| Summary: | Let <i>G</i> be a nontrivial connected graph. For a set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>, we define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>D</mi><mo>¯</mo></mover><mo>=</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∖</mo><mi>D</mi></mrow></semantics></math></inline-formula>. The set <i>D</i> is a total outer-independent dominating set of <i>G</i> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>∩</mo><mi>D</mi><mo>|</mo><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> for every vertex <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> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>D</mi><mo>¯</mo></mover></semantics></math></inline-formula> is an independent set of <i>G</i>. Moreover, <i>D</i> is a double outer-independent dominating set of <i>G</i> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>N</mi><mo>[</mo><mi>v</mi><mo>]</mo><mo>∩</mo><mi>D</mi><mo>|</mo><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> for every vertex <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> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>D</mi><mo>¯</mo></mover></semantics></math></inline-formula> is an independent set of <i>G</i>. In addition, <i>D</i> is a 2-outer-independent dominating set of <i>G</i> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>N</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>∩</mo><mi>D</mi><mo>|</mo><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> for every vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mover><mi>D</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover><mi>D</mi><mo>¯</mo></mover></semantics></math></inline-formula> is an independent set of <i>G</i>. The total, double or 2-outer-independent 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></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mo>×</mo><mn>2</mn></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, is the minimum cardinality among all total, double or 2-outer-independent dominating sets of <i>G</i>, respectively. In this paper, we first show that for any cactus graph <i>G</i> of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> cycles, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>l</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mn>2</mn></mfrac><mo>+</mo><mi>k</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mi>t</mi></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>2</mn><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mn>3</mn></mfrac><mo>+</mo><mi>k</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mo>×</mo><mn>2</mn></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>2</mn><mi>n</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>l</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mn>3</mn></mfrac><mo>+</mo><mi>k</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> represent the number of leaves and the number of support vertices of <i>G</i>, respectively. These previous bounds extend three known results given for trees. In addition, we characterize the trees <i>T</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mo>×</mo><mn>2</mn></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mi>γ</mi><mrow><mi>t</mi></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Moreover, we show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>γ</mi><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mi>i</mi></mrow></msubsup><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≥</mo><mfrac><mrow><mi>n</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>+</mo><mi>l</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mi>s</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></semantics></math></inline-formula> for any tree <i>T</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>. Finally, we give a constructive characterization of the trees <i>T</i> that satisfy the equality above.
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