The symmetric N-matrix completion problem

An n × n matrix is called an N-matrix if all its principal minors are negative. In this paper, we are interested in the symmetric N-matrix completion problem, that is, when a partial symmetric N-matrix has a symmetric N-matrix completion. Here, we prove that a partial symmetric N-matrix has a symmetric N-matrix completion if the graph of its specified entries is chordal. Furthermore, if this graph is not chordal, then examples exist without symmetric N-matrix completions. Necessary and sufficient conditions for the existence of a symmetric N-matrix completion of a partial symmetric N-matrix whose associated graph is a cycle are given.