Critical Graphs for the Positive Definite Completion Problem

Abstract

Among various matrix completion problems that have been considered in recent years, the positive definite completion problem seems to have received the most attention. Indeed, in addition to being a problem of great interest, it is related to various applications as well as other completion problems. It may also be viewed as a fundamental problem in Euclidean geometry. A partial positive definite matrix A is "critical" if A has no positive definite completion, though every proper principal submatrix does. The graph G is called critical for the positive definite completion problem if there is a critical partial positive definite matrix A, the graph of whose specified entries is G. Complete analytical understanding of the general positive definite completion problem reduces to understanding the problem for critical graphs. Thus, it is important to try to characterize such graphs. The first, crucial step toward that understanding is taken here. A novel and restrictive topological graph theoretic condition necessary for criticality is identified. The condition, which may also be of interest on pure graph theoretic grounds, is also shown to be sufficient for criticality of graphs on fewer than 7 vertices, and the authors suspect it to be sufficient in general. In any event, the condition, which may be efficiently verified, dramatically narrows the class of graphs for which completability conditions on the specified data are needed. The concept of criticality and the graph theoretic condition extend to other completion problems, such as that for Euclidean distance matrices.