On 3-connected minors of 3-connected matroids and graphs

Costalonga, João Paulo

doi:10.1016/j.ejc.2011.08.003

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Cited by 4 publications

(2 citation statements)

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“…Now, it is easy to verify that H ∼ = K 1,1 3,3 and conclude the lemma. The next result is Corollary 1.8 of [2]. So, there is e ∈ E (G) and X ⊆ E (G) such that G\X /e ∼ = K 1,1 3,3 .…”

Section: Proofs For the Theoremsmentioning

confidence: 86%

On $K_5$ and $K_{3,3}$-minors of graphs and regular matroids

Costalonga

2013

Preprint

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In this paper we prove two main results about obstruction to graph planarity. One is that, if G is a 3-connected graph with a K 5 -minor andOther is that if G is a 3-connected simple non-planar graph not isomorphic to K 5 and e, f ∈ E (G), then G has a minor H such that e, f ∈ E (H) and, up to isomorphisms, H is one of the four non-isomorphic simple graphs obtained from K 3,3 by the addiction of 0, 1 or 2 edges. We generalize this second result to the class of the regular matroids.

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Section: Proofs For the Theoremsmentioning

confidence: 86%

On $K_5$ and $K_{3,3}$-minors of graphs and regular matroids

Costalonga

2013

Preprint

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“…What can we say about their distribution?". In particular, we generalize the following theorem: [18](k = 1, 2) and Costalonga [4](k = 3)) Let k ∈ {1, 2, 3} and let M be a 3-connected matroid with a 3-connected simple minor N such that r (M) − r (N ) ≥ k. Then, M has a k-independent set of vertically N -contractible elements.…”

Section: Introductionmentioning

confidence: 99%

A splitter theorem on 3-connected matroids

Costalonga¹

2018

European Journal of Combinatorics

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ABSTRACT. We establish the following splitter theorem for graphs and its generalization for matroids: Let G and H be 3-connected simple graphs such that G has an H -minor and k := |V (G)| − |V (H )| ≥ 2. Let n := ⌈k/2⌉ + 1. Then there are pairwise disjoint sets X 1 , . . . , X n ⊆ E (G) such that each G/X i is a 3-connected graph with an H -minor, each X i is a singleton set or the edge set of a triangle of G with 3 degree-3 vertices and X 1 ∪ · · · ∪ X n contains no edge sets of circuits of G other than the X i 's. This result extends previous ones of Whittle (for k = 1, 2) and Costalonga (for k = 3).

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Triangle-roundedness in matroids

Costalonga

Zhou

2020

Discrete Mathematics

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On 3-connected minors of 3-connected matroids and graphs

Cited by 4 publications

References 8 publications

On $K_5$ and $K_{3,3}$-minors of graphs and regular matroids

On $K_5$ and $K_{3,3}$-minors of graphs and regular matroids

A splitter theorem on 3-connected matroids

Triangle-roundedness in matroids

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