The nonnegative Q−matrix completion problem

In this paper, the nonnegative Q-matrix completion problem is studied. A real n × n matrix is a Q-matrix if for k 2 {1, . . . , n}, the sum of all k × k principal minors is positive. A digraph D is said to have nonnegative Q-completion if every partial nonnegative Q-matrix specifying D can be completed to a nonnegative Q-matrix. For nonnegative Q-completion problem, necessary conditions and sufficient conditions for a digraph to have nonnegative Q-completion are obtained. Further, the digraphs of order at most four that have nonnegative Q-completion have been studied.