| Summary: | Domination plays an indispensable role in graph theory. Various types of domination explore various types of applications. Equal-status people work together and interlace with each other easily. In this paper, the paired equitable domination of a graph, its inflated graph, and its complement of an inflated graph were studied. The relationship between the domination number of the graph, the equitable domination number, and the paired equitable domination number of complements of the inflated graph were established. Furthermore, we proved the Nordhaus–Gaddum-type inequality, that is, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>p</mi><mi>r</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>p</mi><mi>r</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>6</mn></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow></semantics></math></inline-formula> is a graph with <i>m</i> nodes where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≡</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo stretchy="false">(</mo><mi>m</mi><mi>o</mi><mi>d</mi><mo> </mo><mn>8</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> = (<i>m</i>/2) for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>. The challenges and limitations of this parameter of paired equitable and equitable domination depends on the degree of the vertex of the graph. Practical applications are discussed in various fields and illustrated using the studied parameter.
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