Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201Nonnegative signed total Roman domination in graphs1391551399210.22049/cco.2019.26599.1124ENNasrinDehgardiSirjan University of Technology, Sirjan 78137, IranLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20190625<br />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 $gamma^{NN}_{stR}(G)ge frac{3}{2}sqrt{4n+1}-1)-n$.http://comb-opt.azaruniv.ac.ir/article_13992_95a6741eac9ee78064b669bd3e8a9b20.pdf