Dominating the Direct Product of Two Graphs through Total Roman Strategies

Given a graphGwithout isolated vertices, a total Roman dominating function forGis a function f:V(G)->{0,1,2}such that every vertexuwithf(u)=0is adjacent to a vertexvwithf(v)=2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number gamma tR(G)ofGis the smallest possible value of n-ary sumation v is an element of V(G)f(v)among all total Roman dominating functionsf. The total Roman domination number of the direct productGxHof the graphsGandHis studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between gamma tR(GxH)and some classical domination parameters for the factors are given. Characterizations of the direct product graphsGxHachieving small values (<= 7) for gamma tR(GxH)are presented, and exact values for gamma tR(GxH)are deduced, while considering various specific direct product classes