Countably determined ends and graphs

The directions of an infinite graph G are a tangle-like description its ends: they choice functions that choose component G−X for all finite vertex sets X⊆V(G) in compatible manner. Although every direction is induced by ray, there exist graphs not uniquely determined any countable subset their choices. We characterise these and countably counterparts terms star-like substructures or rays the graph. Curiously, whose but which cannot be distinguished at once many structurally can choices, we this Our characterisations phrased normal trees tree-decompositions. four (sub)structural imply combinatorial classes defined first second axiom countability applied to end spaces: two spaces countable, respectively, complements classes.