Inseparability graphs of oriented matroids

Roudneff, Jean-Pierre

doi:10.1007/bf02122686

Cited by 7 publications

(9 citation statements)

References 7 publications

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“…Kelly and Moser [20]) of the point e in a realization of M. From this it will be derived in §2 that if the map 3ï(M) -► 3ê(M\e) is surjective (in which case we call e reducible), then the two spaces are homotopy equivalent. Moreover, relating reducibility to the combinatorial concept of sign-invariant pairs in oriented matroids [3,10,32], we obtain a generalization of a result of J. P. Roundneff [33, Proposition 5.1] which relates the realizability of rank 3 oriented matroids to the existence of sign-invariant pairs.…”

Section: Introductionmentioning

confidence: 79%

“…The covariant pairs of M are contravariant in M* and vice versa. The graph 1G(M) on E whose edges are the sign-invariant pairs of M is the inseparability graph of M [3,10,32]. A sign-invariant pair of M is sign-invariant in every minor of M, and IG(M) = IG(M*).…”

Section: Some Construction Techniques For Oriented Matroidsmentioning

confidence: 99%

“…. In [32] J.-P. Roundneff gave a complete classification of all inseparability graphs of uniform oriented matroids, and he showed that all such graphs can be realized in oriented matroids arising from simplicial polytopes. In [33, Proposition 5.1] he also proved that a rank 3 oriented matroid M is realizable if every minor of M has sign-invariant pairs.…”

Section: Some Construction Techniques For Oriented Matroidsmentioning

confidence: 99%

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On the topology and geometric construction of oriented matroids and convex polytopes

Richter¹,

Sturmfels²

1991

Trans. Amer. Math. Soc.

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Abstract. This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M . We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank 3 oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces 32(M) are path-connected.We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes.

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Section: Introductionmentioning

confidence: 79%

Section: Some Construction Techniques For Oriented Matroidsmentioning

confidence: 99%

Section: Some Construction Techniques For Oriented Matroidsmentioning

confidence: 99%

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On the topology and geometric construction of oriented matroids and convex polytopes

Richter¹,

Sturmfels²

1991

Trans. Amer. Math. Soc.

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“…For an oriented matroid M, the inseparability graph of M is the graph with vertex set E(M) in which e, f ∈ E(M) are connected by an edge if and only if they are inseparable in M. Inseparability graphs were studied in [4,5,12]. In this section, we study cliques in the inseparability graph, in other words, we study pairwise inseparable sets in oriented matroids.…”

Section: Pairwise Inseparable Setsmentioning

confidence: 97%

Adjacency, Inseparability, and Base Orderability in Matroids

Keijsper

Pendavingh

Schrijver

2000

European Journal of Combinatorics

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Two elements in an oriented matroid are inseparable if they have either the same sign in every signed circuit containing them both or opposite signs in every signed circuit containing them both. Two elements of a matroid are adjacent if there is no M(K 4 )-minor using them both, and in which they correspond to a matching of K 4 .We prove that two elements e, f of an oriented matroid are inseparable if and only if e, f are inseparable in every M(K 4 ) or U 2 4 -minor containing them. This provides a link between inseparability in oriented matroids (introduced by Bland and Las Vergnas) and adjacency in binary matroids (introduced by Seymour).We define the concepts of base orderable and strongly base orderable subsets of a matroid, generalizing the definitions of base orderable and strongly base orderable matroids. Strongly base orderable subsets can be used to obtain packing and covering results, generalizing results of Davies and McDiarmid, as was shown in a previous paper.In this paper, we prove that any pairwise inseparable subset of an oriented matroid is base orderable. For binary matroids we derive the following characterization: a subset is strongly base orderable if and only if it is pairwise adjacent.

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“…We describe a possible construction of all these odd dimensional cyclic polytopes. Making use of the some results on 'inseparability graphs' of oriented matroids [11,32], we prove two results that emphasize the very special place of alternating oriented matroids among realizable cyclic matroid polytopes of even rank. To conclude, we provide a characterization of the admissible orderings, i.e., the linear orderings of the vertices of a cyclic (matroid) polytope such that Gale's evenness criterion holds.…”

Section: Introductionmentioning

confidence: 98%

Cyclic Polytopes and Oriented Matroids

Cordovil¹,

Duchet

2000

European Journal of Combinatorics

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Consider the moment curve in the real euclidean space R d defined parametrically by the mapis the convex hull of n > d different points on this curve. The matroidal analogs are the alternating oriented uniform matroids. A polytope (resp. matroid polytope) is called cyclic if its face lattice is isomorphic to that of C d (t 1 , . . . , t n ). We give combinatorial and geometrical characterizations of cyclic (matroid) polytopes. A simple evenness criterion determining the facets of C d (t 1 , . . . , t n ) was given by Gale [21]. We characterize the admissible orderings of the vertices of the cyclic polytope, i.e., those linear orderings of the vertices for which Gale's evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes.

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Inseparability graphs of oriented matroids

Cited by 7 publications

References 7 publications

On the topology and geometric construction of oriented matroids and convex polytopes

On the topology and geometric construction of oriented matroids and convex polytopes

Adjacency, Inseparability, and Base Orderability in Matroids

Cyclic Polytopes and Oriented Matroids

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