Comparative Analysis Of Roman And Inverse Roman Domination Numbers Across Graph Families.

Abstract

This research paper delves into the intriguing realm of Roman domination and its inverse counterpart within various graph structures. Initially defining Roman domination as a graph theory concept where vertices are covered by distinct dominating sets, and inversely, inverse Roman domination as a novel extension where vertices outside the dominating set are considered, the study progresses to explore these concepts across a spectrum of graph families. The investigation begins with cubic graphs, exploring their Roman and inverse Roman domination numbers, followed by an analysis of cubic symmetric graphs, Platonic graphs, Holt graphs, Folkman graphs, Frucht graphs, and culminating with the Levi graph. For each graph family, meticulous calculations and comparisons are conducted to ascertain the Roman Domination number and inverse Roman domination numbers, shedding light on their inherent characteristics and relationships.