N-matrix completion problem

An n×n matrix is called an N-matrix if all principal minors are negative. In this paper, we are interested in N-matrix completion problems, that is, when a partial N-matrix has an N-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial N-matrix does not have an N-matrix completion. Here we prove that a combinatorially symmetric partial N-matrix has an N-matrix completion if the graph of its specified entries is a 1-chordal graph. We also prove that there exists an N-matrix completion for a partial N-matrix whose associated graph is an undirected cycle.