Nonreversible trees having a removable edge

Abstract

A relational structure is said to be reversible iff every bijective homomorphism (condensation) of that structure is an automorphism. In the case of a binary structure X = ?X,??, that is equivalent to the following statement: whenever we remove finite or infinite number of edges from X, thus obtaining the structure X', we have that X'?/ X. In this paper, we prove that if a nonreversible tree X = ?X,?? has a removable edge (i.e. if there is ?x,?? ? ? such that ?X,?? ? ?X,?\{?x,y?}?, then it has infinitely many removable edges. We also show that the same is not true for arbitrary binary structure by constructing nonreversible digraphs having exactly n removable edges, for n ? N.