N_0 completions on partial matrices

Abstract

An $n\times n$ matrix is called an $N_0$-matrix if all its principal minors are nonpositive. In this paper, we are interested in $N_0$-matrix completion problems, that is, when a partial $N_0$-matrix has an $N_0$-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial $N_0$-matrix does not have an $N_0$-matrix completion. Here, we prove that a combinatorially symmetric partial $N_0$-matrix, with no null main diagonal entries, has an $N_0$-matrix completion if the graph of its specified entries is a 1-chordal graph or a cycle. We also analyze the mentioned problem when the partial matrix has some null main diagonal entries.