Sum choice number of generalized θ-graphs

Let G = (V,E) be a simple graph and for every vertex v ∈ V let L(v) be a set (list) of available colors. G is called L-colorable if there is a proper coloring φ of the vertices with φ(v) ∈ L(v) for all v ∈ V . A function f : V → N is called a choice function of G and G is said to be f -list colorable if G is L-colorable for every list assignment L with |L(v)| = f(v) for all v ∈ V . The size of a choice function is defined by size(f) = ∑