Total Roman Domination Number of Rooted Product Graphs

• Main Authors: Abel Cabrera Martínez, Suitberto Cabrera García, Andrés Carrión García, Frank A. Hernández Mira
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Mathematics
• Subjects: total Roman domination; total domination; rooted product graph
• Online Access: https://www.mdpi.com/2227-7390/8/10/1850

Let <i>G</i> be a graph with no isolated vertex and <inline-formula><math 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> a function. If <i>f</i> satisfies that every vertex in the set <inline-formula><math display="inline"><semantics><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo lspace="0pt">:</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> is adjacent to at least one vertex in the set <inline-formula><math display="inline"><semantics><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo lspace="0pt">:</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>2</mn><mo>}</mo></mrow></semantics></math></inline-formula>, and if the subgraph induced by the set <inline-formula><math display="inline"><semantics><mrow><mo>{</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo lspace="0pt">:</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> has no isolated vertex, then we say that <i>f</i> is a total Roman dominating function on <i>G</i>. The minimum weight <inline-formula><math display="inline"><semantics><mrow><mi>ω</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> among all total Roman dominating functions <i>f</i> on <i>G</i> is the total Roman domination number of <i>G</i>. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.