On the Roman domination number of generalized Sierpinski graphs

TL;DR

This paper investigates the Roman domination number of generalized Sierpinski graphs, providing an upper bound and analyzing specific cases like paths, cycles, and complete graphs to understand its properties.

Contribution

It introduces a general upper bound for the Roman domination number of generalized Sierpinski graphs and examines its tightness across various base graphs.

Findings

Established a general upper bound for the Roman domination number of S(G,t).

Analyzed the bound's tightness for specific base graphs such as paths and cycles.

Provided insights into how the structure of the base graph influences the Roman domination number.

Abstract

A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G=(V,E)$ if for every vertex $v\in V$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating function is given by $f(V) =\sum_{u\in V}f(u)$. The minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$. In this article we study the Roman domination number of Generalized Sierpi\'{n}ski graphs $S(G,t)$. More precisely, we obtain a general upper bound on the Roman domination number of $S(G,t)$ and we discuss the tightness of this bound. In particular, we focus on the cases in which the base graph $G$ is a path, a cycle, a complete graph or a graph having exactly one universal vertex.