On Independent Domination in Direct Products

Abstract

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