The Excluded Minors for GF(4)-Representable Matroids

Geelen, Jim; Gerards, A.M.H.; Kapoor, Ajai

doi:10.1006/jctb.2000.1963

Cited by 81 publications

(183 citation statements)

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“…See (2.1) and Seymour [16]; the claim follows from the fact that U 2,4 is the only forbidden minor for binary matroids. On the other hand, since we know the seven forbidden minors [3] for matroid representability over the field GF (4), it follows from Theorem 3.1 that the matroid N -minor problem is hard for at least some N . Such an argument, however, does not apply to minors N that are cycle matroids of graphs (graphic N ), and hence representable over all fields.…”

Section: Matroid Minorsmentioning

confidence: 99%

On Matroid Representability and Minor Problems

Hliněný

2006

Lecture Notes in Computer Science

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Abstract. In this paper we look at complexity aspects of the following problem (matroid representability) which seems to play an important role in structural matroid theory: Given a rational matrix representing the matroid M , the question is whether M can be represented also over another specific finite field. We prove this problem is hard, and so is the related problem of minor testing in rational matroids. The results hold even if we restrict to matroids of branch-width three.Keywords: matroid representability, minor, finite field, spike, swirl. 2000 Math subject classification: 05B35, 68Q17, 68R05. IntroductionWe postpone necessary formal definitions until later sections. Matroids present a wide combinatorial generalization of graphs. A useful geometric essence of a matroid is shown in its vector representation over a field ; the elements-vectors of the representation can be viewed as points in the projective geometry over . Not all matroids, however, have vector representations. That is why the question of -representability of a matroid is important to solve. Another motivation for our research lies in a current hot trend in structural matroid theory; work of Geelen, Gerards and Whittle, e.g. [4,5] extending significant portion of the Robertson-Seymour's Graph Minors project [15] to matroids. It turns out that matroids represented over finite fields play a crucial role in that research, analogous to the role played by graphs embedded on surfaces in Graph Minors. Such a role is further justified by related works concerning logic and complexity aspects of matroids, e.g. our [8,10], and by a somehow surprising connection of binary matroids with graph rank-width [1] of Courcelle and Oum.In this paper we prove that it is hard to decide whether a matroid given by a vector representation over the rational numbers, has a vector representation over a specific finite field (Theorems 3.1 and 4.1). In particular this result implies that also the problem of minor testing in rational matroids is generally hard. We moreover prove that the minor testing problem is hard even for a certain small planar minor (Theorem 5.6). Matroids and Vector RepresentationsWe refer to Oxley's book [12]. Since matroid theory seems not widely known among computer scientists, we should briefly review some basic terms here: A matroid is a pair M = (E, B) where E = E(M ) is the finite ground set of M (elements of M ), and B ⊆ 2 E is a nonempty collection of bases of M , no two of which are in an inclusion. Moreover, matroid bases satisfy the "exchange axiom": if B 1 , B 2 ∈ B and x ∈ B 1 \ B 2 , then there is y ∈ B 2 \ B 1 such that (B 1 \ {x}) ∪ {y} ∈ B. Subsets of bases are called independent sets, and the remaining sets are dependent. Minimal dependent sets are called circuits. All bases have the same cardinality called the rank r(M ) of the matroid. The rank function r M (X) in M assigns the maximal cardinality of an independent subset of a set X ⊆ E(M ). A set X is spanning if r M (X) = r(M ), and maximal non-spanning sets are called hyperp...

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Section: Matroid Minorsmentioning

confidence: 99%

On Matroid Representability and Minor Problems

Hliněný

2006

Lecture Notes in Computer Science

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“…Then Now if we look at the projective space P G(2, 3) from the vector space V (3, Z 3 ), we can see that there are 13 points and there exist four hyperplanes passing through each point in P G (2,3). Also each hyperplane contains four points.…”

Section: Representationsmentioning

confidence: 99%

“…Outer four points in the diagram are in a hyperplane of P G (2,3). By deleting these points, we have the following geometric representation of AG (2,3).…”

Section: Representationsmentioning

confidence: 99%

“…By deleting these points, we have the following geometric representation of AG (2,3). Let's take a maximal tree …”

Section: Representationsmentioning

confidence: 99%

“…The representability of U m,n are studied by many authors. ( [1], [2], [5], [7]) Now, if F is a finite field, then F has exactly p k -elements for some prime p. When k > 1, GF (p k ) can be constructed as follows. Let h(w) be a irreducible polynomial of degree k with coefficients Z p .…”

Section: Introductionmentioning

confidence: 99%

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