Total Weak Roman Domination in Graphs

Given a graph G=(V,E), a function f:V ->{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, where N(v) denotes the open neighbourhood of v. Let Vi={x is an element of V:f(x)=i}. We say that a function f:V ->{0,1,2} is a total weak Roman dominating function if f is a total dominating function and for every vertex v is an element of V0 there exists u is an element of N(v)(V1V2) 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\{u,v}, is a total dominating function as well. The weight of a function f is defined to be w(f)=Sigma v is an element of Vf(v). In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by gamma tr(G), which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on gamma tr(G) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.