On regular trees defined from unfoldings and coverings

Abstract

We study the infinite trees that arise, first as complete unfoldings of finite weighted directed graphs, and second, as universal coverings of finite weighted undirected graphs. They are respectively the regular rooted trees and the strongly regular trees, a new notion. A rooted tree is regular if it has finitely many subtrees up to isomorphism. A tree (without root) is strongly regular if it has finitely many rooted trees, up to isomorphism, obtained by taking each of its nodes as a root. We prove the first-order definability of each regular or strongly regular tree with respect to the class of trees (that is not itself first-order definable). We characterize the strongly regular trees among the regular ones and we establish several decidability results.