Forbidden Submatrices Research Articles

Almost integral polyhedra related to certain combinatorial optimization problems

The notion of an almost integral polyhedron is introduced and used to obtain a new proof of the characterization of perfect zero-one matrices which relies only on standard arguments from linear algebra and convexity. The characterization of perfect zero-one matrices in terms of forbidden submatrices is then used to derived the perfect-graph theorem due to Fulkerson and Lovász. Furthermore, a characterization of antiblocking pairs of zero-one matrices by means of a strengthened version of the max-max inequality due to Fulkerson is obtained which entails Lovász's recent characterization of perfect graphs.

A note on the total unimodularity of matrices

A necessary and sufficient characterization of totally unimodular matrices is given which is derived from a necessary condition for total unimodularity due to Camion. This characterization is then used in connection with a theorem of Hoffman and Kruskal to provide an elementary proof of the characterization of totally unimodular matrices in terms of forbidden submatrices due to Camion.

A structure theorem for the consecutive 1's property

A (0, 1)-matrix M has the consecutive 1's property for columns if the rows of M can be permuted so that the 1's in each column appear consecutively. A graph-theoretic approach is used to characterize matrices with this property in terms of forbidden submatrices. Graphs whose adjacency matrix has this property are also characterized.