Triangle-roundedness in matroids
A matroid N is said to be triangle-rounded in a class of matroids ℳ if each 3-connected matroid M∈ℳ with a triangle T and an N-minor has an N-minor with T as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that M is a binary 3-connected matroid with a 3-connected minor N, T is a triangle of M and e∈T∩E(N); then M has a 3-connected minor M′ with an N-minor such that T is a triangle of M′ and |E(M′)|≤|E(N)|+2. We strengthen this result by dropping the condition that such element e exists and proving that there is a 3-connected minor M′ of M with an N-minor N′ such that T is a triangle of M′ and E(M′)−E(N′)⊆T. This result is extended to the non-binary case and, as an application, we prove that M(K5) is triangle-rounded in the class of the regular matroids.
& Zhou, X.
(2020). Triangle-roundedness in matroids. Discrete Mathematics, 343 (2).