Saturation of non-strongly indecomposable matrices

Characterize non-strongly indecomposable forbidden 0-1 matrices for which horizontal and vertical witnesses can be combined to construct a witness and thereby yield bounded saturation functions.

Background

Theorem 3.6 characterizes bounded saturation for strongly indecomposable matrices using compatible horizontal and vertical witnesses. The construction can nevertheless work for some matrices that are not strongly indecomposable, such as Q4. A general categorization of these exceptional matrices is left unresolved.