The generating function of Whitworth runs

Let G be a graph. The number of ways of selecting k vertices in G such that the subgraph induced by the k selected vertices containing l edges may be considered as Whitworth runs. For two arbitrary graphs G₁ and G₂ we show that the generating function of G₁ can be written as a sum of the generating function of G₂. As an application we derive a difference equation satisfied by the generating function of a line graph and that of a cycle graph. Two independent solutions in the closed-form are found. One is equivalent to the Whitworth bracelet problem with two colors. Furthermore, a line and a cycle graph and a two-line graph have been studied. We show that all solutions can be written as a sum of the solutions for single-line cases.