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=G 1k 1□⋅□G mk m is the prime factor decomposition of a connected graph then Det(G)=max{Det(G ik 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)=⌈log2n⌉+1 which matches the lower bound, and that Det(K 3n)=⌈log3(2n+1)⌉+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(logn). © 2009 Wiley Periodicals, Inc. J Graph Theory