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Free products, principal sums, bipartition complete matroids and the linking operation (ℳSP ∨ ℳPQ) × (S uplus Q)

  • H. Narayanan
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Abstract

In this paper we examine the special class of bipartition complete matroids, which includes free products and principal sums of matroids (Crapo and Schmitt, 2005; Bonin and Kung, 2015). A matroid ℳSQ on S uplus Q is said to be {S,Q}--complete iff whenever b1S uplus b1Q, b1S uplus b2Q, and b2S uplus b1Q are bases of ℳSQ, we also have b2S uplus b2Q as a base of ℳSQ. We use the linking operation ℳSP ↔ℳ ℳPQ :≡ (ℳSP ∨ ℳPQ) × (S uplus Q) (Theja and Narayanan, 2014) for this study. We construct the {S, Q}--augmentation of an arbitrary matroid and show that this augmentation is again a matroid, that the dual of the augmentation is the augmentation of the dual matroid and that the {S, Q}--augmentation of an {S, Q}--complete matroid leaves it unchanged. We also show that bipartition completeness is stable under minors in the appropriate bipartitioned sense. Finally, we show that free products and principal sums of matroids ℳS,ℳQ are {S, Q}--complete and give explicit minimal decompositions for them.

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Cite

BibTeXRIS
BibTeX
@misc{narayanan2026free,
  title = {{Free products, principal sums, bipartition complete matroids and the linking operation $(\mathcalM_SP \vee \mathcalM_PQ) \times (S \uplus Q)$}},
  author = {H. Narayanan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14111},
  eprint = {2610.14111},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14111},
}
RIS
TY  - UNPB
TI  - Free products, principal sums, bipartition complete matroids and the linking operation (ℳ_SP ∨ ℳ_PQ) × (S uplus Q)
AU  - H. Narayanan
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.14111
UR  - https://arxiv.org/abs/2610.14111
AB  - In this paper we examine the special class of bipartition complete matroids, which includes free products and principal sums of matroids (Crapo and Schmitt, 2005; Bonin and Kung, 2015). A matroid ℳ_SQ on S uplus Q is said to be {S,Q}--complete iff whenever b^1_S uplus b^1_Q, b^1_S uplus b^2_Q, and b^2_S uplus b^1_Q are bases of ℳ_SQ, we also have b^2_S uplus b^2_Q as a base of ℳ_SQ. We use the linking operation ℳ_SP ↔_ℳ ℳ_PQ :≡ (ℳ_SP ∨ ℳ_PQ) × (S uplus Q) (Theja and Narayanan, 2014) for this study. We construct the {S, Q}--augmentation of an arbitrary matroid and show that this augmentation is again a matroid, that the dual of the augmentation is the augmentation of the dual matroid and that the {S, Q}--augmentation of an {S, Q}--complete matroid leaves it unchanged. We also show that bipartition completeness is stable under minors in the appropriate bipartitioned sense. Finally, we show that free products and principal sums of matroids ℳ_S,ℳ_Q are {S, Q}--complete and give explicit minimal decompositions for them.
ER  -