  This report is an improvement of a prior report (Report 813). It sharpens the
principal theorems (Theorems 4.2 and 11.2 of Report 813) while simplifying
their proofs. There are also several minor changes involving clarifications and
corrections of misprints.
  The Abstract of the prior report remains the same as follow: The galaxies of
the nonstandard enlargements of connected, conventionally infinite graphs as
well as of connected transfinite graphs are defined, analyzed, and illustrated
by some examples. It is then shown that any such enlargement either has exactly
one galaxy, its principal one, or it has infinitely many galaxies. In the
latter case, the galaxies are partially ordered by their "closeness" to the
principal galaxy. If an enlargement has a galaxy different from its principal
galaxy, then it has a two-way infinite sequence of galaxies that are totally
ordered according to that "closeness" property. There may be many such totally
ordred sequences.
