P-matrix completions under weak symmetry assumptions

An n-by-n matrix is called a Π-matrix if it is one of (weakly) sign-symmetric, positive, nonnegative P-matrix, (weakly) sign-symmetric, positive, nonnegative P0,1-matrix, or Fischer, or Koteljanskii matrix. In this paper, we are interested in Π-matrix completion problems, that is, when a partial Π-matrix has a Π-matrix completion. Here, we 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 Π-matrix has a Π-matrix completion if the graph of its specified entries is a 1-chordal graph. This condition is also necessary for (weakly) sign-symmetric …