Total Protection of Lexicographic Product Graphs

Martínez, Abel Cabrera ; Rodríguez-Velázquez, Juan Alberto

Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 3, pp. 967-984

Résumé

Given a graph G with vertex set V (G), a function f : V (G) → 0, 1, 2 is said to be a total dominating function if Σ_u∈N(v) f(u) gt; 0 for every v ∈ V (G), where N(v) denotes the open neighbourhood of v. Let V_i = x ∈ V (G) : f(x) = i. A total dominating function f is a total weak Roman dominating function if for every vertex v ∈ V₀ there exists a vertex u ∈ N(v) ∩ (V₁ ∪ V₂) such that the function f′, defined by f′(v) = 1, f′(u) = f(u) − 1 and f′(x) = f(x) whenever x ∈ V (G) u, v, is a total dominating function as well. If f is a total weak Roman dominating function and V₂ = ∅, then we say that f is a secure total dominating function. The weight of a function f is defined to be ω(f) = Σ_v∈V (G) f(v). The total weak Roman domination number (secure total domination number) of a graph G is the minimum weight among all total weak Roman dominating functions (secure total dominating functions) on G. In this article, we show that these two parameters coincide for lexicographic product graphs. Furthermore, we obtain closed formulae and tight bounds for these parameters in terms of invariants of the factor graphs involved in the product.

Détail

Keywords: total weak Roman domination, secure total domination, total domination, lexicographic product