Nonnegative signed total Roman domination in graphs
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) $\sum_{x\in N(v)}f(x)\ge 0$ for each $v\in V(G)$, where $N(v)$ is the open n...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Azarbaijan Shahide Madani University
2020-12-01
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| Series: | Communications in Combinatorics and Optimization |
| Subjects: |
| Summary: | 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) $\sum_{x\in N(v)}f(x)\ge 0$ for each
$v\in V(G)$, 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 $\omega(f)=\sum_{v\in V (G)}f(v)$.
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 $\gamma^{NN}_{stR}(G)\ge \frac{3}{2}(\sqrt{4n+1}-1)-n$. |
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| ISSN: | 2538-2128 2538-2136 |