thread-safe graphics access

  In a book on Java game programming, I found a game that uses a
  component (called ?this? below), which is a JPanel.
  This component has a custom thread to update it.
  This thread, simplified, does this to update the game display:

this.getGraphics().drawImage( gameImage, 0, 0, null );

  I notice that this is an access of a Swing component from
  within the custom thread, outside of the EDT, though I am not
  sure what constitutes the actual offending access
  (?getGraphics? or ?drawImage?).

  An older Sun tutorial says:

      ?Swing components can be accessed by only one thread at a time.
      Generally, this thread is the event-dispatching thread.?

  So, can I get by without using ?InvokeAndWait?, when I am
  confident that no one else is accessing the graphics object of
  my custom component at this time?

      - I can make sure that no other thread of my program
        accesses this JPanel.

      - I do not expect Swing to draw to the graphics object
        of this component.

  But what happens, if a Swing menu is added to the program,
  gets openend and then overlaps this custom component?

  So, in the end, I should wrap this access with an invoke
  method?

  The whole game is a kind of loop:

while( running)
{ update( gameImage );
  this.getGraphics().drawImage( gameImage, 0, 0, null ); }

  So, if I have to wrap the final statement, how do I do this?
  My first impression is that this needs ?InvokeAndWait?,
  because this is most close to the original behavior as shown
  above. Or may/should I use ?InvokeLater? in this case?