Edge-Ends in Countable Graphs

The notion of ends equivalence classes on the set of rays (one-way infinite paths) of a graph is one of the most studied topics in infinite graph theory. An introduction to this theory and basic results can be found in Halin [3]. Halin defined two rays to be equivalent if no finite set of vertices can separate an infinite part of the first ray from an infinite part of the second one. In particular, Halin proved that in a countable connected graph G, the end-structure can be represented by a kind of spanning tree that he called faithful. Such a tree is defined by the property that from any given end of G it contains exactly one ray originating at x, for any x # V(G)). A natural and, as will be seen in this paper, very useful property of ends is the domination property. An end : is dominated if for some ray R (and so for all rays) in : there exists a vertex x which cannot be separated from an infinite part of R by any finite set of vertices. Intuitively, undominated rays are those one normally has in mind when thinking about infinite paths as ``going to infinity,'' whereas dominated rays appear to be ``trapped'' by the vertices that dominate them (and thus are not ``really'' rays). article no. TB961734