# Determining and Distinguishing within the Cartesian Product

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Debra Boutin

Hamilton College

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**Minisymposium:**
GRAPH PRODUCTS

**Content:**
A {\it determining set} $S$ is a set of vertices with the property that each automorphism of the graph is uniquely identified by its action on $S$. The {\it distinguishing number} is the smallest number of colors necessary to color the vertices so that no nontrivial automorphism preserves the color classes. If a graph can be distinguished with two colors, the {\it distinguishing cost} is the smallest possible size of the smaller color class. We will examine each of these parameters for Cartesian products and powers of graphs.