# Nonnegative signed total Roman domination in graphs

Document Type: Original paper

Authors

1 Sirjan University of Technology, Sirjan 78137, Iran

2 RWTH Aachen University

Abstract

‎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\$ 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\$‎.

Keywords