On Independent Domination in Direct Products

In [16] Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that i (G x H) >= i(G)i(H) where i (G) is the independent domination number of G and G x H is the direct product of graphs G and H. We show this conjecture is false, and, in fact, construct pairs of graphs for which min{i (G), i (H)} - i (G x H) is arbitrarily large. We also give the exact value of i (G x K-n) when G is either a path or a cycle.