The Highly Connected Even-Cycle and Even-Cut Matroids

Abstract

The classes of even-cycle matroids, even-cycle matroids with a blocking pair, and even-cut matroids each have hundreds of excluded minors. We show that the number of excluded minors for these classes can be drastically reduced if we consider in each class only the highly connected matroids of sufficient size.

The Templates for Some Classes of Quaternary Matroids

“…We also give excluded minor characterizations of the highly connected members of these classes of quaternary matroids; these results are analogous to the main results of [16]. The Pappus matroid and the Fano matroid F 7 are wellknown.…”

“…We remark that Lemmas 8.2 and 8.3 provide an improvement to the results in Section 9 of [16], although those results are not incorrect. Namely, the matroids L 19 , M * (K 6 ), and M (K 6 ) can be removed form the statements of Theorems 9.2-9.4, respectively, of [16]. Consequently, M (K 6 ) can also be removed form the statement of Corollary 9.5 of [16].…”

“…Namely, the matroids L 19 , M * (K 6 ), and M (K 6 ) can be removed form the statements of Theorems 9.2-9.4, respectively, of [16]. Consequently, M (K 6 ) can also be removed form the statement of Corollary 9.5 of [16].…”