Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids

We introduce “matroid parse trees” which, using only a limited amount of information at each node, can build up the vector representations of matroids of bounded branch-width over a finite field. We prove that if M is a family of matroids described by a sentence in the monadic second-order logic of matroids, then there is a finite tree automaton accepting exactly those parse trees which build vector representations of the bounded-branch-width representable members of M. Since the cycle matroids of graphs are representable over any field, our result directly extends the so called “MS2-theorem” for graphs of bounded tree-width by Courcelle, and others. Moreover, applications and relations in areas other than matroid theory can be found, like for rank-width of graphs, or in the coding theory.