Secure domination and secure total domination in graphs

Abstract

A secure (total ) dominating set of a graph G = (V, E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V − X, there exists x ∈ X adjacent to u such that (X − {x}) ∪ {u} is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total ) domination number γ s (G) (γ st (G)). We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then γ st (G) ≤ γ s (G). We also show that γ st (G) is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.

On Secure Domination in Graphs

“…Then we prove that for every graph G, γ s (G) ≤ γ (G) + β 0 (G) − 1, which improves a previous bound of Klostermeyer and Mynhardt [7]. Bounds relating the secure domination to the independence number are also presented for triangle-free graphs.…”

“…As we shall show, we extend these two bounds to triangle-free graphs and graphs with girth at least six, respectively. Moreover, Klostermeyer and Mynhardt [7] observed that for every graph G, γ s (G) ≤ 2β 0 (G), and posed the problem of finding graphs for which the bound is sharp. In the next, we show that it is possible to improve Klostermeyer and Mynhardt's bound which will explain in some sense why there has been a difficulty finding a graph reaching their bound.…”

“…The 2-domination number γ 2 (G) and the double domination number γ ×2 (G) represent the minimum cardinality of a secure dominating set of G. Secure domination was introduced by Cockayne et al [5], and is studied, for example, in [5,7,9]. For any parameter μ(G) associated with a graph property P, we refer to a set of vertices with Property P and cardinality μ(G) as a μ(G)-set.…”

On Semitotal Domination in Graphs

“…secure total domination number ) of G. A secure dominating set of cardinality γ s (G) is called a γ s -set. Secure domination and secure total domination in graphs have been studied in [4,5,10,12,19].…”

Protection of lexicographic product graphs

“…For instance, for the graph shown in Figure 1, on the right, a γ s (G)-function (set) can place one guard at each white-coloured vertex. This concept of protection was introduced by Cockayne et al in [9], and studied further in [3,4,6,8,19,26]. The problem of computing γ r (G) is NP-hard, even when restricted to bipartite or chordal graphs [16], and the problem of computing γ s (G) is also NP-Hard, even when restricted to split graphs [3].…”