Roman domination and Mycieleki's structure in graphs

TL;DR

This paper investigates the Roman domination number in graphs, establishing bounds for Mycielekian graphs, characterizing cases of equality, and computing the number for m-Mycieleskian graphs of special Roman graphs.

Contribution

It provides new bounds for Roman domination numbers in Mycielekian graphs and characterizes graphs that attain these bounds, extending understanding of Roman domination in graph theory.

Findings

Bounds for _{R}((G)) are _{R}(G)+1 and _{R}(G)+2.

Characterization of graphs achieving equality in the bounds.

Explicit computation of _{R}(_{m}(G)) for special Roman graphs.

Abstract

For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$ is the value $f(V)=\sum_{v\in V}f(v)$. The minimum weight of an RDF of $G$ is its Roman domination number and denoted by $\gamma_ R(G)$. In this paper, we first show that $\gamma_{R}(G)+1\leq \gamma_{R}(\mu (G))\leq \gamma_{R}(G)+2$, where $\mu (G)$ is the Mycielekian graph of $G$, and then characterize the graphs achieving equality in these bounds. Then for any positive integer $m$, we compute the Roman domination number of the $m$-Mycieleskian $\mu_{m}(G)$ of a special Roman graph $G$ in terms on $\gamma_R(G)$. Finally we present several graphs to illustrate the discussed graphs.