Optimizing Communication in Complete Binary Trees Using Variant Double Roman Domination with Python-Based Pseudocode

In modern communication networks and distributed computing, hierarchical structures like complete binary trees play a crucial role in ensuring efficient data transmission, fault tolerance, and resource allocation. A double Roman dominating function on a graph <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{G}=(\mathrm{V},\mathrm{E})$</tex> is defined as a function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{g}(\mathrm{V}_{{0}}, \mathrm{V}_{{1}}, \mathrm{V}_{{2}}, \mathrm{V}_{{3}})$</tex> that satisfies the following conditions that if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{g}{({v})}{=0}$</tex>, then vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${v}$</tex> must be adjacent to at least one vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${u}$</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{g}{({u})}{=3}$</tex> or two vertices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${v}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${w}$</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{g}{({v})}=\mathrm{g}{({w})}{=2}$</tex> if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{g}{({v})}{=1}$</tex>, then vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${v}$</tex> must be adjacent to at least one vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${w}$</tex> for which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{g}{({w})}{=3}$</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{g}({w}){=2}$</tex>. The weight of a double Roman dominating function is given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{w}(\mathrm{g})= {\Sigma}_{\mathrm{v} {\in} \mathrm{V}(\mathrm{G})}\mathrm{g}(\mathrm{v})$</tex>, and the minimum weight of a double Roman dominating function on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${G}$</tex> is called is the double Roman domination number, denoted as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${\gamma}_{\mathrm{d} \mathrm{R}}{({G})}$</tex>. We explore the double Roman, maximal double Roman, signed double Roman, and inverse double Roman domination number in complete binary trees to enhance network resilience and security. A Python-based pseudocode is developed to compute these parameters, facilitating algorithmic implementation in large-scale hierarchical networks.