The determining number of a Cartesian product

Abstract

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 ifm is the prime factor decomposition of a connected graph then Det(G) = max{Det(G k i 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(Q n ) = log 2 n +1 which matches the lower bound, and that Det(K n 3 ) = log 3 (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(H n ) = (log n).᭧

Base size, metric dimension and other invariants of groups and graphs

“…Our use of this result in Corollary 2.18 gives two more of Boutin's results in [20]: her Theorem 3, expressed as the determining number of the hypercube, is that b(S 2 S d ) = 1 + log 2 d , while her Corollary 6.1, expressed as the determining number of K d 3 , is that b(S 3 S d ) = 1 + log 3 (2d + 1) .…”

“…A deeper result is [20,Theorem 5]. This states that, for a given integer k, the maximum d for which the base size of Aut(H(d, m)) is k is equal to the number of equivalence classes of m × k covering matrices.…”

Small label classes in 2-distinguishing labelings

“…By the remarks following Theorem 2, every distinguishing class for Q n , n ≥ 4, is also a determining set. By [5] a smallest such set for Q n has size log 2 n + 1. This provides the lower bound.…”

“…When d = 2 this translates as: a graph is 2-distinguishable if and only if it has a determining set with the property that any automorphism that preserves the set must fix it pointwise. This shows that ρ(G) is bounded below by the size of a smallest determining set for G. For Q n this bound is log 2 n + 1 [5]. We will use the connection between determining sets and distinguishing labelings, along with particular determining sets found in [5], to create a distinguishing class for Q n that is smaller than twice this lower bound.…”

On the controllability and observability of Cartesian product networks

“…[19] Let G = G k1 1 · · · G km m be the prime factor decomposition for a connected graph G. Then Det(G) = max{Det(G ki i )}. We now have the required ground work to state a consequence of the graph automorphism structure of the graph pertaining to Theorem IV.1.…”