Total Protection of Lexicographic Product Graphs

Identifier:  imarina:7980012

Handle: https://hdl.handle.net/20.500.11797/imarina7980012

Authors:  Cabrera Martinez, Abel; Alberto Rodriguez-Velazquez, Juan

Abstract:
© (V1 â V2) such that the function f′, defined by f′(v) = 1, f′(u) = f(u)-1 and f′(x) = f(x) whenever x â V (G) \ {u, v}, is a total dominating function as well. If f is a total weak Roman dominating function and V2 = â, then we say that f is a secure total dominating function. The weight of a function f is defined to be ω(f) = ςvâV(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. 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 ςuâN(v) f(u) > 0 for every v â V (G), where N(v) denotes the open neighbourhood of v. Let Vi = {x â V (G): F(x) = i}. A total dominating function f is a total weak Roman dominating function if for every vertex v â V0 there exists a vertex u â N(v) â.