M. Conforti, G. Cornuejols, M.R. Rao, Decomposition of Balanced Matrices, Journal of Combinatorial Theory, Series B 77, 292-406 (1999).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 double star cutset. Following Burlet and Uhry s work on parity graphs [15] many people [14, 20, 21, 27, 37, 19] tried to ....

M. Conforti, G. Cornuejols, and M. R. Rao, Decomposition of balanced matrices, J. Combin. Theory Ser. B 77 (1999), 292--406.

....of all labels in C is 1 mod 2, which implies that C has an odd number of edges. Thus C is an odd wheel or a 3 odd path con guration, a contradiction. 2 14 Decomposition Theorem In this section, we present a decomposition theorem for balanced 0; 1 matrices due to Conforti, Cornu ejols and Rao [20] and Conforti, Cornu ejols, Kapoor and Vu skovi c [18] and we give an outline of its proof. By the result of Section 12, it suces to decompose balanceable 0,1 matrices. We state the decomposition theorem in terms of the bipartite representation, as de ned in Section 10. 14.1 Cutsets A set S of ....

....x. In a bipartite graph, an extended star is de ned by disjoint subsets T , A, N of V (G) and a node x 2 T such that (i) A 6= and A [ N N(x) ii) every node of A is adjacent to every node of T , iii) if jT j 2, then jAj 2. This concept was introduced by Conforti, Cornu ejols and Rao [20] and is illustrated in Figure 2. An extended star cutset is one where T [ A [ N is a node cutset. An extended star cutset with N = is called a biclique cutset. An extended star cutset having T = fxg is called a star cutset. Note that a star cutset is a special case of a biclique cutset. A graph ....

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M. Conforti, G. Cornuejols, M. R. Rao, Decomposition of balanced matrices, Journal of Combinatorial Theory B 77 (1999) 292-406.

....recognition algorithm, 2 join, 6 join, extended star cutset Running head: Recognition of balanced 0# Sigma1 matrices 1 Introduction A0# Sigma1 matrix is balanced if, in every square submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. In [3], Conforti, Cornu ejols and Rao prove a decomposition theorem for balanced 0# 1 matrices and they use it to obtain a polynomial time recognition algorithm for these matrices. In this paper, using a similar approach, we give a polynomial time recognition algorithm for balanced 0# Sigma1 matrices, ....

....4. Hence if (H#x) is an odd wheel, the hole H has weight 2 modulo 4. So a signed bipartite graph that contains an odd wheel is not balanced. 2.1 2 Join Decomposition A 2 join E(KA 1 A 2 ) E(KB 1 B 2 )isrigid if A 1 [B 1 or A 2 [B 2 induces a biclique. The following easy result was proved in [3]. Lemma 2.2 Let G beabipartite graph that has no extended star cutset. Then G has no rigid 2 join. Let KA 1 A 2 and KB 1 B 2 define a 2 join of G that is not rigid. The blocks G 1 and G 2 of the 2 join decomposition are defined as follows. For i =1# 2, let G 0 i be the subgraph of G n (E(KA 1 ....

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M. Conforti, G. Cornu'ejols, M. R. Rao, Decomposition of balanced matrices, Journal of Combinatorial TheoryB77(1999) 292-406.

....the strong perfect graph conjecture. So, part of the motivation for this researchistodevelop techniques that may then be used to study odd hole free graphs. 2 It is also worth pointing out that decompositions similar to the ones used here led to the recognition algorithm for balanced matrices [8], 4] 1.2 Notation and Background In this paper we use standard graph theory notation (see for example [15] Given a node set S and a graph G, GnS denotes the subgraph of G obtained by removing the node set S and the edges with at least one node in S. S V (G)isanode cutset of a connected ....

....1 and H 2 . Also, for i =1# 2, jH i j 2andifA and C (resp. B and D) are both of cardinality 1, then the graph induced by H 1 (resp. H 2 )isnotachordless path. D H 1 H 2 A C B Figure 1: 2 join Star cutsets were introduced byChv atal [3] and 2 joins byCornu ejols and Cunningham [10] In [8] and [4] 2 joins, star and double star cutsets are used for recognizing balanced 0# 1 matrices and, together with another edge cutset, the 6 join, for recognizing balanced 0# Sigma1 matrices. Wenowintroduce two classes of graphs that have no 2 join and no star, double star or triple star ....

M. Conforti, G. Cornu'ejols and M.R. Rao, Decomposition of balanced matrices, Journal of Combinatorial Theory B 77 (1999) 292-406.

....to 0# Sigma1 matrices byTruemper [16] A 0# Sigma1 matrix is balanced if, in every square submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This paper extends the decomposition of balanced 0# 1 matrices obtained byConforti, Cornu ejols and Rao [8] to the class of balanced 0# Sigma1 matrices. As a consequence, we obtain a polynomial time algorithm for recognizing balanced 0# Sigma1 matrices. The algorithm is discussed in a sequel paper. Dipartimento di Matematica Pura ed Applicata, Universit adiPadova, Via Belzoni 7, 35131 Padova, Italy. ....

....star (x# T # A# R) is defined by disjoint subsets T , A, R of V (G) and a node x 2 T such that (i) A [ R N(x) ii) the node set T [ A induces a biclique (with node set T on one side of the bipartition and node set A on the other) iii) if jTj2, then jAj2. This concept was introduced in [8]. An extended star cutset is an extended star (x# T # A# R) where T [A[R is a node cutset. When R = # the extended star is a biclique, and the cutset is called a biclique cutset. 6 Joins Let G be a connected bipartite graph containing a biclique KA 1 A 2 with the property that its edge set E(KA ....

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M. Conforti, G. Cornu'ejols and M. R. Rao, Decomposition of balanced matrices, Journal of Combinatorial TheoryB77(1999) 292-406.

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M. Conforti, G. Cornuejols, M.R. Rao, Decomposition of Balanced Matrices, Journal of Combinatorial Theory, Series B 77, 292-406 (1999).

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