The Tube Inspectors puzzle

The newly appointed tube inspector has a problem.
 He has to inspect each of the tube
stations shown below.  He wants to
arrange his journey so that he travels
as little as possible.  He may begin
and end where he likes. What is his
shortest possible route.  It will be
found he must travel through some
stations more than once, for instance
if each station is 1 kilometer apart he
will have to travel more than 17 kilometers
(the 17 lines owned by the rail company
that connect the stations).


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