The completion problem for N-matrices ∗

An n×nmatrix is called anN-matrix if all principal minors are negative. In this paper, we are interested in N-matrix completions 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 combi-natorially symmetric partial N-matrix has an N-matrix completion if the graph of its specified en-tries 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. Key-Words: Partial matrix, completion, combina-torially symmetric, N-matrix, 1-chordal graph, cy-cles.