The nonnegative Q−matrix completion problem

Abstract

In this paper, the nonnegative $Q$-matrix completion problem is studied. A real $n\times n$ matrix is a $Q$-matrix if for $k\in \{1,\ldots, n\}$, the sum of all $k \times k$ 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.