Scaffoldings of totally positive matrices and line insertion

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