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The positive Q-matrix completion problem

A real n × n matrix is a Q-matrix if for k = 1,…,n the sum of all k × k principal minors is positive. A digraph D is said to have positive Q-completion if every partial positive Q-matrix specifying D can be completed to a positive Q-matrix. In this paper, necessary conditions for a digraph to have positive Q-completion are obtained and sufficient conditions for a digraph to have positive Q-completion are provided. The digraphs of order at most 4 that include all loops and have positive Q-completion are characterized. Tournaments whose complements have positive Q-completion are singled out. Further, some comparisons between the Q-matrix and positive Q-matrix completion problems have been made.