Fig. 3.

Graph structures for application examples, showing partitions into $V_0$ (white box) and $V_1$ (gray box), with the splitting edges in magenta and blue and labels omitted for clarity. In the applications cited in the text, the input variable occurs to degree 1 in the labels on magenta edges and nowhere else, although our results offer greater generality. Dashed edges indicate potential intervening vertices. (A) A sequence graph, as used in refs. 8, 16, and 52, to which Proposition 1 applies. (B) Parallel sequences provide a model of single-molecule enzyme kinetics, as explained in the text and used in refs. 7 and 11–14. If thermodynamic equilibrium is assumed, Proposition 1 applies. Away from equilibrium, additional parametric assumptions give rise to the graphs shown in C–F (SI Appendix, Table S1 and Fig. S2.) Proposition 2 applies to C and D, Proposition 3 applies to E, and Proposition 4 applies to F.