%0 Journal Article
%T Nonnegative signed total Roman domination in graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Dehgardi, Nasrin
%A Volkmann, Lutz
%D 2020
%\ 12/01/2020
%V 5
%N 2
%P 139-155
%! Nonnegative signed total Roman domination in graphs
%K nonnegative signed total Roman dominating function
%K nonnegative signed total Roman domination
%K signed total Roman k-domination
%R 10.22049/cco.2019.26599.1124
%X Let $G$ be a finite and simple graph with vertex set $V(G)$. A nonnegative signed total Roman dominating function (NNSTRDF) on a graph $G$ is a function $f:V(G)rightarrow{-1, 1, 2}$ satisfying the conditions that (i) $sum_{xin N(v)}f(x)ge 0$ for each $vin V(G)$, 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 $omega(f)=sum_{vin V (G)}f(v)$. The nonnegative signed total Roman domination number $gamma^{NN}_{stR}(G)$ 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 different bounds on $gamma^{NN}_{stR}(G)$. 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 $gamma^{NN}_{stR}(G)ge frac{3}{4}(sqrt{8n+1}+1)-n$ and $gamma^{NN}_{stR}(G)ge (10n-12m)/5$. In addition, if $G$ is a bipartite graph of order $n$, then we prove that $gamma^{NN}_{stR}(G)ge frac{3}{2}sqrt{4n+1}-1)-n$.
%U http://comb-opt.azaruniv.ac.ir/article_13992_95a6741eac9ee78064b669bd3e8a9b20.pdf