Q:

When is a triangular matrix totally unimodular?

I have an {0,1}, invertible, triangular matrix, that I would like to show to be totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?

A:

Seymour's decomposition theorem for regular matroids yields a polynomial-time algorithm for testing if any {0,1,-1} matrix is totally unimodular.  Unfortunately, due to the sound of paint drying on this question of Gordon Royle, it seems as if no one has yet implemented it.  

