[Paper Review] Counting non-isomorphic maximal independent sets of the n-cycle graph

Raymond Bisdorff, Jean‐Luc Marichal|arXiv (Cornell University)|Jan 23, 2007

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper provides exact combinatorial formulas for counting non-isomorphic and unlabeled maximal independent sets (MISs) in the n-cycle graph $C_n$, using group action of the dihedral group $D_{2n}$ on the set of MISs. It derives closed-form expressions involving the Perrin and Padovan sequences, showing that the number of non-isomorphic MISs is given by $\mathrm{orb}(n) = \frac{1}{2}\left(r(n) + \frac{1}{n}(p*\phi)(n)\right)$, where $p(n)$ is the Perrin sequence and $r(n)$ counts symmetric MISs.

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.

Motivation & Objective

• To enumerate non-isomorphic maximal independent sets (MISs) in the n-cycle graph $C_n$ under dihedral symmetry (rotations and reflections).
• To derive exact formulas for the number of orbits (non-isomorphic MISs) and the number of orbits with a given number of isomorphic representatives.
• To extend the analysis to unlabeled MISs, defined up to rotation only, and provide corresponding exact formulas.
• To establish connections between combinatorial enumeration of MISs and well-known integer sequences, particularly the Perrin and Padovan sequences.

Proposed method

• Apply the action of the dihedral group $D_{2n}$ on the family of MISs of $C_n$, partitioning them into orbits under rotation and reflection.
• Use the orbit-stabilizer theorem to relate orbit size and stabilizer size, with $|\mathrm{Orb}(X)| \times |\mathrm{Stab}(X)| = 2n$.
• Define $\mathrm{orb}(n)$ as the total number of non-isomorphic MISs (orbits), and $\mathrm{orb}_d(n)$ as the number of orbits with stabilizer size $d$, using multiplicative arithmetic functions.
• Express $\mathrm{orb}(n)$ via a formula combining the Perrin sequence $p(n)$ and Euler's totient function $\phi(n)$: $\mathrm{orb}(n) = \frac{1}{2}\left(r(n) + \frac{1}{n}(p*\phi)(n)\right)$.
• Introduce the concept of unlabeled MISs (up to rotation only), and define $\mathrm{orb}^\sigma(n)$, showing it equals $2\mathrm{orb}(n) - r(n)$.
• Use Dirichlet convolution and Möbius inversion to derive $\mathrm{orb}_1^\sigma(n)$, the number of unlabeled MISs with no rotational symmetry.

Experimental results

Research questions

• RQ1 How many non-isomorphic maximal independent sets exist in the $n$-cycle graph $C_n$ under dihedral symmetry?
• RQ2 What is the number of orbits (non-isomorphic MISs) that have exactly $k$ isomorphic representatives under the action of $D_{2n}$?
• RQ3 How many unlabeled maximal independent sets (up to rotation only) exist in $C_n$, and how many of them have $k$ isomorphic representatives?
• RQ4 What is the relationship between the enumeration of MISs and the Perrin and Padovan sequences?

Key findings

• The total number of non-isomorphic MISs in $C_n$, denoted $\mathrm{orb}(n)$, is given by $\mathrm{orb}(n) = \frac{1}{2}\left(r(n) + \frac{1}{n}(p*\phi)(n)\right)$, where $p(n)$ is the Perrin sequence and $r(n)$ counts MISs fixed by reflection.
• The number of non-isomorphic MISs with no symmetry axis (i.e., stabilizer size 1) is $\mathrm{orb}_1(n) = \mathrm{orb}(n) - r(n)$, which equals the number of cyclic, non-palindromic compositions of $n$ with parts 2 and 3.
• The number of unlabeled MISs (up to rotation only) is $\mathrm{orb}^\sigma(n) = 2\mathrm{orb}(n) - r(n)$, and this sequence matches the OEIS sequence A127687.
• The number of unlabeled MISs with exactly $n/d$ isomorphic representatives is given by $\mathrm{orb}_d^\sigma(n)$, which can be computed from $\mathrm{orb}_1^\sigma(n)$ via Dirichlet convolution.
• For prime $n$, $\mathrm{orb}(n) = \mathrm{orb}_1(n) + \mathrm{orb}_2(n)$, with $\mathrm{orb}_2(n) = r(n)$, and $\mathrm{orb}^\sigma(n) = \frac{1}{n}p(n)$, reflecting high symmetry and uniform orbit structure.
• The first 40 values of $\mathrm{orb}(n)$ and $\mathrm{orb}^\sigma(n)$ are tabulated, showing that $\mathrm{orb}(n)$ and $\mathrm{orb}^\sigma(n)$ grow in line with the Perrin and Padovan sequences, respectively.