Further Results on the Total Roman Domination in Graphs

Abel Cabrera Martínez, Suitberto Cabrera García and Andrés Carrión García
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Abel Cabrera Martínez: Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain
Suitberto Cabrera García: Departamento de Estadística e Investigación Operativa Aplicadas y Calidad, Universitat Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain
Andrés Carrión García: Departamento de Estadística e Investigación Operativa Aplicadas y Calidad, Universitat Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain

Mathematics, 2020, vol. 8, issue 3, 1-8

Abstract: Let G be a graph without isolated vertices. A function f : V ( G ) → { 0 , 1 , 2 } is a total Roman dominating function on G if every vertex v ∈ V ( G ) for which f ( v ) = 0 is adjacent to at least one vertex u ∈ V ( G ) such that f ( u ) = 2 , and if the subgraph induced by the set { v ∈ V ( G ) : f ( v ) ≥ 1 } has no isolated vertices. The total Roman domination number of G , denoted γ t R ( G ) , is the minimum weight ω ( f ) = ∑ v ∈ V ( G ) f ( v ) among all total Roman dominating functions f on G . In this article we obtain new tight lower and upper bounds for γ t R ( G ) which improve the well-known bounds 2 γ ( G ) ≤ γ t R ( G ) ≤ 3 γ ( G ) , where γ ( G ) represents the classical domination number. In addition, we characterize the graphs that achieve equality in the previous lower bound and we give necessary conditions for the graphs which satisfy the equality in the upper bound above.

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