On the Total Version of Triple Roman Domination in Graphs
In this paper, we describe the study of total triple Roman domination. Total triple Roman domination is an assignment of labels from <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>{</mo>&l...
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2025-04-01
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| author | Juan Carlos Valenzuela-Tripodoro Maria Antonia Mateos-Camacho Martin Cera Maria Pilar Alvarez-Ruiz |
| author_facet | Juan Carlos Valenzuela-Tripodoro Maria Antonia Mateos-Camacho Martin Cera Maria Pilar Alvarez-Ruiz |
| author_sort | Juan Carlos Valenzuela-Tripodoro |
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| description | In this paper, we describe the study of total triple Roman domination. Total triple Roman domination is an assignment of labels from <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></semantics></math></inline-formula> 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 <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>(</mo><mi>N</mi><mo>[</mo><mi>v</mi><mo>]</mo><mo>)</mo><mo>≥</mo><mo stretchy="false">|</mo><mi>A</mi><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo stretchy="false">|</mo><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> denotes the set of active neighbors of vertex <i>v</i>, 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. |
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| spelling | doaj-art-278446d3f78745da85c7c309e4ef2d9d2025-08-20T02:28:15ZengMDPI AGMathematics2227-73902025-04-01138127710.3390/math13081277On the Total Version of Triple Roman Domination in GraphsJuan Carlos Valenzuela-Tripodoro0Maria Antonia Mateos-Camacho1Martin Cera2Maria Pilar Alvarez-Ruiz3Escuela Técnica Superior de Ingeniería de Algeciras, Universidad de Cádiz, 11202 Algeciras, SpainEscuela Internacional de Doctorado, Universidad de Sevilla, 41013 Sevilla, SpainEscuela Técnica Superior de Ingeniería Agronómica, Universidad de Sevilla, 41005 Sevilla, SpainEscuela Técnica Superior de Ingeniería de Algeciras, Universidad de Cádiz, 11202 Algeciras, SpainIn this paper, we describe the study of total triple Roman domination. Total triple Roman domination is an assignment of labels from <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></semantics></math></inline-formula> 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 <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>(</mo><mi>N</mi><mo>[</mo><mi>v</mi><mo>]</mo><mo>)</mo><mo>≥</mo><mo stretchy="false">|</mo><mi>A</mi><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo stretchy="false">|</mo><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> denotes the set of active neighbors of vertex <i>v</i>, 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.https://www.mdpi.com/2227-7390/13/8/1277Roman dominationtotal Roman dominationtriple Roman dominationtotal triple Roman domination |
| spellingShingle | Juan Carlos Valenzuela-Tripodoro Maria Antonia Mateos-Camacho Martin Cera Maria Pilar Alvarez-Ruiz On the Total Version of Triple Roman Domination in Graphs Mathematics Roman domination total Roman domination triple Roman domination total triple Roman domination |
| title | On the Total Version of Triple Roman Domination in Graphs |
| title_full | On the Total Version of Triple Roman Domination in Graphs |
| title_fullStr | On the Total Version of Triple Roman Domination in Graphs |
| title_full_unstemmed | On the Total Version of Triple Roman Domination in Graphs |
| title_short | On the Total Version of Triple Roman Domination in Graphs |
| title_sort | on the total version of triple roman domination in graphs |
| topic | Roman domination total Roman domination triple Roman domination total triple Roman domination |
| url | https://www.mdpi.com/2227-7390/13/8/1277 |
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