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

English

Bisdorff, Raymond[University of Luxembourg > Faculty of Law, Economics and Finance > Applied Mathematics Unit (SMA)]

Marichal, Jean-Luc[University of Luxembourg > Faculty of Law, Economics and Finance > Applied Mathematics Unit (SMA)]

Jan-2007

1

No

No

International

21st Annual Conf. of the Belgian Operational Research Society (ORBEL 21), Luxembourg, Jan. 18-19, 2007

from 18-01-2007 to 19-01-2007

Raymond Bisdorff

Claude Lamboray

Jean-Luc Marichal

Patrick Meyer

Luxembourg

Luxembourg

[en] maximal independent set ; cycle graph ; combinatorial enumeration ; dihedral group ; group action ; cyclic and palindromic composition of integers ; Perrin and Padovan sequences

[en] It is known that the number of maximal independent sets of the $n$-cycle graph $C_n$ is given by the $n$th 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 unlabelled (i.e., defined up to a rotation) maximal independent sets and the number of unlabelled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences.