Can I delete one specific line from an axes

My axes has 2 lines in it. I'm using a spinner to move a vertical reference marker. Below is a snippet from the spinner callback. In a different axes object I want to keep the plot of the data and move the reference line. What I'm doing now is finding any lines in the axes and deleting all of them. Then I replot both of them, using an updated value for the x position. Seems pretty inelegant. I tried creating a handle for the specific reference to use delete() just on that one but I can't make that work.

axes(handles.LProfile);
  h = findobj(gca, 'Type', 'line');
  if ishandle(h)
      delete(h)
  end
  plot(SW.LProfile,'b-')
 line(handles.LProfile,[newTxprofile, newTxprofile],handles.LProfile.YLim)

[Woody]

You can add a tag on your reference line, and delete based on the tag

line(x,y,'tag','reference');
delete(findobj('tag','reference'));