<jats:p>A total Roman <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
<mfenced open="{" close="}" separators="|">
<mrow>
<mn>2</mn>
</mrow>
</mfenced>
</math>
</jats:inline-formula>-dominating function (TR2DF) on a graph <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
<mi mathvariant="normal">Γ</mi>
<mo>=</mo>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>V</mi>
<mo>,</mo>
<mi>E</mi>
</mrow>
</mfenced>
</math>
</jats:inline-formula> is a function <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
<mi>l</mi>
<mo>:</mo>
<mi>V</mi>
<mo>⟶</mo>
<mfenced open="{" close="}" separators="|">
<mrow>
<mn>0,1,2</mn>
</mrow>
</mfenced>
</math>
</jats:inline-formula>, satisfying the conditions that (i) for every vertex <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
<mi>y</mi>
<mo>∈</mo>
<mi>V</mi>
</math>
</jats:inline-formula> with <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
<mi>l</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>y</mi>
</mrow>
</mfenced>
<mo>=</mo>
<mn>0</mn>
</math>
</jats:inline-formula>, either <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
<mi>y</mi>
</math>
</jats:inline-formula> is adjacent to a vertex labeled 2 under <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
<mi>l</mi>
</math>
</jats:inline-formula>, or <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
<mi>y</mi>
</math>
</jats:inline-formula> is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
<mi>l</mi>
</math>
</jats:inline-formula> is the value <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
<mstyle displaystyle="true">
<msub>
<mrow>
<mo stretchy="false">∑</mo>
</mrow>
<mrow>
<mi>y</mi>
<mo>∈</mo>
<mi>V</mi>
</mrow>
</msub>
<mrow>
<mi>l</mi>
<mfenced open="(" close=")" separators="|">
<mrow>
<mi>y</mi>
</mrow>
</mfenced>
</mrow>
</mstyle>
</math>
</jats:inline-formula>. The total Roman <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
<mfenced open="{" close="}" separators="|">
<mrow>
<mn>2</mn>
</mrow>
</mfenced>
</math>
</jats:inline-formula>-domination number (TR2D-number) of a graph <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
<mi mathvariant="normal">Γ</mi>
</math>
</jats:inline-formula> is the minimum weight of a TR2DF on <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M14">
<mi mathvariant="normal">Γ</mi>
</math>
</jats:inline-formula>. The total Roman <jats:inline-formula>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="M15">
<mfenced open="{" close="}" separators="|">
<mrow>
<mn>2</mn>
</mrow>
</mfenced>
</math>
</jats:inline-formula>-reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of graphs.</jats:p>
[1]
Stephen T. Hedetniemi,et al.
Total domination in graphs
,
1980,
Networks.
[2]
Mustapha Chellali,et al.
Outer independent double Roman domination
,
2020,
Appl. Math. Comput..
[3]
Guoliang Hao,et al.
Italian Reinforcement Number in Graphs
,
2019,
IEEE Access.
[4]
Michael A. Henning,et al.
Italian domination in trees
,
2017,
Discret. Appl. Math..
[5]
Seyed Mahmoud Sheikholeslami,et al.
Total Roman Reinforcement in Graphs
,
2019,
Discuss. Math. Graph Theory.
[6]
Lutz Volkmann.
The Italian domatic number of a digraph
,
2019
.
[7]
Juan Carlos Valenzuela Tripodoro,et al.
Total Roman {2}-Dominating Functions in Graphs
,
2020,
Discuss. Math. Graph Theory.
[9]
Tingting Xu,et al.
Bagging Approach for Italian Domination in $C_{n} \square\,P_{m}$
,
2019,
IEEE Access.
[10]
Michael A. Henning,et al.
Perfect Italian domination in trees
,
2019,
Discret. Appl. Math..
[11]
N. Sridharan,et al.
Total Reinforcement Number of a Graph
,
2020
.
[12]
Michael A. Henning,et al.
Graphs with Large Italian Domination Number
,
2020,
Bulletin of the Malaysian Mathematical Sciences Society.
[13]
Teresa W. Haynes,et al.
Roman {2}-domination
,
2016,
Discret. Appl. Math..
[14]
Noor A'lawiah Abd Aziz,et al.
On the outer-independent double Italian domination number
,
2022,
Electron. J. Graph Theory Appl..
[15]
Seyed Mahmoud Sheikholeslami,et al.
Varieties of Roman domination II
,
2020,
AKCE Int. J. Graphs Comb..
[16]
Mustapha Chellali,et al.
Independent Roman -domination in graphs
,
2018,
Discret. Appl. Math..