Arguably the simplest non-trivial example of an unlabelled simple undirected graph which is not reconstructible from its vertex-deleted subgraphs is the -regular tree, illustrated in parts by
Needfull to say: that you only see five neighbors at each vertex is an arbitrary concession to the finiteness of the technological means used to produce this picture. In reality, this tree has precisely neighbors at each vertex. (This is what the ” -regular ” above means.)
Needless to say, this is not a rayless tree. From a constructivist point of view, one might be inclined to say that it is a rayfull tree.