Deterministic inverse zero-patterns Original Research Article

Given a collection of positions in an n-by-n matrix A and another collection in the n-by-n B, when does it happen that, whenever the values of the entries permit B to be A−1, all the remaining entries of A (and B) are uniquely determined? We consider the case in which the off-diagonal positions in A and B are the same, all the diagonal entries of A are identified while none of those of B are, and all the identified entries of B are zero. In this event, the arrangement of positions may be described by a directed graph whose edges correspond the identified (off-diagonal) entries of A (or B). When the required property holds, we call the corresponding directed graph deterministic. Here, we identify several graph–theoretic conditions that are sufficient for a strongly connected directed graph to be deterministic, and we conjecture that one of them is necessary. These conditions naturally generalize the notion of chordality for undirected graphs, as chordal graphs have the desired property in the undirected case.