Counting non-isomorphic maximal independent sets of the n-cycle graph

TL;DR

This paper derives exact formulas for counting non-isomorphic maximal independent sets in cycle graphs, considering symmetries like rotations and reflections, and relates these counts to Perrin and Padovan sequences.

Contribution

It provides new explicit formulas for counting orbits and unlabeled sets of maximal independent sets in cycle graphs, incorporating automorphism group actions.

Findings

Formulas involve Perrin and Padovan sequences.

Exact counts for orbits under automorphisms.

Counts for unlabeled maximal independent sets.

Abstract

The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabeled (i.e., defined up to a rotation) maximal independent sets and the number of unlabeled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences.