On the security number of the Cartesian product of graphs
Section snippets
Introduction and preliminaries
The concept of secure sets was introduced by Brigham, Dutton, and Hedetniemi [3] by restricting defensive alliances [17] to be more “secure”. Let us first recall the definition of defensive alliances. Let be a graph. (Throughout the paper, all graphs considered are simple, finite, and undirected.) A nonempty set is a defensive alliance of if holds for each , where denotes the closed neighborhood of . For secure sets, we ask such a condition also for each
Upper bounds
We begin this section with an upper bound for the security number of the Cartesian product of two arbitrary graphs.
Proposition 3 If and are arbitrary connected graphs, then
Proof Denote with and , , the vertices of graphs and , respectively. Let be a minimum secure set of , i.e. , and let . We prove that is a secure set of . Suppose that is an arbitrary subset of . For each we define and
The cartesian product of complete graphs
The bound in Theorem 7 can be also very bad. If we take for example the complete graphs and , which also belong to the family , we have and , and the bound in Theorem 7 becomes even worse than the bound in Proposition 3. Therefore, we turn our attention to the Cartesian product of complete graphs and derive a better upper bound for them. For some values of and we even prove exact results. The following theorem gives a better upper bound for the Cartesian product of
Concluding remarks
It would be interesting to find at least one non-trivial lower bound for the security number of the Cartesian product of arbitrary graphs. It is however clear that this would be considerably more difficult than proving the upper bounds which are usually given by constructions. Nevertheless, our investigations lead us to propose a Vizing-type inequality for the security number as an open problem.
Problem 13 Is true for arbitrary connected graphs and ?
A simpler inequality to the one above
Acknowledgments
Marko Jakovac acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and research projects J1-9109, J1-1693, N1-0095). Yota Otachi was partially supported by JSPS, Japan KAKENHI Grant Nos. JP18H04091, JP18K11168, JP18K11169.
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