Totally Positive Shapes and TPk-completable Patterns
Linear Algebra and its Applications
Matrix completion problem, Totally positive matrix, TPk matrix
The notions of total positivity and of TPk are generalized to “shapes” (a generalization of matrices). In particular, the relationship between positivity of “contiguous” minors and all minors is characterized for general shapes and for certain special types of shapes. This and other ideas are used to address the TPk-completion problem and TPk-completable patterns. In case k=2, a near characterization of TP2-completable patterns is given.
Hoff, D.; Johnson, C. R.; and Nasserasr, Shahla, "Totally Positive Shapes and TPk-completable Patterns" (2012). Mathematics Faculty Articles. 127.