Characterization of active paths in complete graphs

Establish a simple criterion determining whether a given sequence of vertices is an active path of the complete graph K_n under the recursive active-path construction defined for a lollipop cycle.

Background

The paper defines active Hamiltonian paths recursively, starting from the two cyclic orientations and repeatedly applying a chord-based transformation. Not every Hamiltonian path beginning at c_1 is active: the authors give an explicit counterexample in K_6 and note that six of the 120 paths beginning at c_1 are non-active. This motivates the search for a direct, simple characterization.