\(P\)-matrix completions under weak symmetry assumptions (Q1573659)

An \(n\times n\) matrix is called a \(\Pi\)-matrix if it is one of (weakly) sign-symmetric, positive, nonnegative \(P\)-matrix, (weakly) sign-symmetric, positive, nonnegative \(P_{0,1}\)-matrix, or Fischer, or Koteljanskii matrix. The paper deals with the \(\Pi\)-matrix completion problems, that is, when a partial \(\Pi\)-matrix has a \(\Pi\)-matrix completion. The authors prove that a combinatorially symmetric partial positive \(P\)-matrix has a positive \(P\)-matrix completion if the graph of its specified entries is an \(n\)-cycle. In general, a combinatorially symmetric partial \(\Pi\)-matrix has a \(\Pi\)-matrix completion if the graph of its specified entries is a 1-chordal graph. This condition is also necessary for (weakly) sign-symmetric \(P_0\)- or \(P_{0,1}\)-matrices.