# [1,2]-Domination in Generalized Petersen Graphs

## TL;DR

This paper determines the minimum size of special dominating sets, called [1,2]-dominating sets, in generalized Petersen graphs P(n,2), advancing understanding of domination parameters in these graphs.

## Contribution

It explicitly calculates the [1,2]-domination and total domination numbers for generalized Petersen graphs P(n,2), a previously unresolved problem.

## Key findings

- Exact formulas for $ ext{γ}_{[1,2]}(P(n,2))$ and $ ext{γ}_{t[1,2]}(P(n,2))$ are provided.
- The results extend the knowledge of domination parameters in Petersen graphs.
- The paper offers new insights into domination theory in specific graph classes.

## Abstract

A vertex subset $S$ of a graph $G=(V,E)$ is a $[1,2]$-dominating set if each vertex of $V\backslash S$ is adjacent to either one or two vertices in $S$. The minimum cardinality of a $[1,2]$-dominating set of $G$, denoted by $\gamma_{[1,2]}(G)$, is called the $[1,2]$-domination number of $G$. In this paper the $[1,2]$-domination and the $[1,2]$-total domination numbers of the generalized Petersen graphs $P(n,2)$ are determined.