Cycle Bases for Lattices of Binary Matroids with No Fano Dual Minor and Their One-Element Extensions

We generalize a minimal 3-connectivity result of Halin from graphs to binary matroids. As applications of this theorem to minimally 3-connected matroids, we obtain new results and short inductive proofs of results of Oxley and Wu. We also give new short inductive proofs of results of Dirac and Halin on minimally k-connected graphs for k ∈ {2,3}.

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A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs

Mathematical Programming ◽

10.1007/s10107-010-0425-z ◽

2010 ◽

Vol 133 (1-2) ◽

pp. 203-225 ◽

Cited By ~ 11

Author(s):

João Gouveia ◽

Monique Laurent ◽

Pablo A. Parrilo ◽

Rekha Thomas

Keyword(s):

Semidefinite Programming ◽

Binary Matroids

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The Penrose Polynomial of Binary Matroids

Monatshefte für Mathematik ◽

10.1007/s006050070020 ◽

2000 ◽

Vol 131 (1) ◽

pp. 1-13 ◽

Cited By ~ 6

Author(s):

Martin Aigner ◽

Hans Mielke

Keyword(s):

Binary Matroids

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Binary Matroids

Combinatorial Geometries ◽

10.1017/cbo9781107325715.004 ◽

1987 ◽

pp. 28-39 ◽

Cited By ~ 1

Author(s):

J.C. Fournier

Keyword(s):

Binary Matroids

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Mathematical models and a constructive heuristic for finding minimum fundamental cycle bases

Yugoslav journal of operations research ◽

10.2298/yjor0501015l ◽

2005 ◽

Vol 15 (1) ◽

pp. 15-24 ◽

Cited By ~ 4

Author(s):

Leo Liberti ◽

Edoardo Amaldi ◽

Francesco Maffioli ◽

Nelson Maculan

Keyword(s):

Linear Programming ◽

Total Cost ◽

Fundamental Cycle ◽

Cycle Basis ◽

Cpu Time ◽

Constructive Heuristic ◽

Cycle Bases ◽

Time Required ◽

Application Fields

The problem of finding a fundamental cycle basis with minimum total cost in a graph arises in many application fields. In this paper we present some integer linear programming formulations and we compare their performances, in terms of instance size, CPU time required for the solution, and quality of the associated lower bound derived by solving the corresponding continuous relaxations. Since only very small instances can be solved to optimality with these formulations and very large instances occur in a number of applications, we present a new constructive heuristic and compare it with alternative heuristics.

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Matchings, Cycle Bases, and the Maximum Genus of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/2479 ◽

2012 ◽

Vol 19 (3) ◽

Cited By ~ 1

Author(s):

Michal Kotrbčík ◽

Martin Škoviera

Keyword(s):

Spanning Tree ◽

Spanning Trees ◽

Perfect Matching ◽

Intersection Graph ◽

Matching Number ◽

Maximum Genus ◽

Cycle Space ◽

Intersection Graphs ◽

Connected Graphs ◽

Cycle Bases

We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is paid to cycle bases consisting of fundamental cycles with respect to a given spanning tree. For $4$-edge-connected graphs, the intersection graph with respect to any spanning tree (and, in fact, with respect to any basis) has either a perfect matching or a matching missing exactly one vertex. We show that if a graph is not $4$-edge-connected, different spanning trees may lead to intersection graphs with different matching numbers. We also show that there exist $2$-edge connected graphs for which the set of values of matching numbers of their intersection graphs contains arbitrarily large gaps.

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Dependencies Among Dependencies in Matroids

The Electronic Journal of Combinatorics ◽

10.37236/8742 ◽

2019 ◽

Vol 26 (3) ◽

Author(s):

James Oxley ◽

Suijie Wang

Keyword(s):

Binary Matroids

In 1971, Rota introduced the concept of derived matroids to investigate "dependencies among dependencies" in matroids. In this paper, we study the derived matroid $\delta M$ of an ${\mathbb F}$-representation of a matroid $M$. The matroid $\delta M$ has a naturally associated ${\mathbb F}$-representation, so we can define a sequence $\delta M$, $\delta^2 M$, \dots . The main result classifies such derived sequences of matroids into three types: finite, cyclic, and divergent. For the first two types, we obtain complete characterizations and thereby resolve some of the questions that Longyear posed in 1980 for binary matroids. For the last type, the divergence is estimated by the coranks of the matroids in the derived sequence.

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