Connected matroids with a small circumference

Removing circuits in 3-connected binary matroids

“…Removable circuits and cocircuits play an important role in studying the structure of graphic matroids (see [11,12,24,25]). There has been much interest in the study of removable circuits and cocircuits in graphs and matroids lately (see [1,[4][5][6][7][8]10,[12][13][14][16][17][18]21,22]).…”

“…Let M be a 2-connected matroid. If M is non-empty, then M has a 2-removable circuit provided: (i) (Lemos and Oxley [14]) |E(M)| ≥ 3r (M); or (ii) (Junior [8]) r (M) ≥ 3 and |E(M)| ≥ 3r (M) − 1; or (iii) (Junior [8]) M is simple, r (M) ≥ 7 and |E(M)| ≥ 3r (M) − 3.…”

Matroids Having Small Circumference

“…Lemos and Oxley [12] proved that a connected matroid M has a circuit C such that M\C is connected provided that |E(M)| > 3r(M). This bound is sharp, but it can be improved provided that some other conditions are imposed on the matroid M. To improve it, Junior [7] needed to know all the matroids that have circumference at most five. These matroids are constructed in [7] and in this paper.…”

“…This bound is sharp, but it can be improved provided that some other conditions are imposed on the matroid M. To improve it, Junior [7] needed to know all the matroids that have circumference at most five. These matroids are constructed in [7] and in this paper. Here, we deal with the most difficult case: the 3-connected matroids having both rank and circumference equal to five.…”

“…, M n ). Junior [7] constructs all the matroids with circumference at most five that are not 3-connected. It is not difficult to see that we need only list the matroids that are connected and simple.…”