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Open AccessArticle

Dominating the Direct Product of Two Graphs through Total Roman Strategies

1
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
*
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
(This article belongs to the Section Mathematics and Computer Science)
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 vV(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
Keywords: total Roman domination; Roman domination; direct product graphs total Roman domination; Roman domination; direct product graphs
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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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