On a list-coloring problem

We study the function f(G) defined for a graph G as the smallest integer k such that the join of G with a stable set of size k is not |V(G)|-choosable. This function was introduced recently in order to describe extremal graphs for a list-coloring version of a famous inequality due to Nordhaus and Gaddum (Dantas et al., Research Report 18, Laboratoire Leibniz-IMAG, Grenoble, 2000). Some bounds and some exact values for f(G) are determined.