Complement minimally non-totally unimodular matrices

Abstract

We prove that, up to row and column permutations and complement operations, the only complement minimally non-totally unimodular matrices are the cycle matrices $C_3$ and $C_5$. This settles a conjecture of Chervet, Grappe, and Vall\'ee. As a consequence, every simplicial cone generated by the rows of a totally equimodular matrix admits a regular unimodular Hilbert triangulation.