On the interplay between graphs and matroids

Oxley, James

doi:10.1017/cbo9780511721328.010

Cited by 17 publications

(15 citation statements)

References 42 publications

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“…[16,125,126]. The natural question arises, whether Feferman-Vaught-type theorems should not be formulated also for matroids, rather than relational structures.…”

Section: Other Structuresmentioning

confidence: 99%

Algorithmic uses of the Feferman–Vaught Theorem

Makowsky

2004

Annals of Pure and Applied Logic

161

222

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The classical Feferman-Vaught Theorem for First Order Logic explains how to compute the truth value of a ÿrst order sentence in a generalized product of ÿrst order structures by reducing this computation to the computation of truth values of other ÿrst order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by L auchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and deÿnability theory.Here we give a uniÿed presentation, including some new results, of how to use the FefermanVaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL.We then extend the technique to graph polynomials where the range of the summation of the monomials is deÿnable in MSOL. Here the Feferman-Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically.Finally, we discuss extensions of MSOL for which the Feferman-Vaught Theorem holds as well. (J.A. Makowsky).1 At the occasion of his 75th birthday. It is also a tribute to Feferman's work in model theory, which went seemingly unnoticed at the celebrations of his 70th birthday. Feferman returned several times to topics related to the preservation of truth under various model theoretic constructions, so in [54][55][56][57][58]60]. The FefermanVaught Theorem can also be viewed as falling into this category.

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“…[16,125,126]. The natural question arises, whether Feferman-Vaught-type theorems should not be formulated also for matroids, rather than relational structures.…”

Section: Other Structuresmentioning

confidence: 99%

Algorithmic uses of the Feferman–Vaught Theorem

Makowsky

2004

Annals of Pure and Applied Logic

161

222

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“…If c 2 ≤ 4, then c ≥ c 1 ≥ 2(c−k+1)−4. From this we obtain that c ≤ 2k+2 < 2k+ 5 2 . Thus we may assume that c 2 ≥ 5.…”

Section: Case 2 Suppose C 2 < Kmentioning

confidence: 77%

Bonds Intersecting Cycles In A Graph

McGuinness

2005

Combinatorica

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“…In this section, we shall use Theorem 1.1 to verify Conjecture 1.1 for cographic matroids (see also [4], Question 3.13).…”

Section: Covering Edges With At Most C * Cyclesmentioning

confidence: 99%

“…Assuming that one could cover the elements of a matroid M with at most c * (M ) circuits so that each element was covered at least twice, we would have the bound 2|e(M )| ≤ cc * , or |e(M )| ≤ 1 2 cc * . In light of this, Oxley [4] posed the following conjecture: Conjecture 1.1. For any connected matroid M with at least 2 elements, one can find a collection of at most c * (M ) circuits which cover each element of M at least twice.…”

Section: Introductionmentioning

confidence: 99%

Circuits Through Cocircuits In A Graph With Extensions To Matroids

McGuinness

2005

Combinatorica

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We show that for any k-connected graph having cocircumference c * , there is a cycle which intersects every cocycle of size c * − k + 2 or greater. We use this to show that in a 2connected graph, there is a family of at most c * cycles for which each edge of the graph belongs to at least two cycles in the family. This settles a question raised by Oxley.A certain result known for cycles and cocycles in graphs is extended to matroids. It is shown that for a k-connected regular matroid having circumference c ≥ 2k if C1 and C2 are disjoint circuits satisfying r(C1) + r(C2) = r(C1 ∪ C2), then |C1| + |C2| ≤ 2(c − k + 1).

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On the interplay between graphs and matroids

Cited by 17 publications

References 42 publications

Algorithmic uses of the Feferman–Vaught Theorem

Algorithmic uses of the Feferman–Vaught Theorem

Bonds Intersecting Cycles In A Graph

Circuits Through Cocircuits In A Graph With Extensions To Matroids

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