From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs

Let G be a graph with no isolated vertex and let N(v) the open neighbourhood of v∈V(G). f:V(G)→{0,1,2} function Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is strongly total Roman dominating on if subgraph induced by V1∪V2 has N(v)∩V2≠∅ v∈V(G)\V2. The domination number G, denoted γtRs(G), defined as minimum weight ω(f)=∑x∈V(G)f(x) among all functions G. This paper devoted to study it contribution Special Issue “Theoretical Computer Science Discrete Mathematics” Symmetry. In particular, we show theory an appropriate framework investigating lexicographic product graphs. also obtain tight bounds this parameter provide closed formulas some Finally consequence study, prove problem computing γtRs(G) NP-hard.