The Q-matrix completion problem

Abstract

A real n x n matrix is a Q-matrix if for every k = 1, 2, . . . ,n the sum of all k x k principal minors is positive. A digraph D is said to have Q-completion if every partial Q-matrix specifying D can be completed to a Q-matrix. For the Q-completion problem, sufficient conditions for a digraph to have Q-completion are given, necessary conditions for a digraph to have Q-completion are provided, and those digraphs of order at most four that have Q-completion are characterized.