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Total Roman {3}-domination in Graphs

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Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
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Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran
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Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
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Symmetry 2020, 12(2), 268; https://doi.org/10.3390/sym12020268
Received: 4 January 2020 / Revised: 2 February 2020 / Accepted: 3 February 2020 / Published: 9 February 2020
For a graph $G = ( V , E )$ with vertex set $V = V ( G )$ and edge set $E = E ( G )$ , a Roman ${ 3 }$ -dominating function (R ${ 3 }$ -DF) is a function $f : V ( G ) → { 0 , 1 , 2 , 3 }$ having the property that $∑ u ∈ N G ( v ) f ( u ) ≥ 3$ , if $f ( v ) = 0$ , and $∑ u ∈ N G ( v ) f ( u ) ≥ 2$ , if $f ( v ) = 1$ for any vertex $v ∈ V ( G )$ . The weight of a Roman ${ 3 }$ -dominating function f is the sum $f ( V ) = ∑ v ∈ V ( G ) f ( v )$ and the minimum weight of a Roman ${ 3 }$ -dominating function on G is the Roman ${ 3 }$ -domination number of G, denoted by $γ { R 3 } ( G )$ . Let G be a graph with no isolated vertices. The total Roman ${ 3 }$ -dominating function on G is an R ${ 3 }$ -DF f on G with the additional property that every vertex $v ∈ V$ with $f ( v ) ≠ 0$ has a neighbor w with $f ( w ) ≠ 0$ . The minimum weight of a total Roman ${ 3 }$ -dominating function on G, is called the total Roman ${ 3 }$ -domination number denoted by $γ t { R 3 } ( G )$ . We initiate the study of total Roman ${ 3 }$ -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman ${ 3 }$ -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman ${ 3 }$ -domination for bipartite graphs. View Full-Text
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MDPI and ACS Style

Shao, Z.; Mojdeh, D.A.; Volkmann, L. Total Roman {3}-domination in Graphs. Symmetry 2020, 12, 268. https://doi.org/10.3390/sym12020268

AMA Style

Shao Z, Mojdeh DA, Volkmann L. Total Roman {3}-domination in Graphs. Symmetry. 2020; 12(2):268. https://doi.org/10.3390/sym12020268

Chicago/Turabian Style

Shao, Zehui, Doost A. Mojdeh, and Lutz Volkmann. 2020. "Total Roman {3}-domination in Graphs" Symmetry 12, no. 2: 268. https://doi.org/10.3390/sym12020268

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