From Total Roman Domination in Lexicographic Product Graphs to Strongly Total Roman Domination in Graphs

• Main Authors: Ana Almerich-Chulia, Abel Cabrera Martínez, Frank Angel Hernández Mira, Pedro Martin-Concepcion
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Symmetry
• Subjects: strongly total Roman domination; total Roman domination; total domination; lexicographic product graph
• Online Access: https://www.mdpi.com/2073-8994/13/7/1282

Let <i>G</i> be a graph with no isolated vertex and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the open neighbourhood of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><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></mrow></semantics></math></inline-formula> be a function and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub><mo>=</mo><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>:</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>i</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula> for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></semantics></math></inline-formula>. We say that <i>f</i> is a strongly total Roman dominating function on <i>G</i> if the subgraph induced by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mn>1</mn></msub><mo>∪</mo><msub><mi>V</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> has no isolated vertex and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∩</mo><msub><mi>V</mi><mn>2</mn></msub><mo>≠</mo><mo>∅</mo></mrow></semantics></math></inline-formula> for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>\</mo><msub><mi>V</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. The strongly total Roman domination number of <i>G</i>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mi>t</mi><mi>R</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, is defined as the minimum weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> among all strongly total Roman dominating functions <i>f</i> on <i>G</i>. This paper is devoted to the study of the strongly total Roman domination number of a graph and it is a contribution to the Special Issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we show that the theory of strongly total Roman domination is an appropriate framework for investigating the total Roman domination number of lexicographic product graphs. We also obtain tight bounds on this parameter and provide closed formulas for some product graphs. Finally and as a consequence of the study, we prove that the problem of computing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mi>t</mi><mi>R</mi></mrow><mi>s</mi></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is NP-hard.