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Dominating the Direct Product of Two Graphs through Total Roman Strategies

Author

Listed:
  • 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
    Institute of Mathematics, Physics and Mechanics, SI-1000 Ljubljana, Slovenia)

  • Ismael G. Yero

    (Departamento de Matemáticas, Universidad de Cádiz, 11202 Algeciras, Spain)

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.

Suggested Citation

  • Abel Cabrera Martínez & Dorota Kuziak & Iztok Peterin & Ismael G. Yero, 2020. "Dominating the Direct Product of Two Graphs through Total Roman Strategies," Mathematics, MDPI, vol. 8(9), pages 1-13, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1438-:d:404911
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    References listed on IDEAS

    as
    1. Chun-Hung Liu & Gerard J. Chang, 2013. "Roman domination on strongly chordal graphs," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 608-619, October.
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