Total Protection of Lexicographic Product Graphs

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 sigma(u)(is an element of)(N)(()(v)()) f(u) > 0 for every v is an element of V (G), where N(v) denotes the open neighbourhood of v. Let V-i = {x is an element of V (G) : f(x) = i}. A total dominating function f is a total weak Roman dominating function if for every vertex v is an element of V-0 there exists a vertex u is an element of N(v) boolean AND (V-1 ? V-2) such that the function f ', defined by f '(v) = 1, f '(u) = f(u) - 1 and f '(x) = f(x) whenever x is an element of V (G) \ {u, v}, is a total dominating function as well. If f is a total weak Roman dominating function and V-2 = null , then we say that f is a secure total dominating function. The weight of a function f is defined to be omega(f) = sigma(v)(is an element of)(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.