Dominating the Direct Product of Two Graphs through Total Roman Strategies

Abel Cabrera Martínez, Dorota Kuziak, Iztok Peterin and Ismael G. Yero
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Abel Cabrera Martínez: Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
Dorota Kuziak: Departamento de Estadística e Investigación Operativa, Universidad de Cádiz, 11202 Algeciras, Spain
Iztok Peterin: Faculty of Electrical Engineering and Computer Science, University of Maribor, 2000 Maribor, Slovenia
Ismael G. Yero: Departamento de Matemáticas, Universidad de Cádiz, 11202 Algeciras, Spain

Mathematics, 2020, vol. 8, issue 9, 1-13

Abstract: 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 γ t R ( 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 γ t R ( 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 γ t R ( G × H ) are presented, and exact values for γ t R ( G × H ) are deduced, while considering various specific direct product classes.

Keywords: total Roman domination; Roman domination; direct product graphs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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