13 research outputs found
Triangle-roundedness in matroids
A matroid is said to be triangle-rounded in a class of matroids
if each -connected matroid with a triangle
and an -minor has an -minor with as triangle. Reid gave a result
useful to identify such matroids as stated next: suppose that is a binary
-connected matroid with a -connected minor , is a triangle of
and ; then has a -connected minor with an
-minor such that is a triangle of and . We
strengthen this result by dropping the condition that such element exists
and proving that there is a -connected minor of with an -minor
such that is a triangle of and . This
result is extended to the non-binary case and, as an application, we prove that
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 mÃnimo 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 capÃtulo 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