On the Total Version of Triple Roman Domination in Graphs

Juan Carlos Valenzuela-Tripodoro (), Maria Antonia Mateos-Camacho, Martin Cera and Maria Pilar Alvarez-Ruiz
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Juan Carlos Valenzuela-Tripodoro: Escuela Técnica Superior de Ingeniería de Algeciras, Universidad de Cádiz, 11202 Algeciras, Spain
Maria Antonia Mateos-Camacho: Escuela Internacional de Doctorado, Universidad de Sevilla, 41013 Sevilla, Spain
Martin Cera: Escuela Técnica Superior de Ingeniería Agronómica, Universidad de Sevilla, 41005 Sevilla, Spain
Maria Pilar Alvarez-Ruiz: Escuela Técnica Superior de Ingeniería de Algeciras, Universidad de Cádiz, 11202 Algeciras, Spain

Mathematics, 2025, vol. 13, issue 8, 1-19

Abstract: In this paper, we describe the study of total triple Roman domination. Total triple Roman domination is an assignment of labels from { 0 , 1 , 2 , 3 , 4 } to the vertices of a graph such that every vertex is protected by at least three units either on itself or its neighbors while ensuring that none of its neighbors remains unprotected. Formally, a total triple Roman dominating function is a function f : V ( G ) → { 0 , 1 , 2 , 3 , 4 } such that f ( N [ v ] ) ≥ | A N ( v ) | + 3 , where A N ( v ) denotes the set of active neighbors of vertex v , i.e., those assigned a positive label. We investigate the algorithmic complexity of the associated decision problem, establish sharp bounds regarding graph structural parameters, and obtain the exact values for several graph families.

Keywords: Roman domination; total Roman domination; triple Roman domination; total triple Roman domination (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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