Total Roman domination number of rooted product graphs

Abstract

Let G be a graph with no isolated vertex and f: V(G) → {0, 1, 2} a function. If f satisfies that every vertex in the set {v ∈ V(G): f (v) = 0} is adjacent to at least one vertex in the set {v ∈ V(G): f (v) = 2}, and if the subgraph induced by the set {v ∈ V(G): f (v) ≥ 1} has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight ω( f ) = ∑v∈V(G) f (v) among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.