The determining number of a Cartesian product

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G = G1 1 2 · · ·2 Gkm m is the prime factor decomposition of a connected graph then Det(G) = max{Det(Gi i )}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Qn) = dlog2 ne+ 1 which matches the lower bound, and that Det(Kn 3 ) = dlog3(2n + 1)e+ 1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(Hn) = Θ(log n).