﻿ Nonnegative signed total Roman domination in graphs

# Nonnegative signed total Roman domination in graphs

##### چکیده

‎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 conditions‎‎that (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 we‎‎initiate 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 domination‎‎number of some classes of graphs‎. ‎If \$n\$ is the order and \$m\$ the size‎‎of 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 prove‎‎that \$gamma^{NN}_{stR}(G)ge frac{3}{2}(sqrt{4n+1}-1)-n\$‎.

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عنوان ژورنال:

دوره 5  شماره 2

صفحات  139- 155

تاریخ انتشار 2020-12-01

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