Classification of perfect and total perfect codes in generalized Petersen graphs

TL;DR

This paper classifies perfect and total perfect codes within generalized Petersen graphs, providing a comprehensive understanding of their structure and properties in this specific class of graphs.

Contribution

It offers the first complete classification of perfect and total perfect codes in generalized Petersen graphs, expanding knowledge on coding theory in graph structures.

Findings

Complete classification of perfect codes in generalized Petersen graphs

Complete classification of total perfect codes in generalized Petersen graphs

Identification of structural properties enabling these classifications

Abstract

In a graph $\Gamma$, a perfect code is an independent set $C$ with the property that every vertex not in $C$ is adjacent to a unique vertex in $C$, and a total perfect code is a set $C$ of vertices of $\Gamma$ such that every vertex of $\Gamma$ is adjacent to a unique vertex in $C$. We classify these codes for generalized Petersen graphs.