Independence number of generalized Petersen graphs

TL;DR

This paper investigates the independence number of generalized Petersen graphs, providing new bounds, exact values for small cases, and an algorithm with linear time complexity for fixed parameters.

Contribution

It introduces new bounds, verifies a conjecture for certain cases, and develops an efficient algorithm for computing the independence number.

Findings

New upper and lower bounds for (P(n,k))

Verification of a conjecture for n > 3k and n < 78

An O(n) algorithm for fixed k to compute (P(n,k))

Abstract

Determining the size of a maximum independent set of a graph $G$, denoted by $\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\alpha (G)$ for special classes of graphs. This paper is aimed toward studying this problem for the class of generalized Petersen graphs. We find new upper and lower bounds and some exact values for $\alpha(P(n,k))$. With a computer program we have obtained exact values for each $n<78$. In \cite{MR2381433} it is conjectured that $\beta(P(n, k)) \leq n + \lceil\frac{n}{5}\rceil $, for all $n$ and $k$. We prove this conjecture for some cases. In particular, we show that if $ n> 3k$, the conjecture is valid. We checked the conjecture with our table for $n < 78$ and it had no inconsistency. Finally, we show that for every fix $k$, $\alpha(P(n, k))$ can be computed using an algorithm with running time O(n).