Completions of partial \(P\)-matrices with acyclic or non-acyclic associated graph (Q1399230)

The paper deals with the partial \(P\)-matrix completion problem, that is when a partial \(P\)-matrix \(A\) has a \(P\)-matrix completion \(A_c\). The first result asserts that every positive (sign-symmetric) partial \(P\)-matrix \(A\) has a positive (sign-symmetric) \(P\)-matrix completion when the associated graph of the specified entries is acyclic. On the other hand, the authors introduce a new type of graphs: the CDUM graphs; they analyze the existence of a positive (sign-symmetric) \(P\)-matrix completion of a positive (sign-symmetric) partial \(P\)-matrix, whose associated graph is a CDUM graph, block-graph, etc. Finally, they study the minimun number of unspecified entries in order to guarantee the existence of a positive (sign-symmetric) \(P\)-matrix completion of a positive (sign-symmetric) partial \(P\)-matrix.