The positive Q-matrix completion problem

Abstract

A real [Formula: see text] matrix is a [Formula: see text]-matrix if for [Formula: see text] the sum of all [Formula: see text] principal minors is positive. A digraph [Formula: see text] is said to have positive [Formula: see text]-completion if every partial positive [Formula: see text]-matrix specifying [Formula: see text] can be completed to a positive [Formula: see text]-matrix. In this paper, necessary conditions for a digraph to have positive [Formula: see text]-completion are obtained and sufficient conditions for a digraph to have positive [Formula: see text]-completion are provided. The digraphs of order at most 4 that include all loops and have positive [Formula: see text]-completion are characterized. Tournaments whose complements have positive [Formula: see text]-completion are singled out. Further, some comparisons between the [Formula: see text]-matrix and positive [Formula: see text]-matrix completion problems have been made.