Triangle-roundedness in matroids

A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ 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 $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an $N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. 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 $E(M')-E(N')\subseteq T$. This result is extended to the non-binary case and, as an application, we prove that $M(K_5)$ is triangle-rounded in the class of the regular matroids

Constructing Minimally 3-Connected Graphs

A 3-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two operations: adding an edge between non-adjacent vertices and splitting a vertex. To test sets of vertices and edges for 3-compatibility, which depends on the cycles of the graph, we develop a method for obtaining the cycles of G′ from the cycles of G, where G′ is obtained from G by one of the two operations above. We eliminate isomorphic duplicates using certificates generated by McKay’s isomorphism checker nauty. The algorithm consecutively constructs the non-isomorphic minimally 3-connected graphs with n vertices and m edges from the non-isomorphic minimally 3-connected graphs with n−1 vertices and m−2 edges, n−1 vertices and m−3 edges, and n−2 vertices and m−3 edges

Cocircuitos não-separadores que evitam um elemento e graficidade em matroides binárias

Bixby e Cunningham relacionaram graficidade de matroides binárias 3-conexas e cocircuitos não separadores, generalizando um critério de planaridade de grafos 3-conexos de Tutte. Lemos estudou o conjunto de cocircuitos não-separadores que evita um elemento de uma matroide binária 3-conexa e conseguiu outra caracterização: M é gráfica se e só se cada elemento de M evita exatamente r (M)¡1 cocircuitos não separadores. Aqui estudamos o conjunto Y (M), dessas obstruções para graficidade, formado pelos elementos de M que evitam no mínimo r (M) cocircuitos não-separadores. Mostramos que, numa matroide binária 3-conexa existem 3 circuitos contidos em Y (M), cada qual não contido na união dos outros dois. Isso implica numa generalização do resultado de Lemos. No caso em que M não possui menor M¤(K000 3,3) ou M não é regular, conseguimos resultado muito melhor: jE(M)¡Y (M)j · 1. A demonstração desses resultados se baseia numa extensão de alguns resultados de Whittle a respeito demenores de matroide 3-conexas, que também são desenvolvido aqui: Seja M uma matroide binária e 3-conexa com um menor 3-conexo N. Suponha que r (M) ¸ r (N)Å3. Então existe um 3-coindependente I ¤ de M tal que co(M\e) é 3-conexa com menor isomorfo a N para todo e 2 I ¤. No mesmo capítulo desse teorema mostramos ainda uma versão para grafos que, porém, não se extende para matroides binária

On Critical Circuits in k-Connected Matroids

We show that, for every integer k≥ 4 , if M is a k-connected matroid and C is a circuit of M such that for every e∈ C, M\ e is not k-connected, then C meets a cocircuit of size at most 2 k- 3 ; furthermore, if M is binary and k≥ 5 , then C meets a cocircuit of size at most 2 k- 4. It follows from our results and a result of Reid et al [5] that every minimally k-connected matroid has a cocircuit of size at most 2 k- 3 , and every minimally k-connected binary matroid has a cocircuit of size at most 2 k- 4