Exploring Variant Roman Domination Number in Complete Binary Trees Using Python Programming

A Roman Dominating Function (RDF) on a graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> is defined as a function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$g$</tex> that assigns a value of 0, 1, or 2 to each vertex in such a way that any vertex assigned a value of 0 is adjacent to at least one vertex assigned a value of 2. The total weight of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$RDF$</tex>, denoted as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$w(g)$</tex> is the sum of the function values over all vertices. The smallest possible weight of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$RDF$</tex> is referred to as the Roman domination number, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\gamma_{R}(G)$</tex>, of the graph. In this manuscript, we find some variants of Roman Domination number such as the Roman domination number, maximal, signed, inverse and perfect Roman domination numbers in complete binary trees with programming code in Python.