In this paper we consider the monoid $DPC_n$ of all partial isometries of an $n$-cycle graph $C_n$. We show that $DPC_n$ is the submonoid of the monoid of all oriented partial permutations on an $n$-chain whose elements are precisely all restrictions of a dihedral group of order $2n$. Our main aim is to exhibit a presentation of $DPC_n$. We also describe Green's relations of $DPC_n$ and calculate its cardinality and rank.
Transformations, orientation, partial isometries, cycle graphs, rank, presentations
FERNANDES, VITOR H. and PAULISTA, TANIA
"On the monoid of partial isometries of a cycle graph,"
Turkish Journal of Mathematics: Vol. 47:
6, Article 10.
Available at: https://journals.tubitak.gov.tr/math/vol47/iss6/10