Elsevier

Discrete Applied Mathematics

Volume 304, 15 December 2021, Pages 119-128
Discrete Applied Mathematics

On the security number of the Cartesian product of graphs

https://doi.org/10.1016/j.dam.2021.07.030Get rights and content

Abstract

A secure set of a graph is, intuitively, a set that can refute any attack from the neighborhood to its subsets. Formally, it is defined as a set SV(G) such that |N[X]S||N[X]S| for all XS. Although finding a minimum secure set is a computationally intractable problem, the minimum size of secure sets, called the security number, is studied for some specific graphs. Especially, determining the security number of the Cartesian product of graphs is one of the developed directions in this area. In this paper, we present an upper bound on the security number of the Cartesian product of general graphs, which is tight for some sparse graphs. We then determine the security number of K3mK3n, the Cartesian product of complete graphs K3m and K3n, as well as good lower and upper bounds on the security number of the Cartesian product of complete graphs with any number of vertices.

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 G be a graph. (Throughout the paper, all graphs considered are simple, finite, and undirected.) A nonempty set SV(G) is a defensive alliance of G if |N[x]S||N[x]S| holds for each xS, where N[x] denotes the closed neighborhood of x. 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 G and H are arbitrary connected graphs, then s(GH)min{s(G)|V(H)|,|V(G)|s(H)}.

Proof

Denote with V(G)={x1,,xm} and V(H)={y1,,yn}, m,nN, the vertices of graphs G and H, respectively.

Let S be a minimum secure set of G, i.e. |S|=s(G), and let S=S×V(H). We prove that S is a secure set of GH. Suppose that X is an arbitrary subset of S. For each i{1,,n} we define Si=SV(Gyi) and Xi

The cartesian product of complete graphs

The bound in Theorem 7 can be also very bad. If we take for example the complete graphs Km and Kn, which also belong to the family G, we have es(Km)=m and es(Kn)=n, 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 m and n 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 s(GH)s(G)s(H) true for arbitrary connected graphs G and H?

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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