Brick partitions of graphs

For each rational number q = b / c where b ≥ c are positive integers, we define a q -brick of G to be a maximal subgraph H of G such that c H has b edge-disjoint spanning trees, and a q -superbrick of G to be a maximal subgraph H of G such that c H − e has b edge-disjoint spanning trees for all edges e of c H , where c H denotes the graph obtained from H by replacing each edge by c parallel edges. We show that the vertex sets of the q -bricks of G partition the vertex set of G , and that the vertex sets of the q -superbricks of G form a refinement of this partition. The special cases when q = 1 are the partitions given by the connected components and the 2-edge-connected components of G , respectively. We obtain structural results on these partitions and describe their relationship to the principal partitions of a matroid.