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Nonnegative signed total Roman domination in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 5، دوره 5، شماره 2، اسفند 2020، صفحه 139-155 اصل مقاله (424.79 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2019.26599.1124 | ||
| نویسندگان | ||
| Nasrin Dehgardi* 1؛ Lutz Volkmann2 | ||
| 1Sirjan University of Technology, Sirjan 78137, Iran | ||
| 2RWTH Aachen University | ||
| 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 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 $\gamma^{NN}_{stR}(G)\ge \frac{3}{2}\sqrt{4n+1}-1)-n$. | ||
| کلیدواژهها | ||
| nonnegative signed total Roman dominating function؛ nonnegative signed total Roman domination؛ signed total Roman k-domination | ||
| مراجع | ||
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[1] N. Dehgardi and L. Volkmann, Signed total Roman k-domination in directed graphs, Commun. Comb. Optim. 1 (2016), no. 2, 165–178.
[2] N. Dehgardi and L. Volkmann, Nonnegative signed edge domination in graphs, Kragujevac J. Math. 43 (2019), no. 1, 31–47.
[3] N. Dehgardi and L. Volkmann, Nonnegative signed Roman domination in graphs, J. Combin. Math. Combin. Comput. (to appear).
[4] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Domination in Graphs, Advanced Topics, Marcel Dekker, Inc., New York, 1998.
[5] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998. [6] L. Volkmann, Signed total Roman domination in graphs, J. Comb. Optim. 32 (2016), no. 3, 855–871.
[7] L. Volkmann, Signed total Roman domination in digraphs, Discuss. Math. Graph Theory 37 (2017), no. 1, 261–272.
[8] L. Volkmann, Signed total Roman k-domination in graphs, J. Combin. Math. Combin. Comput. 105 (2018), 105–116. | ||
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