A chain theorem for 4-connected matroids
Much attention has been given to the study of 3-connected matroids. Inductive arguments can be made in this class of matroids via the use of Tutte's wheels and whirls theorem and Seymour's splitter theorem. Matroid representation theory requires a higher connectivity and strict 4-connectivity is too strong. Studying weakenings of 4-connectivity is important in furthering the study of matroid representations. In the paper under review, the author focusses on weakening 4-connectivity by focusing on the size of 3-connectors as opposed the structure of 3-connectors. The main theorem of the paper proves that if a 3-connected matroid \(M\) is 4-connected up to 3-separators of size 5, then there is an element \(x\in E(M)\) such that either co\((M\setminus x)\) or si\((M/x)\) is 3-connected and 4-connected up to 3-separators of size 5, and has a cardinality of \(| E(M)| -1\) or \(| E(M)| -2\). In addition, a theorem useful in bounding the size of excluded minors for matroids of branch-width 3 is proven.
scientific article; zbMATH DE number 2132707