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

Scaffoldings of Totally Positive Matrices and Line Insertion