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# Total Weak Roman Domination in Graphs

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Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
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CONACYT Research Fellow—Centro de Investigación en Matemáticas, 36023 Guanajuato, GTO, Mexico
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Author to whom correspondence should be addressed.
Symmetry 2019, 11(6), 831; https://doi.org/10.3390/sym11060831
Received: 17 May 2019 / Revised: 18 June 2019 / Accepted: 19 June 2019 / Published: 24 June 2019
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# Abstract

Given a graph $G = ( V , E )$ , a function $f : V → { 0 , 1 , 2 , ⋯ }$ is said to be a total dominating function if $∑ u ∈ N ( v ) f ( u ) > 0$ for every $v ∈ V$ , where $N ( v )$ denotes the open neighbourhood of v. Let $V i = { x ∈ V : f ( x ) = i }$ . We say that a function $f : V → { 0 , 1 , 2 }$ is a total weak Roman dominating function if f is a total dominating function and for every vertex $v ∈ V 0$ there exists $u ∈ N ( v ) ∩ ( V 1 ∪ V 2 )$ such that the function $f ′$ , defined by $f ′ ( v ) = 1$ , $f ′ ( u ) = f ( u ) − 1$ and $f ′ ( x ) = f ( x )$ whenever $x ∈ V \ { u , v }$ , is a total dominating function as well. The weight of a function f is defined to be $w ( f ) = ∑ v ∈ V f ( v ) .$ In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by $γ t r ( G )$ , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on $γ t r ( G )$ and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard. View Full-Text
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0). MDPI and ACS Style

Cabrera Martínez, A.; Montejano, L.P.; Rodríguez-Velázquez, J.A. Total Weak Roman Domination in Graphs. Symmetry 2019, 11, 831.

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