Nonnegative signed total Roman domination in graphs

Let G be a nite and simple graph with vertex set V (G). A nonnegative signed total Roman dominating function (NNSTRDF) on a graph G is a function f : V (G) ! f 1; 1; 2g satisfying the conditions that (i) Px2N(v) f(x) ≥ 0 for each v 2 V (G), where N(v) is the open neighborhood of v, and (ii) every vertex u for which f(u) = -1 has a neighbor v for which f(v) = 2. The weight of an NNSTRDF f is !(f) =P v2V (G) f(v). The nonnegative signed total Roman domination number NN stR (G) of G is the minimum weight of an NNSTRDF on G. In this paper we initiate the study of the nonnegative signed total Roman domination number of graphs, and we present dierent bounds on NN stR (G). We determine the nonnegative signed total Roman domination number of some classes of graphs. If n is the order and m is the size of the graph G, then we show that NN stR (G) ≥ 3 4 (p8n + 1 + 1) - n and NN stR (G) ≥ (10n - 12m)=5. In addition, if G is a bipartite graph of order n, then we prove that NN stR (G) ≥ 3 2 (p4n + 1 - 1) - n.