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The secure domination numbers of the Cartesian products of two small graphswith paths or cycles is determined, as well as for Mobius ladder graphs. Priorto this work, in all cases where the secure domination number has beendetermined, the proof has either been trivial, or has been derived from lowerbounds established by considering different forms of domination. However, thelatter mode of proof is not applicable for most graphs, including thoseconsidered here. Hence, this work represents the first attempt to determinesecure domination numbers via the properties of secure domination itself, andit is expected that these methods may be used to determine further results inthe future.

The Secure Domination Number of Cartesian Products of Small Graphs with Paths and Cycles