Asymmetrization of infinite trees

AbstractAn asymmetrizing set of a tree T is a set A of vertices of T such that the identity is the only automorphism of T which stabilizes A. The asymmetrizing number of T is the cardinality of the set of orbits of asymmetrizing sets of T. We complete the results of Polat and Sabidussi (1991) by characterizing the asymmetrizable trees containing a double ray, and by proving that the asymmetrizing number of such a tree T (resp. of any tree T containing a ray but no double ray) is the product of the asymmetrizing numbers of the components of T\T∗, where T∗ is the union of all double rays (resp. is some ray) of T. We show that, if the asymmetrizing number of a tree is infinite, then it is of the form 2κ for some cardinal κ. Besides, given two cardinals κ and λ with κ⩽λ we present, on the one hand, a set of 2λ non-isomorphic rayless trees of cardinality λ and asymmetrizing number 2κ, and on the other hand, a set of 2κ non-isomorphic asymmetric (i.e. having no non-identity automorphism) rayless graphs of cardinality κ