Decomposition of binary matroids
The first part studies balanced sets in a matroid: If a matroid on E with rank function \(\rho\) is induced by an integer polymatroid \(\mu\) (as a submodular set function) then \(A\subseteq E\) is \(\mu\)-balanced if \(\mu A=\rho A_ iA\) is balanced if it is \(\mu\)-balanced for every such \(\mu\). This concept was introduced by the author in J. Math. Anal. Appl. 95, 214-222 (1983; Zbl 0515.05025), and it is then a useful tool in deriving a series of results on sum decompositions of binary matroids. Among them: If a cosimple binary matroid is the sum of \(M_ 1\) and \(M_ 2\) then \(M_ 1\) and \(M_ 2\) are cosimple too [conjectured by \textit{A. Recski}, to appear in Proc. Matroid Theory Conf. Szeged 1982]. Every cosimple binary matroid has a unique sum decomposition into irreducible matroids [conjectured by \textit{W. H. Cunningham} in Q. J. Math., Oxf. II. Ser. 30, 271-281 (1979; Zbl 0416.05026)]. The third part is devoted to the concept of freedom \(\| A\|\) in a matroid M generalizing it to subsets: \(\| A\| =\max \mu (A)\) for integer polymatroids inducing M. Theorem: For a binary matroid there is a unique maximal integer polymatroid inducing it. This polymatroid is characterized.
scientific article; zbMATH DE number 3918382
scientific article