Systematic connection between decorated orientations and kinematic flow

Investigate systematically the connection between decorated orientations of graphs and the basis functions appearing in the kinematic flow.

Background

The paper introduces cut tubings by combining edge cuts, represented by binary tubes, with unary tubings on the resulting connected components. Each cut tubing induces a decorated orientation in which graph edges may be broken, unoriented, or oriented. The authors observe that the number of distinct decorated orientations agrees with the number of basis functions in the kinematic flow described in prior work, and they provide a traversability-based rule for identifying valid decorated orientations.

Although this agreement and the associated selection rule are established through examples and structural arguments, the paper does not develop a systematic account of why decorated orientations correspond to the kinematic-flow basis functions. The authors explicitly leave that investigation unresolved.