123-Forcing matrices

A permutation σ of {1, 2,...,n} contains a 123-pattern provided it contains an increasing subsequence of length 3 and, otherwise, is 123-avoiding. In terms of the n × n permutation matrix P corresponding to σ, P contains a 123-pattern provided the 3 × 3 identity matrix I3 is a submatrix of P. If A is an n × n (0, 1)-matrix, then A is 123-forcing provided every permutation matrix P ≤ A contains a 123-pattern. The main purpose of this paper is to characterize such matrices A with the minimum number of 0’s.