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# Dominating the Direct Product of Two Graphs through Total Roman Strategies

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Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, 11202 Algeciras, Spain
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Faculty of Electrical Engineering and Computer Science, University of Maribor, 2000 Maribor, Slovenia
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Institute of Mathematics, Physics and Mechanics, SI-1000 Ljubljana, Slovenia
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Departamento de Matemáticas, Universidad de Cádiz, 11202 Algeciras, Spain
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(9), 1438; https://doi.org/10.3390/math8091438
Received: 30 July 2020 / Revised: 21 August 2020 / Accepted: 24 August 2020 / Published: 27 August 2020
Given a graph G without isolated vertices, a total Roman dominating function for G is a function $f:V(G)→{0,1,2}$ such that every vertex u with $f(u)=0$ is adjacent to a vertex v with $f(v)=2$, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number $γtR(G)$ of G is the smallest possible value of $∑v∈V(G)f(v)$ among all total Roman dominating functions f. The total Roman domination number of the direct product $G×H$ of the graphs G and H is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between $γtR(G×H)$ and some classical domination parameters for the factors are given. Characterizations of the direct product graphs $G×H$ achieving small values ($≤7$) for $γtR(G×H)$ are presented, and exact values for $γtR(G×H)$ are deduced, while considering various specific direct product classes. View Full-Text
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MDPI and ACS Style

Martínez, A.C.; Kuziak, D.; Peterin, I.; Yero, I.G. Dominating the Direct Product of Two Graphs through Total Roman Strategies. Mathematics 2020, 8, 1438. https://doi.org/10.3390/math8091438

AMA Style

Martínez AC, Kuziak D, Peterin I, Yero IG. Dominating the Direct Product of Two Graphs through Total Roman Strategies. Mathematics. 2020; 8(9):1438. https://doi.org/10.3390/math8091438

Chicago/Turabian Style

Martínez, Abel C.; Kuziak, Dorota; Peterin, Iztok; Yero, Ismael G. 2020. "Dominating the Direct Product of Two Graphs through Total Roman Strategies" Mathematics 8, no. 9: 1438. https://doi.org/10.3390/math8091438

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