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

TL;DR

This paper determines the secure domination numbers for Cartesian products of small graphs with paths, cycles, and Möbius ladders, introducing new methods based on properties of secure domination itself.

Contribution

It presents the first non-trivial determination of secure domination numbers using properties of secure domination, beyond lower bounds and trivial proofs.

Findings

Secure domination numbers for Cartesian products with paths and cycles are established.

Secure domination numbers for Möbius ladder graphs are determined.

New methods based on secure domination properties are introduced for future research.

Abstract

The secure domination numbers of the Cartesian products of two small graphs with paths or cycles is determined, as well as for Mobius ladder graphs. Prior to this work, in all cases where the secure domination number has been determined, the proof has either been trivial, or has been derived from lower bounds established by considering different forms of domination. However, the latter mode of proof is not applicable for most graphs, including those considered here. Hence, this work represents the first attempt to determine secure domination numbers via the properties of secure domination itself, and it is expected that these methods may be used to determine further results in the future.