Decomposition of {0,1}-matrices

A simple decomposition of a r×c{0,1}-matrix is defined in terms of a collection of disjoint submatrices obtained by deleting a “minimal” set of columns. In general, the number of such simple decompositions is Θ(2^r). A class of matrices, namely, vertex-tree graphic, is defined, and it is shown that the number of simple decompositions of a vertex-tree graphic matrix is at most r-1. Finally, the relevance of simple decomposition to the well-known problem of cluster formation on {0,1}-matrices is uncovered, and an O(r² c) time algorithm is given to solve this problem for vertex-tree graphic matrices