A decomposition theory for matroids III. Decomposition conditions

Decomposition Theory for Lattices without Chain Conditions

Branch decomposition heuristics for linear matroids

This paper presents two new heuristics which utilize classification and max-flow algorithm respectively to derive near-optimal branch decompositions for linear matroids. In the literature, there are already excellent heuristics for graphs, however, no practical branch decomposition methods for general linear matroids have been addressed yet. Introducing a “measure” which compares the “similarit...

An Obstacle to a Decomposition Theorem for Near-Regular Matroids

Seymour’s Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obta...

Decomposition width - a new width parameter for matroids

We introduce a new width parameter for matroids called decomposition width and prove that every matroid property expressible in the monadic second order logic can be computed in linear time for matroids with bounded decomposition width if their decomposition is given. Since decompositions of small width for our new notion can be computed in polynomial time for matroids of bounded branch-width r...

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K5\e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four inf...