Mixed dominating matrices

Fischer, Klaus G.

doi:10.1016/s0024-3795(98)80018-9

Cited by 5 publications

(7 citation statements)

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“…The submatrix in rows 1,2 and 4 and columns (1,2), (2,3) and (4,5) has determinant 2. Note also that columns (1,8), (2,9), (2,6) and (1,7) contain a triangular basis with -1s on the diagonal. 7.…”

Section: Existence Of a Separating Path A Directed Pathmentioning

confidence: 99%

“…Complete Intersection Affine Semigroups. In this section we review the results of [4], [5], [6] and [7] on general complete intersection affine semigroups, which we will specialize later. Suppose that T is a set of n nonzero vectors in Q m .…”

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confidence: 99%

“…The mixed dominating property of a matrix was shown to imply a certain block structure, in Theorem 2.2 of [6] (see also [7]): Call the elements of T corresponding to columns of A red and the elements of T corresponding to B green. A geometric interpretation of the decomposition (1) is that the span of the red elements of T intersects the span of the green elements of T in a line, and the intersection of S with this line is generated by an element α of both of the subsemigroups corresponding to the red and the green elements of T .…”

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confidence: 99%

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Acyclic digraphs giving rise to complete intersections

Morris¹

2019

J. Commut. Algebra

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We call a directed acyclic graph a CIdigraph if a certain affine semigroup ring defined by it is a complete intersection. We show that if D is a 2-connected CI-digraph with cycle space of dimension at least 2, then it can be decomposed into two subdigraphs, one of which can be taken to have only one cycle, that are CI-digraphs and are glued together on a directed path. If the arcs of the digraph are the covering relations of a poset, this is the converse of a theorem of Boussicault, Feray, Lascoux and Reiner [2]. The decomposition result shows that CI-digraphs can be easily recognized. 1991 AMS Mathematics subject classification. 14M10, 05C25, 05C20. Keywords and phrases. complete intersection, cycle basis, directed graph.

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Section: Existence Of a Separating Path A Directed Pathmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Acyclic digraphs giving rise to complete intersections

Morris¹

2019

J. Commut. Algebra

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“…For checking this equality we use Theorem 2.4. It is worth pointing out that in [7] the authors give a polynomial algorithm to decide if a matrix is dominating; thus one can check if the equality P G = (B w 1 , . .…”

Section: The Connected Components Of G Are Odd Bands or Even Möbius B...mentioning

confidence: 99%

Graphs and complete intersection toric ideals

2015

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Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal P G is a complete intersection or not. Whenever P G is a complete intersection, the algorithm also returns a minimal set of generators of P G . Moreover, we prove that if G is a connected graph and P G is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

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“…(c) AD has no mixed cycle for each strict signing such that AD is row-mixed. Theorem 2.2 (see [2], [3]). If a strictly row-mixable matrix A has signed nullspace, then there exist matrices B and C (possibly with no rows) and nonzero vectors b and c such that B and C are strictly row-mixable matrices with signed null-spaces, The converse also holds.…”

Section: I K and Distinctmentioning

confidence: 99%