Spanning trees of countable graphs omitting sets of dominated ends Original Research Article

Generalizing the well-known theorem of Halin (1964) that a countable connected graph G contains an end-faithful spanning tree (i.e., an end-preserving tree that omits no end of G), we establish some results about the existence of end-preserving spanning trees omitting some prescribed set of ends. We remark that if such a tree exists, the omitted ends must all be dominated, and even then counterexamples exist. We then give sufficient conditions for the existence of such trees, generalizing a result of Siran (1991) that guarantees their existence if the set of ‘desired’ ends is countable.