  Given a totally positive matrix, can one insert a line (row or column)
between two given lines while maintaining total positivity? This question was
first posed and solved by Johnson and Smith who gave an algorithm that results
in one possible line insertion. In this work we revisit this problem. First we
show that every totally positive matrix can be associated to a certain
vertex-weighted graph in such a way that the entries of the matrix are equal to
sums over certain paths in this graph. We call this graph a scaffolding of the
matrix. We then use this to give a complete characterization of all possible
line insertions as the strongly positive solutions to a given homogeneous
system of linear equations.
