Total Perfect Roman Domination.

A total perfect Roman dominating function (TPRDF) on a graph G = (V , E) is a function f from V to { 0 , 1 , 2 } satisfying (i) every vertex v with f (v) = 0 is a neighbor of exactly one vertex u with f (u) = 2 ; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is ∑ v ∈ V f (v) . The total perfect Roman domination number of G, denoted by γ t R p (G) , is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible γ t R p (G) . We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related γ t R p (G) to perfect domination γ p (G) by proving γ t R p (G) ≤ 3 γ p (G) for every graph G, and we characterized trees T of order n ≥ 3 for which γ t R p (T) = 3 γ p (T) . This notion can be utilized to develop a defensive strategy with some properties. [ABSTRACT FROM AUTHOR]