Optimizing Communication in Complete Binary Trees Using Variant Double Roman Domination with Python-Based Pseudocode
J. Meena, T. N. M. Malini Mai · 2025
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$\mathrm{G}=(\mathrm{V},\mathrm{E})$is defined as a function$\mathrm{g}(\mathrm{V}_{{0}}, \mathrm{V}_{{1}}, \mathrm{V}_{{2}}, \mathrm{V}_{{3}})$that satisfies the following conditions that if$\mathrm{g}{({v})}{=0}$, then vertex${v}$must be adjacent to at least one vertex${u}$where$\mathrm{g}{({u})}{=3}$or two vertices${v}$and${w}$where$\mathrm{g}{({v})}=\mathrm{g}{({w})}{=2}$if$\mathrm{g}{({v})}{=1}$, then vertex${v}$must be adjacent to at least one vertex${w}$for which$\mathrm{g}{({w})}{=3}$or$\mathrm{g}({w}){=2}$. The weight of a double Roman dominating function is given by$\mathrm{w}(\mathrm{g})= {\Sigma}_{\mathrm{v} {\in} \mathrm{V}(\mathrm{G})}\mathrm{g}(\mathrm{v})$, and the minimum weight of a double Roman dominating function on${G}$is called is the double Roman domination number, denoted as${\gamma}_{\mathrm{d} \mathrm{R}}{({G})}$. 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.