A matrix description of weakly bipartitive and bipartitive families

The notions of weakly bipartitive and bipartitive families were introduced by Montgolfier (2003) as a general tool for studying some decomposition of graphs and other combinatorial structures. One way to construct such families comes from a result of Loewy (1986): Given an irreducible n×n matrix A over a field, the family of partitions {X,Y} of {1,…,n} such that the submatrices A[X,Y] and A[Y,X] have a rank at most 1 is weakly bipartitive. In this paper, we show that this family is bipartitive when A is symmetric. In the converse direction, we prove that weakly bipartitive and bipartitive families are all obtained via the construction above.