The Domination Number of Generalized Petersen Graphs with a Faulty Vertex

TL;DR

This paper studies how the domination number of generalized Petersen graphs P(n, 2) changes when a vertex becomes faulty, revealing specific conditions under which it decreases by one.

Contribution

It provides a precise characterization of the change in domination number of P(n, 2) with a faulty vertex based on the value of n.

Findings

The domination number decreases by one when n=5k+1 or 5k+2.

The domination number remains unchanged for other values of n.

The results depend on the modular relationship of n with respect to 5.

Abstract

In this paper, we investigate the domination number of generalized Petersen graphs P(n, 2) when there is a faulty vertex. Denote by $\gamma(P(n,2))$ the domination number of P(n,2) and $\gamma(P_f(n,2))$ the domination number of P(n,2) with a faulty vertex $u_f$. We show that $\gamma(P_f(n,2))=\gamma(P(n,2))-1$ when $n=5k+1$ or $5k+2$ and $\gamma(P_f(n,2))=\gamma(P(n,2))$ for the other cases.