose rows span the circuits of M , then by definition of sources, P ort(M; e) is a source of N . Thus, Remark 3.4. F 7 has a unique ....

....A matroid M has a k separation E 1 ; E 2 if and only if its dual M does (Oxley [21] 4.2.7) A matroid is k connected if it has no (k 1) separation and is internally k connected if it has no strict (k 1) separation. A 2 connected matroid is simply said to be connected. We now follow Seymour [24] when presenting k sums. Let M 1 ; M 2 be binary matroids whose element sets E(M 1 ) E(M 2 ) may intersect. We define M 1 4M 2 to be the binary matroid on E(M 1 ) 4 E(M 2 ) where the cycles are all the subsets of E(M 1 ) 4 E(M 2 ) of the form C 1 4 C 2 where C i is a cycle of M i , i = 1; 2. The ....

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P. D. Seymour. Decomposition of regular matroids. J. of Combinatorial Theory B, 28:305--359, 1979.

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P.D. Seymour. Decomposition of regular matroids. Journal of Combinatorial Theory (B), 28:305--359, 1980.

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P.D. Seymour. Decomposition of regular matroids. Journal of Combinatorial Theory (B), 28:305--359, 1980.

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P.D. Seymour. Decomposition of regular matroids. Journal of Combinatorial Theory (B), 28:305--359, 1980.

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P.D. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory (B), 28 (1980), 305-359.

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Seymour, P. D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305--359. 26

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Seymour, P. D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305--359. 35

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Seymour, P. D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305--359.

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P. D. Seymour. Decomposition of regular matroids. J. Combin. Theory Ser. B, 28(3):305--359, 1980.

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P.D. Seymour, Decomposition of Regular Matroids, J. Combin. Theory Ser. B 28 (1980), 305--359.

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P.D. Seymour, Decomposition of Regular Matroids, J. Combin. Theory Ser. B 28 (1980), 305-359.

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P .D. Seymour, Decomposition of Regular Matroids, J. Comb. Theory, Ser. B., 28, (1980), 305-359.

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Seymour, P.D., Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305--359.

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P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B, 28 (1980), 305--359.

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