Roman domination number of Generalized Petersen Graphs P(n,2)

TL;DR

This paper determines the exact Roman domination number for generalized Petersen graphs P(n,2), establishing a formula that applies for all n ≥ 5, thereby advancing understanding of domination parameters in these graphs.

Contribution

The paper provides a precise formula for the Roman domination number of P(n,2), a significant step in graph domination theory for generalized Petersen graphs.

Findings

Roman domination number of P(n,2) is eil(8n/7)or n 

Established a formula applicable for all n 

Contributed to graph domination parameter knowledge

Abstract

A $Roman\ domination\ function$ on a graph $G=(V, E)$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The $weight$ of a Roman domination function $f$ is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the $Roman\ domination\ number$ of $G$, denoted by $\gamma_{R}(G)$. In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that $\gamma_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n \geq 5)$.